# Weak Poincaré Inequalities for Markov chains: theory and applications

Christophe Andrieu, Anthony Lee, Sam Power, Andi Q. Wang
School of Mathematics, University of Bristol and University of Warwick

###### Abstract

We investigate the application of Weak Poincaré Inequalities (WPI) to Markov
chains to study their rates of convergence and to derive complexity bounds. At
a theoretical level we investigate the necessity of the existence of WPIs to
ensure $\mathrm{L}^{2}$-convergence, in particular by establishing equivalence
with the Resolvent Uniform Positivity-Improving (RUPI) condition and providing
a counterexample. From a more practical perspective, we extend the celebrated
Cheeger’s inequalities to the subgeometric setting, and further apply these
techniques to study random-walk Metropolis algorithms for heavy-tailed target
distributions and to obtain lower bounds on pseudo-marginal algorithms.

###### Contents

  1. 1 Introduction
  2. 2 Weak Poincaré inequalities
    1. 2.1 Basic definitions
    2. 2.2 Optimal $\alpha,\beta$
    3. 2.3 Ordering of $\alpha$’s, $\beta$’s and $\gamma$’s and Peskun–Tierney ordering
    4. 2.4 Lower bounds on convergence rates
    5. 2.5 Other types of convergence from $\big{(}\left\|\cdot\right\|_{\mathrm{osc}}^{2},\gamma\big{)}$ convergence
    6. 2.6 ${\rm L}^{2}$ and ${\rm L}^{1}$ subgeometric convergence rates can be different and CLT
  3. 3 Existence of WPIs
    1. 3.1 Weak Cheeger inequalities
    2. 3.2 $\mu$–irreducibility, aperiodicity and WPIs
      1. 3.2.1 Equivalence of $\left\|\cdot\right\|_{{\rm osc}}^{2}$_–_ WPI and RUPI
      2. 3.2.2 WPIs and convergence under (PH)
      3. 3.2.3 WPIs and convergence in general
    3. 3.3 WPIs from Close Coupling and Isoperimetry
      1. 3.3.1 Main results
      2. 3.3.2 Examples of (Convex) Isoperimetric Profiles
  4. 4 Application to Pseudo-marginal MCMC
  5. 5 Applications to RWM on heavy-tailed targets
    1. 5.1 Mixing times of RWM for $L$-smooth targets
    2. 5.2 Mixing times: examples
    3. 5.3 Asymptotic variance: examples
  6. 6 Discussion
  7. A Notation
  8. B Dirichlet forms of indicators
  9. C Improved sharpness of $\mathrm{L}^{1}$_–_ WPI result
  10. D Auxiliary results for examples
    1. D.1 Examples 92, 93
    2. D.2 Example 82

##  1 Introduction

This paper is concerned with the characterization of the convergence to
equilibrium of (discrete) time homogeneous Markov chains with transition
kernel $P$ to their invariant distributions. Such a distribution is assumed
throughout to exist and be unique, and will be denoted by $\mu$. Numerous
techniques and metrics are possible for this and we focus here on functional
analytic tools to characterize a type of $\mathrm{L}^{2}$-convergence to
equilibrium using functional inequalities, particularly Poincaré Inequalities.
More precisely for a $\mu$-invariant kernel $P$ we aim to characterize
scenarios where

$\left\|P^{n}f-\mu\left(f\right)\right\|_{2}^{2}\leqslant\gamma\left(n\right)\cdot\Psi\left(f-\mu\left(f\right)\right)\,,$

for a vanishing sequence $\gamma=\left\\{\gamma\left(n\right)\colon
n\in\mathbb{N}\right\\}$ and a type of functional $\Psi$ e.g.
$\Psi\left(\cdot\right)=\left\|\cdot\right\|_{2}^{2}$ or
$\Psi\left(\cdot\right)=\left\|\cdot\right\|_{\mathrm{osc}}^{2}$ – notation
definitions are detailed in Appendix A. In this paper we will refer to such
types of convergence as $\left(\Psi,\gamma\right)$-convergence (Definition 7).
The use of Strong Poincaré Inequalities (SPIs) for the analysis of Markov
chains is well-developed and understood, allowing for the characterization of
$\mathrm{L}^{2}-$geometric convergence, that is
$\big{(}\left\|\cdot\right\|_{2}^{2},\left\\{\rho^{n}:\rho\in\left[0,1\right),n\in\mathbb{N}\right\\}\big{)}$-convergence,
or providing bounds on the asymptotic variance of ergodic averages.

However it has been realised only recently that such tools can be particularly
powerful to establish results beyond the reach of, for example,
drift/minorization approaches [29, 18]. For instance, (a) in [2], leveraging
isoperimetric inequalities and building on [19, 16], quantitative SPIs have
been established, providing the first rigorous proof that the spectral gap of
the random walk Metropolis on $\mathbb{R}^{d}$ vanishes as $d^{-1}$ for a
broad class of probability distributions, (b) in [1], in the scenario where
$\mathrm{L}^{2}$-geometric convergence does not hold, comparison tools based
on Weak Poincaré Inequalities (WPIs) have been proposed and developed,
allowing for a fine and natural characterization of pseudo-marginal algorithms
in relation to their marginal counterparts.

The use and theory of WPIs in the Markov chain context are largely
underdeveloped in comparison to the Markov process scenario. This paper aims
to fill this gap by developing practical tools to establish such functional
inequalities, while also exploring the equivalence between
$\left(\Psi,\gamma\right)$-convergence and the existence of WPIs. More
specifically,

In Section 2 we provide a precise overview of the basic properties of WPIs for
Markov chains and their use in deducing
$\left(\Psi,\gamma\right)$-convergence, complete with various novel notions of
comparison and optimality leading, for example, to new techniques to lower
bound convergence rates. In this same section, we also discuss equivalence
with convergence in other standard metrics and provide a counterexample.

Section 3 is concerned with the existence of WPIs. Our first result is a Weak
Cheeger Inequality (WCI) for Markov chains which provides a theoretically and
practically useful tool to establish WPIs, linking the probabilistic and
functional analytic worlds. We further investigate the necessity of the
existence of a non-trivial WPI for a Markov chain to be
$\left(\Psi,\gamma\right)$-convergent. We show equivalence of WPIs with the
more probabilistic Resolvent Uniform Positivity-Improving (RUPI) condition of
[20]. In turn we show that RUPI is itself implied by $\mu$-irreducibility and
a positive holding probability condition. In addition, we provide a
counterexample of an
$\big{(}\left\|\cdot\right\|_{\mathrm{osc}}^{2},\left\\{\rho^{n}:\rho\in\left[0,1\right),n\in\mathbb{N}\right\\}\big{)}$-convergent
Markov chain which fails to satisfy a WPI. We finish Section 3 by showing that
WCIs can be obtained from isoperimetric properties of the target distribution
$\mu$ and a close coupling condition of the Markov kernel $P$, opening the way
to quantitative WPIs and estimates of $\mathrm{L}^{2}$ rates of convergence.

In Section 4 we study pseudo-marginal MCMC methods, and give lower bounds on
their convergence rates, complementing the previous upper bounds derived in
[1]. In Section 5 we derive specific WPIs for the Random Walk Metropolis (RWM)
MCMC algorithm, as applied to simulation from heavy-tailed probability
measures on Euclidean spaces and deduce complexity bounds. As a consequence,
we obtain both positive and negative results on their convergence to
equilibrium, providing an understanding of the cost incurred when sampling
from such distributions with this type of algorithm.

In Section 6 we give some concluding discussion regarding related literature.
Some of the results in this manuscript appeared in a preliminary form in our
technical report [3].

##### Notation

A list of notation can be found in Appendix A.

##  2 Weak Poincaré inequalities

###  2.1 Basic definitions

We first review the basic definitions needed to define weak Poincaré
inequalities; see also [3].

Throughout this section, for $\mu$ a probability measure defined on
$\left(\mathsf{E},\mathscr{E}\right)$, we fix a $\mu$-invariant Markov kernel
$P\colon\mathsf{E}\times\mathscr{E}\rightarrow\left[0,1\right]$.

######  Definition 1 (Sieve).

We call a functional
$\Psi:\mathrm{L}^{2}\bigl{(}\mu\bigr{)}\to\left[0,\infty\right]$ a sieve
functional, or sieve, if

  1. a).

for any $f\in\mathrm{L}^{2}\left(\mu\right)$, $c>0$, it holds that
$\Psi\left(c\cdot f\right)=c^{2}\cdot\Psi\left(f\right)$,

  2. b).

$\mathfrak{a}:=\sup_{f\in\mathrm{L}^{2}_{0}\left(\mu\right)\backslash\left\\{0\right\\}}\left\|f\right\|_{2}^{2}/\Psi\left(f\right)<\infty$.

We will furthermore describe $\Psi$ as $P$-non-expansive if
$\Psi\left(Pf\right)\leqslant\Psi\left(f\right)$ for
$f\in\mathrm{L}^{2}_{0}\left(\mu\right)$.

Note that the second property implies that if $\Psi\left(f\right)=0$ for some
$f\in\mathrm{L}^{2}_{0}\left(\mu\right)$ then $f=0$, which, as we shall see,
is a requirement for a weak Poincaré inequality to hold.

######  Example 2.

Typically, we will work with the sieve
$\Psi:\mathrm{L}^{2}\left(\mu\right)\to\left[0,\infty\right]$ given by
$\Psi:=\left\|\cdot\right\|_{{\rm osc}}^{2}$, which satisfies the requirements
of Definition 1 with $\mathfrak{a}\leqslant\frac{1}{4}$.

In Section 2.5 we will consider other choices of functional $\Psi$.

######  Definition 3 (Weak Poincaré Inequalities).

For a sieve $\Psi$, we say that a $\mu$-invariant kernel $T$ satisfies

  1. a).

a $\left(\Psi,\alpha\right)$–weak Poincaré inequality, abbreviated
$\left(\Psi,\alpha\right)$–WPI, if for a decreasing function
$\alpha:\left(0,\infty\right)\to\left[0,\infty\right)$,

$\left\|f\right\|_{2}^{2}\leqslant\alpha(r)\cdot\mathcal{E}\left(T,f\right)+r\cdot\Psi\left(f\right),\quad\forall
r>0,f\in\mathrm{L}^{2}_{0}\left(\mu\right),$ (1)

  2. b).

a $\left(\Psi,\beta\right)$_–WPI_ if for
$\beta:\left(0,\infty\right)\to\left[0,\infty\right)$ a decreasing function
with $\beta\left(s\right)\to 0$ as $s\to\infty$,

$\left\|f\right\|_{2}^{2}\leqslant
s\cdot\mathcal{E}\left(T,f\right)+\beta\left(s\right)\cdot\Psi\left(f\right),\quad\forall
s>0,f\in\mathrm{L}^{2}_{0}\left(\mu\right),$ (2)

  3. c).

a $\left(\Psi,K^{*}\right)$_–WPI_ if for a nonnegative, convex, strictly
increasing function
$K^{*}:\left[0,\mathfrak{a}\right)\to\left[0,\infty\right)$ satisfying
$K^{*}\left(0\right)=0$,

$\frac{\mathcal{E}\left(T,f\right)}{\Psi\left(f\right)}\geqslant
K^{*}\left(\frac{\left\|f\right\|_{2}^{2}}{\Psi\left(f\right)}\right),\quad
f\in\mathrm{L}^{2}_{0}\left(\mu\right),0<\Psi\left(f\right)<\infty\,.$

The condition
$\mathfrak{a}:=\sup_{f\in\mathrm{L}^{2}_{0}\left(\mu\right)\backslash\left\\{0\right\\}}\left\|f\right\|_{2}^{2}/\Psi\left(f\right)<\infty$
implies that

  1. a).

if a $\left(\Psi,\alpha\right)$_–_ WPI holds for some $\alpha$, then it
necessarily holds that $\alpha\left(r\right)>0$ for
$r\in\left(0,\mathfrak{a}\right)$,

  2. b).

if a $\left(\Psi,\alpha\right)$_–_ WPI holds for some $\alpha$, then there
also holds a $\left(\Psi,\alpha^{{}^{\prime}}\right)$_–_ WPI with
$\alpha^{{}^{\prime}}\left(r\right):=\alpha\left(r\right)\cdot\mathbf{1}_{\left[0,\mathfrak{a}\right)}\left(r\right)$,
and

  3. c).

if a $\left(\Psi,\beta\right)$_–_ WPI holds for some $\beta$, then there also
holds a $\left(\Psi,\beta^{{}^{\prime}}\right)$_–_ WPI with
$\beta^{{}^{\prime}}\left(s\right):=\min\left\\{\beta\left(s\right),\mathfrak{a}\right\\}$.

The subsequent Proposition 4 establishes that any one WPI holding implies that
all of them hold, and hence we can refer to a $\Psi$–WPI when the
parametrization is unimportant. The proposition also provides a recipe to move
between parametrizations. As we shall see, the $K^{*}$ formulation leads to
quantitative characterizations of convergence to equilibrium, but the $\alpha$
or $\beta$ parametrizations are often more convenient to establish in
practice.

For any decreasing function $f\colon\mathbb{R}_{+}\rightarrow\mathbb{R}$ we
let $f^{\mathrm{\shortminussymb}}:\mathbb{R}\to\left[0,\infty\right]$ be its
generalised inverse, given by
$f^{\mathrm{\shortminussymb}}\left(x\right):=\inf\left\\{y>0\colon
f\left(y\right)\leqslant x\right\\}$ for $x\in\mathbb{R}$.

######  Proposition 4.

Let $T$ be a Markov kernel on $\big{(}\mathsf{E},\mathscr{E}\big{)}$, $\Psi$
be a sieve, and
$\mathfrak{a}:=\sup_{f\in\mathrm{L}^{2}_{0}\left(\mu\right)\backslash\left\\{0\right\\}}\left\|f\right\|_{2}^{2}/\Psi\left(f\right)<\infty$.

  1. a).

If a $\left(\Psi,\alpha\right)$–WPI holds with $\alpha\left(r\right)=0$ for
$r\geqslant\mathfrak{a}$, then a $\left(\Psi,\beta\right)$–WPI, with
$\beta:=\alpha^{\mathrm{\shortminussymb}}$ on $\left(0,\infty\right)$, holds.

  2. b).

If a $\left(\Psi,\beta\right)$–WPI holds with $\beta\leqslant\mathfrak{a}$,
then a $\left(\Psi,\alpha\right)$–WPI holds, with
$\alpha:=\beta^{\mathrm{\shortminussymb}}$ on $\left(0,\infty\right)$.

  3. c).

Further, if $\alpha$ (resp. $\beta$) is right continuous then
$\left(\alpha^{\mathrm{\shortminussymb}}\right)^{\mathrm{\shortminussymb}}=\alpha$
(resp.
$\left(\beta^{\mathrm{\shortminussymb}}\right)^{\mathrm{\shortminussymb}}=\beta$);
the two parametrisations are thus equivalent.

  4. d).

If a $\left(\Psi,\beta\right)$–WPI holds then a $\left(\Psi,K^{*}\right)$–WPI
holds with

$K^{*}(v):=\sup_{u\geqslant 0}\left\\{u\cdot v-K\left(u\right)\right\\},\qquad
v\geqslant 0,$ (3)

the convex conjugate of
$K\colon\left[0,\infty\right)\rightarrow\left[0,\infty\right)$ given by
$K\left(u\right):=u\,\beta\left(1/u\right)$ for $u>0$ and
$K\left(0\right):=0$.

  5. e).

If a $\left(\Psi,K^{*}\right)$–WPI holds then a $\left(\Psi,\beta\right)$–WPI
holds with $\beta\left(s\right)=s\cdot K\left(s^{-1}\right)$ for $s>0$ where

$K\left(u\right):=\sup_{0\leqslant v\leqslant\mathfrak{a}}\left\\{u\cdot
v-K^{*}\left(v\right)\right\\},\qquad u\geqslant 0,$

is the convex conjugate of $K^{*}$.

###### Proof.

The proof of a)-c) can be found in [3, Proposition 5]. The proof of d) can be
found in [1, Lemma 1, Supp. material].

So we focus now on e). For $f\in\mathrm{L}^{2}_{0}\left(\mu\right)$ such that
$0<\Psi\left(f\right)<\infty$ and any $u>0$, we have that

$\displaystyle\frac{\mathcal{E}\left(T,f\right)}{\Psi\left(f\right)}$
$\displaystyle\geqslant
K^{*}\left(\frac{\left\|f\right\|_{2}^{2}}{\Psi\left(f\right)}\right)\geqslant
u\cdot\frac{\left\|f\right\|_{2}^{2}}{\Psi\left(f\right)}-K\left(u\right),$

and by rearranging,

$\left\|f\right\|_{2}^{2}\leqslant
u^{-1}\cdot\mathcal{E}\left(T,f\right)+u^{-1}\cdot
K\left(u\right)\cdot\Psi\left(f\right)\,.$

For $u>0$, the function

$u\mapsto u^{-1}\cdot
K\left(u\right)=\sup_{0<v\leqslant\mathfrak{a}}\left\\{v-u^{-1}\cdot
K^{*}\left(v\right)\right\\}$

is strictly increasing since for $0<u_{0}<u_{1}$,

$\sup_{0<v\leqslant\mathfrak{a}}\left\\{v-u_{0}^{-1}\cdot
K^{*}\left(v\right)\right\\}<\sup_{0<v\leqslant\mathfrak{a}}\left\\{v-u_{1}^{-1}\cdot
K^{*}\left(v\right)\right\\}\,.$

We show that $\lim_{u\downarrow 0}u^{-1}\cdot K\left(u\right)=0$, which
implies that $\beta\left(s\right)\to 0$ as $s\to\infty$. For $u>0$,

$\displaystyle u^{-1}\cdot K\left(u\right)$
$\displaystyle\leqslant\sup\left\\{v\colon
0<v\leqslant\mathfrak{a},v^{-1}\cdot K^{*}\left(v\right)\leqslant
u\right\\}=:\kappa\left(u\right)\,.$

Since $v\mapsto v^{-1}K^{*}\left(v\right)$ is increasing, it follows that
$u\mapsto\kappa\left(u\right)$ is increasing. Now suppose that
$\ell:=\lim_{v\downarrow 0}v^{-1}\cdot K^{*}\left(v\right)>0$. Then for
$0<u<\ell$ we have directly that $u^{-1}\cdot
K\left(u\right)=\sup_{0<v\leqslant\mathfrak{a}}\left\\{v-u^{-1}\cdot
K^{*}\left(v\right)\right\\}=\sup_{0<v\leqslant\mathfrak{a}}v\cdot\left\\{1-u^{-1}\cdot
v^{-1}\cdot K^{*}\left(v\right)\right\\}$ must be 0. Otherwise if $\ell=0$,
then for any $\bar{v}\in\left(0,\mathfrak{a}\right]$ and any $0\leqslant
u<\bar{v}^{-1}K^{*}\left(\bar{v}\right)$ we have
$\kappa\left(u\right)<\bar{v}$.

The final statement follows from the definition of the convex conjugate $K$ of
$K^{*}$. ∎

######  Definition 5.

In the situation where a $\left(\Psi,\alpha\right)$–WPI (resp.
$\left(\Psi,\beta\right)$–WPI) holds for a right-continuous function $\alpha$
(resp. $\beta$), we refer to it as a $\left(\Psi,\alpha,\beta\right)$–WPI
where $\beta=\alpha^{\mathrm{\shortminussymb}}$ (resp.
$\alpha=\beta^{\mathrm{\shortminussymb}}$).

######  Remark 6.

As remarked in Appendix A, it is straightforward to see that for all
$p\geqslant 1$, there holds the equality
$\mathcal{E}_{p}\left(T,f\right)=\mathcal{E}_{p}\left(S,f\right)$, defined in
(28), where $S:=\frac{1}{2}\left(T+T^{*}\right)$ is the additive
reversibilization of $T$. Hence, WPIs are generally only properties of the
reversible part of $T$, a property used in some of our later arguments.

In practice, we are interested in bounding the convergence rate to equilibrium
of a given $\mu$-invariant Markov kernel $P$, as measured in
$\mathrm{L}^{2}\left(\mu\right)$.

######  Definition 7 ($\big{(}\Psi,\gamma\big{)}$–convergence).

Given a $\mu$-invariant Markov kernel, $P$ and a sieve $\Psi$, we say that $P$
is $\left(\Psi,\gamma\right)$-convergent or simply $\Psi$-convergent if for
all $n\in\mathbb{N}$ and $f\in\mathrm{L}^{2}_{0}\left(\mu\right)$ such that
$0<\Psi\left(f\right)<\infty$,

$\left\|P^{n}f\right\|_{2}^{2}\leqslant\gamma\left(n\right)\cdot\Psi\left(f\right),$
(4)

holds for a sequence
$\left\\{\gamma\left(n\right):n\in\mathbb{N}\right\\}\subseteq\left(0,\infty\right)$
such that $\lim_{n\rightarrow\infty}\gamma\left(n\right)=0$.

$\big{(}\Psi,\gamma\big{)}$-convergence can be obtained in the framework of
Definition 3 by taking $T=P^{*}P$ due to the following convergence result.

######  Theorem 8.

Let $P$ be a $\mu$-invariant Markov kernel on
$\left(\mathsf{E},\mathscr{E}\right)$ and assume that $T:=P^{*}P$ satisfies a
$\left(\Psi,K^{*}\right)-$WPI, where $\Psi$ is a $P$-non-expansive sieve.
Define $F\colon\left(0,\mathfrak{a}\right]\rightarrow\mathbb{R}$ to be the
strictly decreasing, convex and invertible function

$F\left(x\right):=\int_{x}^{\mathfrak{a}}\frac{{\rm
d}v}{K^{*}\left(v\right)}\,.$

Then

  1. a).

for any $\varepsilon\in\left(0,\mathfrak{a}\right]$, with

$n\left(\varepsilon;\Psi\right):=\inf\left\\{n\in\mathbb{N}\colon\left\|P^{n}f\right\|_{2}^{2}\leqslant\varepsilon\cdot\Psi\left(f\right)\,\forall
f\in\mathrm{L}^{2}_{0}\left(\mu\right)\right\\}\,,$

it holds that

$n\left(\varepsilon;\Psi\right)\leqslant F\left(\varepsilon\right)\,.$

Therefore for $n>F\left(\varepsilon\right)$,
$\left\|P^{n}f\right\|_{2}^{2}\leqslant\varepsilon\cdot\Psi\left(f\right)\,\forall
f\in\mathrm{L}^{2}_{0}\left(\mu\right)$.

  2. b).

$P$ is $\big{(}\Psi,\gamma\big{)}$-convergent with
$\gamma\left(n\right):=F^{-1}\left(n\right)$.

###### Proof.

For a), from the proof of [1, Proof of Theorem 8, Supplementary material] we
have for any $n\geqslant 1$ and $\Psi\left(f\right)>0$,

$n\leqslant
F\left(\frac{\left\|P^{n}f\right\|_{2}^{2}}{\Psi\left(f\right)}\right)-F\left(\frac{\left\|f\right\|_{2}^{2}}{\Psi\left(f\right)}\right)\,.$
(5)

Let $\varepsilon>0$. Since
$n\mapsto\left\|P^{n}f\right\|_{2}^{2}/\Psi\left(f\right)$ is decreasing and
$F$ is strictly decreasing, it then follows that
$\left\|P^{n}f\right\|_{2}^{2}/\Psi\left(f\right)>\varepsilon$ implies
$n\leqslant
F\left(\varepsilon\right)-F\left(\left\|f\right\|_{2}^{2}/\Psi\left(f\right)\right)$
and as a result for
$n>F\left(\varepsilon\right)-F\big{(}\left\|f\right\|_{2}^{2}/\Psi\left(f\right)\big{)}$
we must have that
$\left\|P^{n}f\right\|_{2}^{2}/\Psi\left(f\right)\leqslant\varepsilon$. From
the first statement we conclude that $n\left(\varepsilon;\Psi\right)\leqslant
F\left(\varepsilon\right)$ noting that there is no loss in this bound since

$\inf_{\Psi\left(f\right)\neq
0}F\left(\frac{\left\|f\right\|_{2}^{2}}{\Psi\left(f\right)}\right)=F\left(\sup_{\Psi\left(f\right)\neq
0}\frac{\left\|f\right\|_{2}^{2}}{\Psi\left(f\right)}\right)=0\,.$

For b) simply invert (5). ∎

######  Example 9 ([1, Lemma 14]).

If for some $c_{0},c_{1}>0$ $\beta\left(s\right)=c_{0}s^{-c_{1}}$ then
$K^{*}(v)=C\left(c_{0},c_{1}\right)v^{1+c_{1}^{-1}}$ for some
$C\left(c_{0},c_{1}\right)>0$ and

$F^{-1}\left(n\right)\leqslant c_{0}\cdot\left(1+c_{1}\right)^{1+c_{1}}\cdot
n^{-c_{1}}.$

######  Remark 10.

In the case where $P$ is reversible and
$\Psi=\left\|\cdot\right\|_{\mathrm{osc}}^{2}$, the conclusions of Theorem 8
allow one to make statements about the convergence of the probability measures
$\nu P^{n}$ to equilibrium for suitably regular initial $\nu$; we return to
this point in Section 2.5 and Section 5 for our main application.

###  2.2 Optimal $\alpha,\beta$

It is clear from the definition of a Poincaré inequality that if a kernel
satisfies a $\left(\Psi,\alpha\right)$-weak Poincaré inequality, then it also
satisfies a $\left(\Psi,\alpha^{\prime}\right)$-weak Poincaré inequality, for
any decreasing $\alpha^{\prime}$ which satisfies
$\alpha^{\prime}\geqslant\alpha$ pointwise. It is thus natural to introduce a
notion of a minimal or optimal weak Poincaré inequality.

By studying these extremal cases, we will be able to assume without a loss of
generality that our $\left(\alpha,\beta,K^{*}\right)$ satisfy some natural
structural properties, thus facilitating our subsequent technical
developments. Moreover, given the task of comparing the convergence behaviour
of a pair of Markov chains $\left(P_{1},P_{2}\right)$, matters become
substantially more transparent by comparing optimal inequalities.

######  Definition 11.

For a $\mu$-invariant Markov kernel $T$ and sieve $\Psi$ define,

  1. a).

for any $r>0$,

$\alpha^{\star}\left(r;\Psi\right):=\sup\left\\{\frac{\left\|g\right\|_{2}^{2}}{\mathcal{E}\left(T,g\right)}\cdot\left(1-\frac{r}{\left\|g\right\|_{2}^{2}}\right)\colon
g\in\mathrm{L}^{2}_{0}\left(\mu\right),\Psi\left(g\right)=1\right\\}\vee 0,$

noting that if $r\geqslant\mathfrak{a}$, then
$\alpha^{\star}\left(r;\Psi\right)=0$;

  2. b).

for any $s>0$,

$\beta^{\star}\left(s;\Psi\right):=\sup\left\\{\left\|g\right\|_{2}^{2}-s\cdot\mathcal{E}\left(T,g\right)\colon
g\in\mathrm{L}^{2}_{0}\left(\mu\right),\Psi\left(g\right)=1\right\\}\vee 0,$

noting that for $s>0$ $\beta^{\star}\left(s;\Psi\right)\leqslant\mathfrak{a}$.

When $\Psi=\left\|\cdot\right\|_{\mathrm{osc}}^{2}$ we shall plainly write
$\alpha^{\star}\left(\cdot\right):=\alpha^{\star}\left(\cdot;\Psi\right)$ and
$\beta^{\star}\left(\cdot\right):=\beta^{\star}\left(\cdot;\Psi\right)$.

An important point is that $\alpha^{\star}$ and $\beta^{\star}$ do not
necessarily define valid WPIs. Indeed as shown in Proposition 66 the Markov
chain of Example 65 is such that $P^{*}P$ does not satisfy a WPI, although it
can be shown to be $\|\cdot\|_{{\rm osc}}$-convergent; in fact inspection of
the proof of Proposition 66 allows one to deduce that in this specific example
$\inf_{s>0}\beta^{\star}\left(s\right)>0$.

######  Theorem 12.

Suppose that the $\mu$-invariant kernel $T$ possesses some
$\left(\Psi,\alpha\right)$– or $\left(\Psi,\beta\right)$–WPI for some
functions $\alpha$ or $\beta$. Then

  1. a).

$\alpha^{\star}\left(\cdot;\Psi\right)\leqslant\alpha\left(\cdot\right)$
defines a $\left(\Psi,\alpha^{\star}\right)\mathrm{\shortminussymb}$WPI and
$\beta^{\star}\left(\cdot;\Psi\right)\leqslant\beta\left(\cdot\right)$ defines
a $\left(\Psi,\beta^{\star}\right)$–WPI;

  2. b).

the functions
$\alpha^{\star}\left(\cdot;\Psi\right):\left(0,\mathfrak{a}\right]\to\left[0,\infty\right)$
and
$\beta^{\star}\left(\cdot;\Psi\right):\left[0,\infty\right)\to\left[0,\mathfrak{a}\right]$
are convex and continuous;

  3. c).

$\beta^{\star}$ is strictly decreasing to $0$ and
$\alpha^{\star}=\left(\beta^{\star}\right)^{-1}$ is the inverse function,
which is well-defined on $\left(0,\mathfrak{a}\right]$ and strictly
decreasing.

###### Proof.

We consider the $\beta$ formulation, and drop explicit reference to the fixed
$\Psi$ under consideration; the $\alpha$ formulation is analogous. By
assumption, we know that $T$ possesses a $\left(\Psi,\beta\right)$_–_ WPI, for
some function $\beta$ as in Definition 3 (c.f. [3, Proposition 5]). By
definition of $\beta^{\star}$, we have that
$0\leqslant\beta^{\star}\leqslant\beta$ pointwise and so
$\beta^{\star}\left(s\right)\to 0$ as $s\to\infty$. Since the pointwise
supremum of affine functions (of $s$) is convex, we obtain convexity and
continuity of $\beta^{\star}$, from the fact that it is the composition of a
nondecreasing convex continuous function, $s\mapsto\max\left\\{0,s\right\\}$,
with a convex function. We observe that
$\beta^{\star}\left(0\right)=\mathfrak{a}$. Now, let
$s_{0}:=\inf\left\\{s>0:\beta^{\star}\left(s\right)=0\right\\}$, which may be
infinite. Since $\beta^{\star}$ is convex and continuous, it is strictly
decreasing on $\left(0,s_{0}\right)$. It follows that $\beta^{\star}$ is
invertible on $\left(0,s_{0}\right)$ with inverse
$\left(\beta^{\star}\right)^{-1}:\left(0,\mathfrak{a}\right]\to\left[0,\infty\right)$
that is also convex and strictly decreasing.

Now we show that $\alpha^{\star}=\left(\beta^{\star}\right)^{-1}$. For
$r\in\left(0,\mathfrak{a}\right]$, let
$s:=\left(\beta^{\star}\right)^{-1}\left(r\right)$. For any
$f\in\mathrm{L}^{2}_{0}\left(\mu\right)$ with $\Psi\left(f\right)=1$ we have

$\left\|f\right\|_{2}^{2}-s\cdot\mathcal{E}\left(T,f\right)\leqslant\beta^{\star}\left(s\right)=r,$

and this implies

$\alpha^{\star}\left(r\right)=\sup_{f:\Psi\left(f\right)=1}\left\\{\frac{\left\|f\right\|_{2}^{2}}{\mathcal{E}\left(T,f\right)}-\frac{r}{\mathcal{E}\left(T,f\right)}\right\\}\leqslant
s.$

Assume for the sake of contradiction that $\alpha^{\star}\left(r\right)=t<s$.
For any $f\in\mathrm{L}^{2}_{0}\left(\mu\right)$ with $\Psi\left(f\right)=1$
we have

$\frac{\left\|f\right\|_{2}^{2}}{\mathcal{E}\left(T,f\right)}-\frac{r}{\mathcal{E}\left(T,f\right)}\leqslant
t,$

and so

$\beta^{\star}\left(t\right)=\sup_{f:\Psi\left(f\right)=1}\left\\{\left\|f\right\|_{2}^{2}-t\cdot\mathcal{E}\left(T,f\right)\right\\}\leqslant
r=\beta^{\star}\left(s\right),$

which is a contradiction since $\beta^{\star}$ is decreasing, and we conclude.
∎

###  2.3 Ordering of $\alpha$’s, $\beta$’s and $\gamma$’s and Peskun–Tierney
ordering

######  Theorem 13.

Let $P_{1}$ and $P_{2}$ be $\mu$-invariant Markov kernels such that for a
sieve $\Psi$, $P_{1}^{*}P_{1}$ satisfies a
$\left(\Psi,\alpha_{1},\beta_{1}\right)$–WPI and $P_{2}^{*}P_{2}$ a
$\left(\Psi,\alpha_{2},\beta_{2}\right)$–WPI respectively. Let
$\gamma_{i}=F_{i}^{-1}$ be the respective convergence bound functions as
defined in Theorem 8. Then we have

  1. a).

$\alpha_{2}\left(\cdot;\Psi\right)\geqslant\alpha_{1}\left(\cdot;\Psi\right)$
if and only if
$\beta_{2}\left(\cdot;\Psi\right)\geqslant\beta_{1}\left(\cdot;\Psi\right)$;

  2. b).

$\beta_{2}\left(\cdot;\Psi\right)\geqslant\beta_{1}\left(\cdot;\Psi\right)$
implies
$\gamma_{2}\left(\cdot;\Psi\right)\geqslant\gamma_{1}\left(\cdot;\Psi\right)$.

###### Proof.

First statement: we drop $\Psi$ for notational simplicity. For the direction
$\left(\implies\right):$ for any $s>0$ we have
$\left\\{r>0:\alpha_{2}\left(r\right)\leqslant
s\right\\}\subset\left\\{r>0:\alpha_{1}\left(r\right)\leqslant s\right\\}$ and
hence
$\beta_{2}=\alpha_{2}^{\mathrm{\shortminussymb}}\geqslant\alpha_{1}^{\mathrm{\shortminussymb}}=\beta_{1}$;
$\left(\Longleftarrow\right)$ follows along the same lines. For the second
statement: from their definitions, $K_{1}\leqslant K_{2}$ and hence
$K_{1}^{*}\geqslant K_{2}^{*}$. As a result, $F_{1}\leqslant F_{2}$ and
consequently $\gamma_{1}:=F_{1}^{-1}\leqslant F_{2}^{-1}=:\gamma_{2}$. ∎

We know from [39] that for $P_{1},P_{2}$ $\mu$-reversible, then
$\mathcal{E}\left(P_{1},g\right)\geqslant\mathcal{E}\left(P_{2},g\right)$ for
any $g\in\mathrm{L}^{2}\left(\mu\right)$ implies ${\rm{\rm
var}}\left(P_{1},f\right)\leqslant{\rm{\rm var}}\left(P_{2},f\right)$ for
$f\in\mathrm{L}^{2}\left(\mu\right)$ and ${\rm
Gap}_{\mathrm{R}}\left(P_{1}\right)\geqslant{\rm
Gap}_{\mathrm{R}}\left(P_{2}\right)$. The latter is useful when ${\rm
Gap}\left(P_{1}\right)={\rm Gap}_{\mathrm{R}}\left(P_{1}\right)$ and ${\rm
Gap}\left(P_{2}\right)={\rm Gap}_{\mathrm{R}}\left(P_{2}\right)$ since this
implies $P_{1}$ has a faster rate of convergence. The following generalizes
the latter statement to the subgeometric setup.

######  Theorem 14.

Let $P_{1},P_{2}$ be $\mu$-invariant Markov kernels such that for a sieve
$\Psi$,

  1. a).

$P_{1}^{*}P_{1}$ (resp. $P_{2}^{*}P_{2}$) satisfies a
$\left(\Psi,\alpha_{1},\beta_{1}\right)$–WPI (resp. a
$\left(\Psi,\alpha_{2},\beta_{2}\right)$–WPI),

  2. b).

$\mathcal{E}\left(P_{1}^{*}P_{1},g\right)\geqslant\mathcal{E}\left(P_{2}^{*}P_{2},g\right)$
for any $g\in\mathrm{L}^{2}\left(\mu\right)$ such that
$\Psi\left(g\right)<\infty$.

Then with $\alpha_{i}^{\star}\left(\cdot;\Psi\right)$ and
$\beta_{i}^{\star}\left(\cdot;\Psi\right)$ for $i=1,2$ defined as in
Definition 11, a $\left(\Psi,\alpha_{i}^{\star},\beta_{i}^{\star}\right)$–WPI
holds for $i=1,2$, there are the orderings
$\alpha_{1}^{\star}\left(\cdot;\Psi\right)\leqslant\alpha_{2}^{\star}\left(\cdot;\Psi\right)$,
$\beta_{1}^{\star}\left(\cdot;\Psi\right)\leqslant\beta_{2}^{\star}\left(\cdot;\Psi\right)$,
and the corresponding respective convergence bounds in Theorem 8 satisfy
$\gamma_{1}^{\star}\leqslant\gamma_{2}^{\star}$.

###### Proof.

From the ordering of Dirichlet forms, we have for any
$g\in\mathrm{L}^{2}\left(\mu\right)$ that

$\left\|g\right\|_{2}^{2}-s\cdot\mathcal{E}\left(P_{1}^{*}P_{1},g\right)\leqslant\left\|g\right\|_{2}^{2}-s\cdot\mathcal{E}\left(P_{2}^{*}P_{2},g\right),$

from Definition 11 we deduce
$\beta_{1}^{\star}\left(\cdot;\Psi\right)\leqslant\beta_{2}^{\star}\left(\cdot;\Psi\right)$;
a similar argument gives that
$\alpha_{1}^{\star}\left(\cdot;\Psi\right)\leqslant\alpha_{2}^{\star}\left(\cdot;\Psi\right)$.
From Theorem 13, we then conclude that
$\gamma_{1}^{\star}\leqslant\gamma_{2}^{\star}$. ∎

###  2.4 Lower bounds on convergence rates

This section is concerned with tools to establish lower bounds on $\beta$
involved in a $\left(\Psi,\beta\right)$_–_ WPI from which one can deduce lower
bounds on the convergence rate $\gamma$ in Theorem 8. This is illustrated with
examples.

A practical difficulty with the result of Theorem 8 and the results presented
so far is that they require manipulating $\mathcal{E}\left(P^{*}P,f\right)$
which is most often not tractable, in contrast with
$\mathcal{E}\left(P,f\right)$. This is addressed extensively in [1, Section
2.2.1 and Theorem 42]. We first show that a lower bound on $\beta_{1}^{\star}$
in a $\left(\Psi,\beta_{1}^{\star}\right)$_–_ WPI for $P$ can imply a lower
bound on $\beta_{2}^{\star}$ in a $\left(\Psi,\beta_{2}^{\star}\right)$_–_ WPI
for $P^{*}P$. We first recall the following result.

######  Lemma 15 ([17, Remark 3.1]).

Let $P$ be $\mu$-invariant. Then

$\mathcal{E}\left(P^{*}P,f\right)\leqslant
2\cdot\mathcal{E}\left(P,f\right),\qquad
f\in\mathrm{L}^{2}_{0}\left(\mu\right).$

######  Remark 16.

If $P$ is $\mu$-reversible, then one can obtain
$\mathcal{E}\left(P^{2},f\right)\leqslant\left(1+\lambda_{\star}\right)\cdot\mathcal{E}\left(P,f\right)$
by using the spectral theorem, where
$\lambda_{\star}=\sup\sigma_{0}\left(P\right)$. However, since the focus here
is on WPIs, the case $\lambda_{\star}<1$ is less relevant. We note also that a
converse of sorts may be obtained when $P$, and therefore $P^{*}$, satisfies
$P\big{(}x,\left\\{x\right\\}\big{)}\geqslant\varepsilon$ on a $\mu$-full set;
see Lemma 57.

######  Lemma 17.

Let $P$ be $\mu$-invariant, and assume it satisfies a
$\left(\Psi,\beta_{1}^{\star}\right)$_–_ WPI, where $\beta_{1}^{\star}$ is
pointwise minimal. Assume $P^{*}P$ satisfies a
$\left(\Psi,\beta_{2}^{\star}\right)$_–_ WPI where $\beta_{2}^{\star}$ is
pointwise minimal. Then
$\beta_{2}^{\star}\left(s\right)\geqslant\beta_{1}^{\star}\left(2\cdot
s\right)$.

###### Proof.

Let
$\mathcal{F}=\left\\{f\in\mathrm{L}^{2}_{0}\left(\mu\right):\Psi\left(f\right)=1\right\\}$.
By Lemma 15 we have that $\mathcal{E}\left(P^{*}P,f\right)\leqslant
2\cdot\mathcal{E}\left(P,f\right)$. We may write

$\beta_{1}^{\star}\left(s\right)=0\vee\sup_{f\in\mathcal{F}}\left\\{\left\|f\right\|_{2}^{2}-s\cdot\mathcal{E}\left(P,f\right)\right\\}.$

We then have that

$\displaystyle\beta_{2}^{\star}\left(s\right)$
$\displaystyle=0\vee\sup_{f\in\mathcal{F}}\left\\{\left\|f\right\|_{2}^{2}-s\cdot\mathcal{E}\left(P^{*}P,f\right)\right\\}$
$\displaystyle\geqslant
0\vee\sup_{f\in\mathcal{F}}\left\\{\left\|f\right\|_{2}^{2}-2\cdot
s\cdot\mathcal{E}\left(P,f\right)\right\\}$
$\displaystyle=\beta_{1}^{\star}\left(2\cdot s\right),$

and we conclude. ∎

In the case where $P$ is $\mu$-reversible, we can then deduce from a
$\left(\Psi,\beta_{1}\right)$–WPI for $P$ a lower bound on a separable rate of
convergence for $\left\|P^{n}f\right\|_{2}$.

######  Proposition 18.

Assume $P$ is $\mu$-reversible, satisfies a $\left(\Psi,\beta\right)$–WPI and
the pointwise minimal $\beta^{\star}$ satisfies
$\beta^{\star}\left(s\right)\in\Omega\left(s^{-p}\right)$ for some $p>0$. Then
it cannot hold that with $q>p$,
$\left\|P^{n}f\right\|_{2}^{2}\in\mathcal{O}\left(n^{-q}\right)$ for all
$f\in\mathrm{L}^{2}_{0}\left(\mu\right)$ with $\Psi\left(f\right)<\infty$.

###### Proof.

If $\beta^{\star}\left(s\right)\in\Omega\left(s^{-p}\right)$ then we may
deduce by Lemma 17 that if $P^{2}$ satisfies (2), its pointwise minimal
$\beta_{2}^{\star}$ also satisfies
$\beta_{2}^{\star}\left(s\right)\in\Omega\left(s^{-p}\right)$. Now assume for
the sake of contradiction that
$\left\|P^{n}f\right\|_{2}^{2}\in\mathcal{O}\left(n^{-q}\right)$ for all
$f\in\mathrm{L}_{0}^{2}\left(\mu\right)$ such that
$\Psi\left(f\right)<\infty$. Then by [1, Proposition 24 and Remark 25], we
deduce that a WPI for $P^{2}$ holds with
$\beta_{2}\left(s\right)\in\mathcal{O}\left(s^{-q}\right)$, which contradicts
$\beta_{2}^{\star}\left(s\right)\in\Omega\left(s^{-p}\right)$ being pointwise
minimal. ∎

In principle, noting that $\alpha^{\star}$ and $\beta^{\star}$ are pointwise
minimal functions, any function $f\in\mathrm{L}^{2}_{0}\left(\mu\right)$ with
$\Psi\left(f\right)=1$ may be used to construct a lower bound on them. For
example, for any such function,

$s\mapsto\beta^{\star}\bigl{(}s\bigr{)}\geqslant\left\|f\right\|_{2}^{2}-s\cdot\mathcal{E}\left(T,f\right),\qquad
s>0.$

In practice, to produce an informative lower bound for the whole function
$\beta^{\star}$, one will need to identify an appropriate subset of
$\mathrm{L}^{2}_{0}\left(\mu\right)$ and consider the supremum of the lower
bound above over such a subset. Indicator functions of measurable sets are
always in $\mathrm{L}^{2}_{0}\left(\mu\right)$, have finite oscillation, and
can provide tractability. For example in such a scenario, for
$A\in\mathscr{E}$, $\mathcal{E}\left(P,{\bf 1}_{A}\right)$ has a natural
probabilistic interpretation (see Lemma 102).

The following result establishes a lower bound on $\beta^{\star}$ for Markov
kernels $P$ that can exhibit sticky behaviour in regions of the state space.
[36, Theorem 5.1] showed that for a $\mu$-invariant Markov kernel $P$ with
$\mu$ not concentrated at a single point, then ${\rm
ess}_{\mu}\sup_{x}P\left(x,\left\\{x\right\\}\right)=1$ implies that $P$
cannot converge geometrically. In [25, Theorem 1] conductance is used to prove
the same when $P$ is $\mu$-reversible, and the following provides a
quantitative refinement. A similar result can be obtained using a Weak Cheeger
inequality established in Section 3.1, Theorem 35-b).

######  Theorem 19.

Let $P$ be $\mu$-invariant satisfying a $\left(\Psi,\beta\right)$–WPI, where
$\Psi\left(\mathbf{1}_{A}\right)\leqslant 1$ for any set $A\in\mathscr{E}$.
For any $\varepsilon>0$, define the set
$A_{\varepsilon}:=\left\\{x\in\mathsf{E}\colon
P\left(x,\left\\{x\right\\}\right)\geqslant 1-\varepsilon\right\\}$. Then

  1. a).

for any $s>0$,

$\beta\left(s;\Psi\right)\geqslant\sup_{\varepsilon\in\left(0,1\right)}\left\\{\mu\left(A_{\varepsilon}\right)\cdot\left(1-s\cdot\varepsilon-\mu\left(A_{\varepsilon}\right)\right)\right\\},$
(6)

this is in particular true for $\beta^{\star}$ given in Definition 11 and $P$
satisfies a $\left(\Psi,\beta^{\star}\right)$-WPI,

  2. b).

if in addition there exist $A\in\mathscr{E}$ such that $\mu\left(A\right)>0$
and $P\left(x,\left\\{x\right\\}\right)<1$ for $x\in A$ then there exists
$\underline{s}>0$ and $C>0$ such that for any $s\geqslant\underline{s},$

$\beta\left(s;\Psi\right)\geqslant
C\cdot\mu\left(A_{\varepsilon\left(s\right)}\right)\,,$

with $\varepsilon\left(s\right):=C/s$.

###### Proof.

For any $A\in\mathscr{E}$, from Lemma 102, we have
$\mathcal{E}\left(P,\mathbf{1}_{A}\right)=\mu\otimes P\left(A\times
A^{\complement}\right)$ and ${\rm
var}\big{(}\mathbf{1}_{A}\big{)}=\mu\otimes\mu\left(A\times
A^{\complement}\right)$. Since
$\Psi\left(\mathbf{1}_{A_{\varepsilon}}\right)\leqslant 1$, for any
$\varepsilon>0$ we have that

$\displaystyle\beta^{\star}\left(s\right)$
$\displaystyle:=\sup\left\\{\left\|f\right\|_{2}^{2}-s\cdot\mathcal{E}\left(P,f\right)\colon
f\in\mathrm{L}^{2}_{0}\left(\mu\right),\Psi\left(f\right)\leqslant 1\right\\}$
$\displaystyle\geqslant{\rm
var}_{\mu}\left(\mathbf{1}_{A_{\varepsilon}}\right)-s\cdot\int\mu\left(\mathrm{d}x\right)\cdot
P\left(x,\mathrm{d}y\right)\cdot\mathbf{1}_{A_{\varepsilon}}\left(x\right)\cdot\mathbf{1}_{A_{\varepsilon}^{\complement}}\left(y\right)$
$\displaystyle\geqslant{\rm
var}_{\mu}\left(\mathbf{1}_{A_{\varepsilon}}\right)-s\cdot\int\mu\left(\mathrm{d}x\right)\cdot
P\big{(}x,\left\\{x\right\\}^{\complement}\big{)}\cdot\mathbf{1}_{A_{\varepsilon}}\left(x\right)$
$\displaystyle\geqslant\mu\left(A_{\varepsilon}\right)\cdot\mu\big{(}A_{\varepsilon}^{\complement}\big{)}-s\cdot\mu\left(A_{\varepsilon}\right)\cdot\varepsilon$
$\displaystyle=\mu\left(A_{\varepsilon}\right)\cdot\left(1-s\cdot\varepsilon-\mu\left(A_{\varepsilon}\right)\right).$

Together with Theorem 12 the first statement follows.

For the second statement, note first that there exists
$\bar{\varepsilon}\in\left(0,1\right)$ such that
$\bar{\mu}\left(\bar{\varepsilon}\right):=\sup\left\\{\mu\big{(}A_{\varepsilon}\big{)}:\varepsilon\in\left[0,\bar{\varepsilon}\right]\right\\}<1$.
Indeed, assume $\bar{\mu}\left(\bar{\varepsilon}\right)=1$ for all
$\bar{\varepsilon}\in\left(0,1\right)$. We can thus find a sequence
$\bar{\varepsilon}_{n}\to 0$ such that
$\bar{\mu}\left(\bar{\varepsilon}_{n}\right)=1$ for all $n$, and hence we can
find $\varepsilon_{n}\in\left[0,\bar{\varepsilon}_{n}\right]$ such that
$\mu\big{(}A_{\varepsilon_{n}}\big{)}\geqslant 1-1/n$ for each $n$. By passing
to a subsequence if necessary, we can assume that $\varepsilon_{n}$ is
decreasing in $n$, so that the $A_{\varepsilon_{n}}$ form a decreasing
sequence of sets (with respect to inclusion). We can thus define
$A_{\infty}:=\cap_{n\in\mathbb{N}}A_{\varepsilon_{n}}$ , and by the decreasing
property, it follows that

$\mu\left(A_{\infty}\right)=\lim_{n\rightarrow\infty}\mu\left(A_{\varepsilon_{n}}\right)=1.$

However, we have

$A_{\infty}=\left\\{x\in\mathsf{E}:P\left(x,\left\\{x\right\\}\right)\geqslant
1-\varepsilon_{n},\forall
n\in\mathbb{N}\right\\}=\left\\{x\in\mathsf{E}:P\left(x,\left\\{x\right\\}\right)=1\right\\},$

from which it follows that $P\left(x,\left\\{x\right\\}\right)=1$ $\mu$-a.e.,
leading to a contradiction.

From the first statement, and with
$\left(\bar{\varepsilon},\bar{\mu}\right)\in\left(0,1\right)$ as above, it
then holds for
$s\in\left(\frac{1-\bar{\mu}}{2\cdot\bar{\varepsilon}},\infty\right)$ that

$\displaystyle\beta\left(s;\Psi\right)$
$\displaystyle\geqslant\sup_{\varepsilon\in\left(0,\varepsilon_{0}\right)}\left\\{\mu\big{(}A_{\varepsilon}\big{)}\cdot\left(1-s\cdot\varepsilon-\mu\big{(}A_{\varepsilon}\big{)}\right)\right\\}$
$\displaystyle\geqslant\sup_{\varepsilon\in\left(0,\varepsilon_{0}\right)}\left\\{\mu\big{(}A_{\varepsilon}\big{)}\cdot\left(1-\bar{\mu}-s\cdot\varepsilon\right)\right\\}$
$\displaystyle\geqslant\frac{1-\bar{\mu}}{2}\cdot\mu\big{(}A_{\varepsilon\left(s\right)}\big{)}\,,$

where $\varepsilon\left(s\right):=\frac{1-\bar{\mu}}{2\cdot s}$. ∎

######  Example 20.

Let $P$ be $\mu$-invariant and satisfy a $\left(\Psi,\beta\right)$–WPI. If
$\mu\big{(}A_{\varepsilon}\big{)}\geqslant c\cdot\varepsilon^{a}$ for
$\varepsilon\in\left[0,\bar{\varepsilon}\right]$ for some
$\bar{\varepsilon}>0$, then there exists $C>0$ such that for large enough
$s>0$

$\beta\left(s;\Psi\right)\geqslant C\cdot s^{-a}\,.$

In particular we deduce that
$\beta^{\star}\left(s;\Psi\right)\in\Omega\left(s^{-a}\right)$, and by Lemma
17, we conclude that if $P^{*}P$ satisfies a
$\left(\Psi,\beta_{2}\right)$–WPI, then
$\beta_{2}^{\star}\in\Omega\left(s^{-a}\right)$ also. If $P$ is additionally
reversible, then application of Proposition 18 shows that $P$ cannot be
$\left(\Psi,\gamma\right)$-convergent with $\gamma\left(n\right)\sim n^{-b}$
for any $b>a$.

######  Example 21.

The $\mu$-reversible independent Metropolis–Hastings (IMH) Markov kernel $P$
with proposal probability measure $\nu$ is

$P\left(x,A\right)=\int\nu\left({\rm
d}y\right)\cdot\left\\{1\wedge\frac{w\left(y\right)}{w\left(x\right)}\right\\}{\bf
1}_{A}\left(y\right)+\int\nu\left({\rm
d}y\right)\cdot\left\\{1-1\wedge\frac{w\left(y\right)}{w\left(x\right)}\right\\},\quad
A\in\mathscr{E},$

where $w={\rm d}\mu/{\rm d}\nu$. One can deduce that

$\displaystyle P\left(x,\left\\{x\right\\}\right)$
$\displaystyle\geqslant\int\nu\left({\rm
d}y\right)\cdot\left\\{1-1\wedge\frac{w\left(y\right)}{w\left(x\right)}\right\\}$
$\displaystyle=1-\int\mu\left({\rm
d}y\right)\cdot\left\\{w\left(x\right)^{-1}\wedge
w\left(y\right)^{-1}\right\\}$ $\displaystyle\geqslant
1-w\left(x\right)^{-1}.$

and as a result for $\varepsilon>0$, one sees that

$B_{\varepsilon}:=\left\\{x\in\mathsf{E}:w\left(x\right)^{-1}\leqslant\varepsilon\right\\}\subset
A_{\varepsilon}=\left\\{x\in\mathsf{E}\colon
P\left(x,\left\\{x\right\\}\right)\geqslant 1-\varepsilon\right\\}.$

From Theorem 19 we deduce the existence of $C>0$ such that for any
$s\geqslant\underline{s}$,

$\displaystyle\beta\left(s;\Psi\right)$ $\displaystyle\geqslant
C\cdot\mu\big{(}A_{\varepsilon\left(s\right)}\big{)}\geqslant
C\cdot\mu\big{(}w\geqslant C^{-1}\cdot s\big{)}.$

Upon noting that the IMH operator is positive, this then implies a lower bound
on the fastest rate of convergence possible.

###  2.5 Other types of convergence from
$\big{(}\left\|\cdot\right\|_{\mathrm{osc}}^{2},\gamma\big{)}$ convergence

In practice it can sometime be difficult to establish that a candidate sieve
$\Psi$ found through calculations is indeed a sieve. In contrast, the specific
choices $\Psi=\left\|\cdot\right\|_{\infty}^{2}$ or
$\Psi=\left\|\cdot\right\|_{\mathrm{osc}}^{2}$ often simplify calculations
greatly. This appears at first sight to be at the expense of generality in
terms of the class of functions or metrics for which convergence can be
established. We present here a series of results on the implications of
$\left(\left\|\cdot\right\|_{{\rm osc}}^{2},\gamma\right)$-convergence in
other metrics.

The following, which generalizes the result of [13, Lemma 5.1] (see the
discussion below), shows that $\left(\left\|\cdot\right\|_{{\rm
osc}}^{2},\gamma\right)$-convergence automatically implies convergence in any
Orlicz norm dominating the $\mathrm{L^{2}}$ norm.

######  Definition 22 (Orlicz norm).

For $\mu$ a probability distribution on $\left(\mathsf{E},\mathscr{E}\right)$
and a convex, lower semicontinuous function
$N:\left[0,\infty\right)\to\left[0,\infty\right)$ satisfying
$N\left(0\right)=0$, the $N$-Orlicz norm $\left\|\cdot\right\|_{N}$ (or
Luxemburg norm) is defined as

$\left\|f\right\|_{N}:=\inf\left\\{t>0:\mu\left(N\left(\frac{\left|f\right|}{t}\right)\right)\leqslant
1\right\\},\forall f\colon\mathsf{E}\rightarrow\mathbb{R}.$

Note that whenever $N_{1}\geqslant N_{2}$, then
$\left\|\cdot\right\|_{N_{1}}\geqslant\left\|\cdot\right\|_{N_{2}}$, and that
the choice $N(x)=x^{2}$ leads to the norm $\left\|\cdot\right\|_{2}$. The
following allows one to establish rates of convergence for Orlicz norms for
which $N\left(x\right)$ grows faster than $x^{2}$ in a suitable sense.

######  Proposition 23.

Let $P$ be a $\mu$-invariant Markov kernel, assumed to be
$\left(\left\|\cdot\right\|_{{\rm osc}}^{2},\gamma\right)$-convergent, and let
$N:\left[0,\infty\right)\to\left[0,\infty\right)$ be a convex, lower
semicontinuous function satisfying

$N\left(0\right)=0,\quad x\mapsto x^{-2}\cdot N\left(x\right)\,\text{is
increasing}.$ (7)

Then $N$ defines an Orlicz norm $\left\|\cdot\right\|_{N}^{2}$ and $P$ is also
$\left(\left\|\cdot\right\|_{N}^{2},\gamma_{N}\right)$-convergent, with

$\gamma_{N}\left(n\right)\leqslant 2^{4}\cdot\gamma\left(n\right)\cdot
N^{-1}\left(\gamma\left(n\right)^{-1}\right)^{2},\quad n\in\mathbb{N}.$

###### Proof.

That $N$ defines an Orlicz norm is straightforward. Let
$f\colon\mathsf{E}\rightarrow\mathbb{R}$ satisfying $\left\|f\right\|_{N}=1$,
which implies $\mu\left(N\left(\left|f\right|\right)\right)\leqslant 1$ since
$N$ is non-decreasing. Let $H>0$, and decompose
$f=f\cdot\mathbf{1}\left[\left|f\right|\leqslant
H\right]+f\cdot\mathbf{1}\left[\left|f\right|>H\right]=f_{\downarrow}+f_{\uparrow}$.
Write that

$\displaystyle\left\|P^{n}f-\mu\left(f\right)\right\|_{2}$
$\displaystyle\leqslant\left\|P^{n}\left(f-f_{\downarrow}\right)\right\|_{2}+\left\|P^{n}f_{\downarrow}-\mu\left(f_{\downarrow}\right)\right\|_{2}+\left\|\mu\left(f_{\downarrow}-f\right)\right\|_{2}$
$\displaystyle=\left\|P^{n}f_{\uparrow}\right\|_{2}+\left\|P^{n}f_{\downarrow}-\mu\left(f_{\downarrow}\right)\right\|_{2}+\left|\mu\left(f_{\uparrow}\right)\right|$
$\displaystyle\leqslant\left\|f_{\uparrow}\right\|_{2}+\left\|P^{n}f_{\downarrow}-\mu\left(f_{\downarrow}\right)\right\|_{2}+\left\|f_{\uparrow}\right\|_{2}.$

Since $f_{\downarrow}$ is bounded, $\left(\left\|\cdot\right\|_{{\rm
osc}}^{2},\gamma\right)$-convergence yields that
$\left\|P^{n}f_{\downarrow}-\mu\left(f_{\downarrow}\right)\right\|_{2}\leqslant
2\cdot H\cdot\left(\gamma\left(n\right)\right)^{1/2}$. For the other term,
using (7) and the fact that
$\mu\left(N\left(\left|f\right|\right)\right)\leqslant 1$ to write

$\displaystyle\left\|f_{\uparrow}\right\|_{2}^{2}$
$\displaystyle=\mu\left(f^{2}\cdot\mathbf{1}\left[\left|f\right|>H\right]\right)$
$\displaystyle=\mu\left(\frac{f^{2}}{N\left(\left|f\right|\right)}\cdot
N\left(\left|f\right|\right)\cdot\mathbf{1}\left[\left|f\right|>H\right]\right)$
$\displaystyle\leqslant\frac{H^{2}}{N\left(H\right)}\cdot\mu\left(N\left(\left|f\right|\right)\cdot\mathbf{1}\left[\left|f\right|>H\right]\right)$
$\displaystyle\leqslant\frac{H^{2}}{N\left(H\right)},$

which yields the bound $\left\|P^{n}f-\mu\left(f\right)\right\|_{2}\leqslant
2\cdot H\cdot\gamma\left(n\right)^{1/2}+2\cdot H\cdot N\left(H\right)^{-1/2}$.
Taking $H=N^{-1}\left(\gamma\left(n\right)^{-1}\right)$ gives that
$\left\|P^{n}f-\mu\left(f\right)\right\|_{2}\leqslant
2^{2}\cdot\gamma\left(n\right)^{1/2}\cdot
N^{-1}\left(\gamma\left(n\right)^{-1}\right)$, from which the result follows.
∎

For examples of the above, consider

  * •

$N\left(x\right)=x^{p}$, $p>2$, so that
$\left\|f\right\|_{N}=\left\|f\right\|_{\mathrm{L}^{p}}$ and
$\gamma_{N}\left(n\right)\leqslant 2^{4}\cdot\gamma\left(n\right)^{1-2/p}$.

  * •

$N\left(x\right)=\exp\left(x^{r}\right)-1$, $r\geqslant 1$, so that

$\gamma_{N}\left(n\right)\leqslant
2^{4}\cdot\gamma\left(n\right)\cdot\left(\log\left(1+\gamma\left(n\right)^{-1}\right)\right)^{2/r}\lesssim\gamma\left(n\right)\cdot\left(\log\left(\frac{1}{\gamma\left(n\right)}\right)\right)^{2/r}\,\cdot$

An alternative strategy to handle broader classes of functions consist of
establishing that a $\big{(}\left\|\cdot\right\|_{{\rm
osc}}^{2},\alpha\big{)}$_–_ WPI or $\big{(}\left\|\cdot\right\|_{{\rm
osc}}^{2},\beta\big{)}$_–_ WPI implies a
$\big{(}\left\|\cdot\right\|_{N}^{2},\beta_{N}\big{)}$_–_ WPI for $N$ defining
an Orlicz norm, see [41, Theorem 37], [10, Sections 8-10], and [1, Proposition
35, Theorems 36, 40]. We do not know whether these approaches are comparable
in general, but have observed that in each case, one recovers similar rates in
the polynomial scenario for $\mathrm{L}^{p}$ norms. We note however that we
have found the approach given in [1] more difficult to use in practice.

Since the bound for $N\left(x\right)=x^{2}$ is not decreasing with $n$, the
above result does not provide an $\mathrm{L}^{2}$ convergence rate for all
$\mathrm{L}^{2}$ functions. This should not be surprising as this would
otherwise imply that $P^{k}$ is a strict contraction for some
$k\in\mathbb{N}$, therefore implying geometric convergence. Note however the
result of [15], which can be adapted to reversible Markov chains and
establishes that
$\big{(}\left\|\cdot\right\|_{2}^{2},\rho^{n}\big{)}$-convergence can be
deduced from $\big{(}\left\|\cdot\right\|_{{\rm
osc}}^{2},\rho^{n}\big{)}$-convergence for $\rho\in\left(0,1\right)$.

The following provides a useful equivalence to
$\big{(}\left\|\cdot\right\|_{{\rm osc}}^{2},\gamma\big{)}$-convergence,
particularly relevant in the reversible scenario, and shows that
$\big{(}\left\|\cdot\right\|,\gamma\big{)}$-convergence implies total
variation convergence for a broad class of initial distributions: the latter
improves on [1, Remark 2, Supp. Material].

######  Proposition 24.

Let $P$ be a $\mu$-invariant Markov kernel. Then for all $n\in\mathbb{N}$, it
holds that

$\gamma_{\star}^{1/2}\left(n\right):=\sup_{f:\left\|f\right\|_{{\rm
osc}}\leqslant
1}\left\|P^{n}f-\mu\left(f\right)\right\|_{2}=\frac{1}{2}\sup_{f:\left\|f\right\|_{2}\leqslant
1}\left\|\left(P^{*}\right)^{n}f-\mu\left(f\right)\right\|_{1}.$

Further, for any $\nu\ll\mu$

$\left\|\nu
P^{n}-\mu\right\|_{\mathrm{TV}}\leqslant\gamma_{\star}^{1/2}\left(n\right)\cdot\chi^{2}\left(\nu,\mu\right)^{1/2}.$

Conversely, if for some
$\gamma\colon\mathbb{N}\rightarrow\left[0,\infty\right)$ for all probabilities
$\nu\ll\mu$ such that $\frac{{\rm d}\nu}{{\rm
d}\mu}\in\mathrm{L}^{2}\left(\mu\right)$

$\left\|\nu
P^{n}-\mu\right\|_{\mathrm{TV}}\leqslant\gamma^{1/2}\left(n\right)\cdot\chi^{2}\left(\nu,\mu\right)^{1/2}$

then
$\gamma_{\star}^{1/2}\left(n\right)\leqslant\sqrt{2}\cdot\gamma^{1/2}\left(n\right)$.

######  Remark 25.

The the first statement of Proposition 24 can be adapted to replace the
$\left\|\cdot\right\|_{1}$ norm with a general Orlicz norms, but we do not
detail this here for brevity. This type of result mirrors the theory on
quantitative convergence obtained via drift/minorization techniques [18,
Theorem 17.2.5].

We note briefly that similar duality-type results are also available for
general norm-based sieves. In particular, given an estimate of the form
$\left\|P^{n}f\right\|_{2}^{2}\leqslant\gamma\left(n\right)\cdot\left\|f\right\|_{\circ}^{2}$
for all suitable $f$, an analogous argument establishes that
$\left\|\left(P^{*}\right)^{n}g\right\|_{\bullet}^{2}\leqslant\gamma\left(n\right)\cdot\left\|g\right\|_{2}^{2}$,
whenever
$\left\\{\left\|\cdot\right\|_{\circ},\left\|\cdot\right\|_{\bullet}\right\\}$
are a dual pair of norms defined on $\mathrm{L}^{2}_{0}\left(\mu\right)$, i.e.
$\left|\left\langle
f,g\right\rangle\right|\leqslant\left\|f\right\|_{\circ}\cdot\left\|g\right\|_{\bullet}$
for all suitable $f,g$.

###### Proof of Proposition 24.

Since for $f\in\mathrm{L}^{2}_{0}\left(\mu\right)$,

$\left\|f\right\|_{2}=\sup\left\\{\left\langle
f,g\right\rangle:g\in\mathrm{L}^{2}_{0}\left(\mu\right),\left\|g\right\|_{2}\leqslant
1\right\\},$

we have

$\displaystyle\sup\big{\\{}\left\|P^{n}f\right\|_{2}$
$\displaystyle:\left\|f\right\|_{\mathrm{osc}}\leqslant 1\big{\\}}$
$\displaystyle=\sup$ $\displaystyle\left\\{\left\langle
P^{n}f,g\right\rangle:f\in\mathrm{L}^{2}_{0}\left(\mu\right),\left\|f\right\|_{\mathrm{osc}}\leqslant
1,g\in\mathrm{L}^{2}_{0}\left(\mu\right),\left\|g\right\|_{2}\leqslant
1\right\\}$ $\displaystyle=\sup$ $\displaystyle\left\\{\left\langle
f,\left(P^{*}\right)^{n}g\right\rangle:f\in\mathrm{L}^{2}_{0}\left(\mu\right),\left\|f\right\|_{\mathrm{osc}}\leqslant
1,g\in\mathrm{L}^{2}_{0}\left(\mu\right),\left\|g\right\|_{2}\leqslant
1\right\\}.$

Now, for any $h\in{\rm L}^{1}\bigl{(}\mu\bigr{)}$, $h\neq 0$, satisfying
$\mu\bigl{(}h\bigr{)}=0$,

$\displaystyle\sup\left\\{\left\langle
f,h\right\rangle:\left\|f\right\|_{\mathrm{osc}}\leqslant 1\right\\}$
$\displaystyle=\frac{1}{2}\cdot\sup\left\\{\left\langle
f,h\right\rangle:\left\|f\right\|_{\infty}\leqslant 1\right\\}$
$\displaystyle=\frac{1}{2}\cdot\mu\left(\left|h\right|\right).$

The second equality follows from the fact that for $\|f\|_{\infty}<\infty$,
$\left\langle
f,h\right\rangle\leqslant\left\langle\left|f\right|,\left|h\right|\right\rangle=\left\langle\left|f\right|\cdot\mathrm{sgn}\left(h\right),h\right\rangle$,
that is, equality holds for $f=g\cdot\mathrm{sgn}\left(h\right)$ with
$0\leqslant g\leqslant 1$ and the supremum is reached for $g=1$. Similarly,
for the first equality, for $\left\|f\right\|_{\mathrm{osc}}\leqslant 1$ the
condition

$\mu\left(h\right)=0\iff\mu\left(h\cdot\mathbf{1}\left[h>0\right]\right)=-\mu\left(h\cdot\mathbf{1}\left[h<0\right]\right)=\frac{1}{2}\mu\left(\left|h\right|\right)$

implies that the supremum is reached, for example, for $g=1/2$. The case $h=0$
is straightforward. Consequently

$\displaystyle\sup\left\\{\left\|P^{n}f\right\|_{2}:f\in\mathrm{L}^{2}_{0}\left(\mu\right),\left\|f\right\|_{\mathrm{osc}}\leqslant
1\right\\}$ $\displaystyle=\frac{1}{2}\cdot\sup$
$\displaystyle\left\\{\mu\left(\left|\left(P^{*}\right)^{n}g\right|\right):g\in\mathrm{L}^{2}_{0}\left(\mu\right),\left\|g\right\|_{2}\leqslant
1\right\\},$

and the first statement follows. For any
$g\in\mathrm{L}^{2}_{0}\left(\mu\right)$ and $n\in\mathbb{N}$, we have

$\frac{1}{2}\cdot\mu\left(\left|\left(P^{*}\right)^{n}g\right|\right)\leqslant\gamma_{\star}^{1/2}\left(n\right)\cdot\left\|g\right\|_{2}\,,$

and for $\nu$ a probability measure such that $\nu\ll\mu$ and
$g:=\frac{\mathrm{d}\nu}{\mathrm{d}\mu}-1\in\mathrm{L}^{2}_{0}\left(\mu\right)$,
we obtain that

$\displaystyle\frac{1}{2}\cdot\mu\left(\left|\left(P^{*}\right)^{n}\left(\frac{\mathrm{d}\nu}{\mathrm{d}\mu}-1\right)\right|\right)$
$\displaystyle=\frac{1}{2}\cdot\mu\left(\left|\frac{\mathrm{d}\left(\nu
P^{n}\right)}{\mathrm{d}\mu}-1\right|\right)$ $\displaystyle=\left\|\nu
P^{n}-\mu\right\|_{\mathrm{TV}}\,,$

and the second statement follows. Assume now that for any $n\in\mathbb{N}$ and
probability $\nu$ obtained as $\nu=g\cdot\mu/\mu\left(g\right)$ for
$g\in\mathrm{L}^{2}\left(\mu\right)$ non-negative, it holds that

$\left\|\nu
P^{n}-\mu\right\|_{\mathrm{TV}}\leqslant\gamma^{1/2}\left(n\right)\cdot\chi^{2}\left(\nu,\mu\right)^{1/2}.$

Noting that ${\rm d}\nu/{\rm d}\mu=g/\mu\left(g\right)$, we can conversely
deduce that

$\frac{1}{2}\cdot\mu\left(\left|\left(P^{*}\right)^{n}g-\mu\left(g\right)\right|\right)\leqslant\gamma^{1/2}\left(n\right)\cdot\left\|g-\mu\left(g\right)\right\|_{2}.$

For $g\in\mathrm{L}^{2}\left(\mu\right)$, taking the decomposition into
positive and negative parts $g=g_{+}-g_{-}$ shows that

$\displaystyle\qquad\frac{1}{4}\cdot\mu\left(\left|\left(P^{*}\right)^{n}\left(g-\mu\left(g\right)\right)\right|\right)^{2}$
$\displaystyle\leqslant\frac{1}{4}\cdot\mu\left(\left|\left(P^{*}\right)^{n}\left(g_{+}-\mu\left(g_{+}\right)\right)\right|+\left|\left(P^{*}\right)^{n}\left(g_{-}-\mu\left(g_{-}\right)\right)\right|\right)^{2}$
$\displaystyle\leqslant\frac{1}{2}\cdot\mu\left(\left|\left(P^{*}\right)^{n}\left(g_{+}-\mu\left(g_{+}\right)\right)\right|\right)^{2}+\frac{1}{2}\cdot\mu\left(\left|\left(P^{*}\right)^{n}\left(g_{-}-\mu\left(g_{-}\right)\right)\right|\right)^{2}$
$\displaystyle\leqslant
2\cdot\gamma\left(n\right)\cdot\left\\{\left\|g_{+}-\mu\left(g_{+}\right)\right\|_{2}^{2}+\left\|g_{-}-\mu\left(g_{-}\right)\right\|_{2}^{2}\right\\}$
$\displaystyle=2\cdot\gamma\left(n\right)\cdot\left\|g-\mu\left(g\right)\right\|_{2}^{2},$

from which we may conclude the last part. ∎

The first part of Proposition 24 seems relevant in the reversible scenario
only. However, as mentioned in [38], the following non quantitative result
holds even without reversibility.

######  Proposition 26.

The following are equivalent:

$\lim_{n\to\infty}\sup_{f:\left\|f\right\|_{2}\leqslant
1}\left\|P^{n}f-\mu\left(f\right)\right\|_{1}=0,$ (8)

and

$\lim_{n\to\infty}\sup_{f:\left\|f\right\|_{\infty}\leqslant
1}\left\|P^{n}f-\mu\left(f\right)\right\|_{2}=0.$ (9)

###### Proof.

We start with (8)$\Rightarrow$ (9). So consider $f$ with
$\left\|f\right\|_{\infty}\leqslant 1$.

$\displaystyle\left\|P^{n}f-\mu\left(f\right)\right\|_{2}^{2}$
$\displaystyle=\int\left|P^{n}f-\mu\left(f\right)\right|\cdot\left|P^{n}f-\mu\left(f\right)\right|\,\mathrm{d}\mu$
$\displaystyle\leqslant
2\cdot\int\left|P^{n}f-\mu\left(f\right)\right|\,\mathrm{d}\mu,$

and this final expression converges uniformly over $f$ to 0 by (8), since
$\left\\{f:\left\|f\right\|_{\infty}\leqslant
1\right\\}\subseteq\left\\{f:\mu\left(f^{2}\right)\leqslant 1\right\\}$. We
now consider the converse, (9)$\Rightarrow$ (8). Without loss of generality,
we may consider $f\in\mathcal{F}:=\left\\{g\in{\rm
L}_{0}^{2}\left(\mu\right):\left\|g\right\|_{2}\leqslant 1\right\\}$. Let
$\epsilon>0$ be arbitrary; we will show that for $n$ large enough,
$\sup_{f:\left\|f\right\|_{2}\leq
1}\int\left|P^{n}f\right|\,\mathrm{d}\mu\leqslant\epsilon$. Take
$K=4/\epsilon$ and $N$ large enough such that

$\sup_{g:\left\|g\right\|_{\infty}\leqslant
K}\left\|P^{N}g-\mu\left(g\right)\right\|_{2}\leqslant\frac{\epsilon}{2},$

which is valid due to (9). Decomposing an arbitrary $f\in\mathcal{F}$ as
$f=f\cdot{\bf 1}_{A}+f\cdot{\bf 1}_{A^{\complement}}$ for $A\in\mathscr{E}$,
we have

$\int\left|P^{N}f\right|\,\mathrm{d}\mu\leqslant\int\left|P^{N}\left(f\cdot{\bf
1}_{A}\right)\right|\,\mathrm{d}\mu+\int\left|P^{N}\left(f\cdot{\bf
1}_{A^{\complement}}\right)\right|\,\mathrm{d}\mu,$

by Minkowski’s inequality. Now by Jensen’s inequality, $\mu$-invariance of
$P^{N}$, and the Cauchy–Schwarz inequality, we see that

$\int\left|P^{N}\left(f\cdot{\bf
1}_{A^{\complement}}\right)\right|\,\mathrm{d}\mu\leqslant\int\left|f\cdot{\bf
1}_{A^{\complement}}\right|\,\mathrm{d}\mu\leqslant\left\|f\right\|_{2}\cdot\mu\left(A^{\complement}\right)^{1/2}\leqslant\mu\left(A^{\complement}\right)^{1/2}.$

Taking $A=\left\\{x\in\mathsf{E}:\left|f\left(x\right)\right|\leqslant
K\right\\}$, we obtain by Markov’s inequality that

$\mu\left(A^{\complement}\right)=\mu\left({\bf
1}_{\left|f\right|^{2}>K^{2}}\right)\leqslant K^{-2}.$

From

$\left|\mu\left(f\cdot{\bf
1}_{A^{\complement}}\right)\right|\leqslant\mu\left(\left|f\cdot{\bf
1}_{A^{\complement}}\right|\right)\leqslant K^{-1},$

and $\mu\left(f\right)=0$, we also obtain that $\left|\mu\left(f\cdot{\bf
1}_{A}\right)\right|\leqslant K^{-1}$. Finally, we deduce that

$\displaystyle\int\left|P^{N}f\right|\,\mathrm{d}\mu$
$\displaystyle\leqslant\int\left|P^{N}\left(f\cdot{\bf
1}_{A}\right)\right|\,\mathrm{d}\mu+\int\left|P^{N}\left(f\cdot{\bf
1}_{A^{\complement}}\right)\right|\,\mathrm{d}\mu.$
$\displaystyle\leqslant\int\left|P^{N}\left(f\cdot{\bf
1}_{A}\right)-\mu\left(f\cdot{\bf
1}_{A}\right)\right|\,\mathrm{d}\mu+\left|\mu\left(f\cdot{\bf
1}_{A}\right)\right|+K^{-1}$ $\displaystyle\leqslant\frac{\epsilon}{2}+2\cdot
K^{-1}$ $\displaystyle\leqslant\epsilon.$

Since $f\in\mathcal{F}$ was arbitrary, the result follows. ∎

###  2.6 ${\rm L}^{2}$ and ${\rm L}^{1}$ subgeometric convergence rates can be
different and CLT

Proposition 24 yields that given a bound on the decay of the total variation
distance of the form $\left\|\nu
P^{n}-\mu\right\|_{\mathrm{TV}}\leqslant\gamma^{1/2}\left(n\right)\cdot\chi^{2}\left(\nu,\mu\right)^{1/2}$,
one can deduce an $\mathrm{L}^{2}$ bound on the convergence of bounded
functions as
$\left\|P^{n}f-\mu\left(f\right)\right\|_{2}\leqslant\sqrt{2}\cdot\gamma^{1/2}\left(n\right)\cdot\left\|f\right\|_{\mathrm{osc}}$.
Note that this transition can be lossy, and that in general, TV and
$\mathrm{L}^{2}$ rates of convergence need not match in the subgeometric
setting. Indeed, in Example 27, we will exhibit a Markov chain whose
(subgeometric) rate of convergence in TV is a strict improvement on what is
implied by its $\mathrm{L}^{2}$ convergence rate. This may be contrasted with
the convergence rates of reversible, geometrically ergodic Markov chains,
which are always the same by [37, Theorem 3].

A secondary point is that standard techniques to derive TV convergence bounds
generally do not lead to tight and convenient estimates of the type given
above. Indeed, consider the scenario where $P$ is
$\mu\mathrm{\shortminussymb}$irreducible and aperiodic, therefore implying
that for $\mu\mathrm{\shortminussymb}$almost all $x\in\mathsf{E}$,
$\lim_{n\rightarrow\infty}\left\|\delta_{x}P^{n}-\mu\right\|_{\mathrm{TV}}=0$.
Recalling that for a signed measure $\nu$ with $\nu\left(1\right)=0$, we have
$\left\|\nu\right\|_{\mathrm{TV}}=\sup\left\\{\nu\left(f\right):\left\|f\right\|_{\mathrm{osc}}\leqslant
1\right\\}$, we see that

$\displaystyle\sup\left\\{\left(P^{n}f\right)\left(x\right)-\mu\left(f\right):\left\|f\right\|_{\mathrm{osc}}\leqslant
1\right\\}$ $\displaystyle=\left\|\delta_{x}P^{n}-\mu\right\|_{\mathrm{TV}}$
$\displaystyle\implies\left\|P^{n}f-\mu\left(f\right)\right\|_{2}$
$\displaystyle\leqslant\mu\left(\left\|\delta_{x}P^{n}-\mu\right\|_{\mathrm{TV}}^{2}\right)^{1/2}\cdot\left\|f\right\|_{\mathrm{osc}}.$

Applying the dominated convergence theorem yields that
$\mu\big{(}\left\|\delta_{x}P^{n}-\mu\right\|_{\mathrm{TV}}^{2}\big{)}\to 0$,
and so we obtain that $\left\|P^{n}f-\mu\left(f\right)\right\|_{2}\to 0$ for
all bounded functions. However, converting this estimate into an explicit
quantitative control on the $\mathrm{L}^{2}$ rate of convergence can prove
challenging in practice. In particular, since TV convergence bounds are often
in the form
$\left\|\delta_{x}P^{n}-\mu\right\|_{\mathrm{TV}}\leqslant\min\left\\{1,V\left(x\right)\cdot\gamma\left(n\right)\right\\}$
for choices of $V$, $\gamma$ resulting from a tradeoff between rate of
convergence and dependence on the initial condition. As a result, capturing
fast decay of
$n\mapsto\mu\big{(}\left\|\delta_{x}P^{n}-\mu\right\|_{\mathrm{TV}}^{2}\big{)}$
is not particularly simple in general. For example, under the assumption that
$\mu\left(V^{2}\right)<\infty$, one can obtain that
$\left\|P^{n}f-\mu\left(f\right)\right\|_{2}\leqslant\frac{1}{2}\cdot\left\|V\right\|_{2}\cdot\gamma\left(n\right)\cdot\left\|f\right\|_{\mathrm{osc}}$,
but experience shows that enforcing this integrability assumption requires
sacrificing rates of convergence, in particular in the subgeometric scenario.

######  Example 27.

Let $\mathsf{E}=\left[1,\infty\right)$, $a>1$,

$P\left(x,\cdot\right)=w\left(x\right)\cdot\nu\left(\cdot\right)+\left\\{1-w\left(x\right)\right\\}\cdot\delta_{x}\left(\cdot\right),$

where $\nu$ is a probability measure with density

$\nu\left(x\right)=\frac{a-1}{x^{a}},\qquad x\in\mathsf{E},$

and $w:\mathsf{E}\to\left(0,1\right]$ given by $w\left(x\right)=x^{-b}$ with
$b\in\left(0,a-1\right)$. Then $P$ is $\mu$-reversible, where the probability
measure $\mu$ satisfies

$\mu\left({\rm d}x\right)=\frac{\nu\left({\rm d}x\right)\cdot
w\left(x\right)^{-1}}{\nu\left(w^{-1}\right)},$

and therefore has density

$\mu\left(x\right)=\frac{a-b-1}{x^{a-b}},\qquad x\in\mathsf{E}.$

The following proposition establishes that for the Markov chain in Example 27,
${\rm L}^{2}$ and ${\rm L}^{1}$ subgeometric convergence rates are different.

######  Proposition 28.

In Example 27, $P$ satisfies:

  1. a).

$n^{s}\cdot\left\|P^{n}\left(x,\cdot\right)-\mu\right\|_{{\rm TV}}\to 0$ for
any $s<\frac{a-b-1}{b}$.

  2. b).

$n^{s}\cdot\sup_{f\in\mathrm{L}^{2}_{0}(\mu):\left\|f\right\|_{{\rm
osc}}=1}\left\|P^{n}f\right\|_{2}<\infty$ if and only if
$s\leqslant\frac{a-b-1}{2b}$.

###### Proof.

This follows from Lemmas 30, 32 and 33. ∎

######  Remark 29.

We end this section with some remarks concerning the existence of a central
limit theorem for trajectories of Markov chains. For
$f\in\mathrm{L}_{0}^{2}\left(\mu\right)$ the standard Maxwell–Woodroofe
condition [27],

$\sum_{n=1}^{\infty}n^{-3/2}\left\|{\textstyle\sum_{k=0}^{n-1}}P^{k}f\right\|_{2}<\infty\,,$
(10)

is sufficient to ensure
$n^{-1/2}\sum_{i=0}^{n-1}f\left(X_{i}\right)\overset{L}{\to}\mathcal{N}\left(0,\sigma^{2}\right)$
for some $\sigma^{2}>0$. $\left(\Psi,\gamma\right)$\- convergence can
therefore provide a direct route to checking this condition in practice.

For example, suppose that $P$ is
$\big{(}\left\|\cdot\right\|_{\mathrm{osc}}^{2},\gamma\big{)}$-convergent.
Then, by combining (10), Proposition 23, and a summation-by-parts, we can
obtain that a central limit theorem will hold for
$f\in\mathrm{L}^{2}_{0}\left(\mu\right)$ with $\left\|f\right\|_{N}<\infty$
provided that

$\sum_{k=1}^{\infty}k^{-1/2}\cdot\gamma_{N}(k)^{1/2}<\infty,$ (11)

where $\gamma_{N}(k)\lesssim\gamma(k)\cdot
N^{-1}\big{(}\gamma(k)^{-1}\big{)}^{2}$.

We offer now a brief comparison with [18, Theorem 21.4.9], a parallel result
inspired by drift and minorization techniques. The starting point of [18] is a
pair of Young functions $\left(\phi,\psi\right)$: $\phi,\psi$ are continuous,
strictly increasing with
$\phi^{-1}\left(x\right)\cdot\psi^{-1}\left(y\right)\leqslant x+y$, and it is
assumed that convergence to equilibrium in total variation distance occurs at
rate $1/\phi$, in the sense that for some constant $C>0$, one can bound
$\int\mu(\mathrm{d}x)\cdot\left\|P^{n}\left(x,\cdot\right)-\mu\right\|_{{\rm
TV}}\leqslant C/\phi\left(n\right)$. The conclusion of [18, Theorem 21.4.9] is
that a central limit theorem holds for functions $f$ satisfying
$\left\|f\right\|_{\tilde{N}}<\infty$ with $\tilde{N}(x)=\psi(x^{2})$.

In our setting, assuming $\left(\Psi,\gamma\right)$\- convergence, application
of Theorem 24 would only yield estimates of the form $\left\|\nu
P^{n}-\mu\right\|_{{\rm
TV}}^{2}\lesssim\phi(k)^{-1}\cdot\chi^{2}\left(\nu,\mu\right)$ and direct
application of [18, Theorem 21.4.9] seems to require additional work. However
one can directly work with condition (11). Setting $\phi(k)=\gamma(k)^{-1}$
and $N(x)=\psi(x^{2})$ a necessary condition for the summability of the series
concerned is that the summands decay faster than $k^{-1}$:

$k^{-1/2}\cdot\phi\left(k\right)^{-1/2}\cdot\psi^{-1}\left(\phi\left(k\right)\right)^{-1/2}\ll
k^{-1}.$

In the polynomial case with rate $\gamma(k)\lesssim k^{-a}$ with $a>1$ and
with $N(x)=x^{p}$, this condition requires $p>\frac{2a}{a-1}=2+\frac{2}{a-1}$,
which precisely matches the threshold obtained in [18, Corollary 21.4.5]. More
generally, this condition is equivalent to

$\phi^{-1}\left(z\right)\cdot\psi^{-1}\left(z\right)\ll z,$

which closely mirrors the Young function condition of [18, Theorem 21.4.9]. We
leave further investigations of this relation to future work.

######  Lemma 30.

In Example 27, $P$ satisfies

$n^{s}\cdot\left\|P^{n}\left(x,\cdot\right)-\mu\right\|_{{\rm TV}}\to 0,$

for any $s<\frac{a-b-1}{b}$.

###### Proof.

We first identify a drift function $V$ such that

$PV\leqslant V-c\cdot V^{\alpha}+b\cdot{\bf 1}_{C},$

with $C$ a small set. Let $k\in\left(0,a-1\right)$, $\alpha=1-b/k$ and define
$V\left(x\right)=x^{k}$. We observe that

$\nu\left(V\right)=\frac{a-1}{a-k-1}<\infty,$

since $k<a-1$. Take $C=\left[1,x_{0}\right]$, where

$x_{0}=\left\\{2\cdot\frac{a-1}{a-k-1}\right\\}^{\frac{1}{\alpha\cdot k}}.$

It follows that for $x\in\mathsf{E}$,

$\displaystyle PV\left(x\right)$
$\displaystyle=V\left(x\right)-w\left(x\right)\cdot
V\left(x\right)+w\left(x\right)\cdot\nu\left(V\right)$ $\displaystyle\leqslant
V\left(x\right)-w\left(x\right)\cdot V\left(x\right)+\nu\left(V\right)$
$\displaystyle=V\left(x\right)-x^{k-b}+\nu\left(V\right)$
$\displaystyle=V\left(x\right)-V\left(x\right)^{\alpha}+\nu\left(V\right)$

The choice of $x_{0}$ implies that for $x\in C^{\complement}$, it holds that

$V\left(x\right)^{\alpha}\geqslant x_{0}^{k\alpha}=2\cdot\nu\left(V\right),$

and hence, we may verify that

$PV\left(x\right)\leqslant
V\left(x\right)-\frac{1}{2}V\left(x\right)^{\alpha}+\nu\left(V\right)\cdot{\bf
1}_{C}\left(x\right).$

Since $P\left(x,\cdot\right)\geqslant
w\left(x_{0}\right)\cdot\nu\left(\cdot\right)$ on $C$, we verify that $C$ is
small. By [21, Theorem 3.6], we may conclude that the rate of convergence in
total variation is given by

$\frac{\alpha}{1-\alpha}=\frac{k-b}{b},$

which can be made arbitrarily close to $\left(a-b-1\right)/b$. ∎

######  Lemma 31.

Let $\nu\ll\mu$ be probability measures and let

$w=\frac{{\rm d}\mu}{{\rm d}\nu},$

with the abuse of notation that we allow $w=\infty$ on regions $A$ where
$\mu\left(A\right)>0$ and $\nu\left(A\right)=0$. Then for any bounded function
$f$, it holds that

${\rm var}_{\mu}\left(f\right)\leqslant s\cdot{\rm
var}_{\nu}\left(f\right)+\mu\left(w>s\right)\cdot\left\|f\right\|_{{\rm
osc}}^{2}.$

###### Proof.

Observe that for any $c\in\mathbb{R}$,

${\rm
var}_{\mu}\left(f\right)=\inf_{t}\mu\left(\left(f-t\right)^{2}\right)\leqslant\mu\left(\left(f-c\right)^{2}\right).$

Take $c=\nu\left(f\right)$ and let $s>0$. Define
$A\left(s\right)=\left\\{x:w\left(x\right)\leqslant s\right\\}$. It follows
that

$\displaystyle\mu\left(\left(f-c\right)^{2}\right)$
$\displaystyle=\mu\left({\bf
1}_{A\left(s\right)}\cdot\left(f-c\right)^{2}\right)+\mu\left({\bf
1}_{A\left(s\right)^{\complement}}\cdot\left(f-c\right)^{2}\right)$
$\displaystyle\leqslant s\cdot\nu\left({\bf
1}_{A\left(s\right)}\cdot\left(f-c\right)^{2}\right)+\mu\left(A\left(s\right)^{\complement}\right)\cdot\left\|f\right\|_{{\rm
osc}}^{2}$ $\displaystyle\leqslant
s\cdot\nu\left(\left(f-c\right)^{2}\right)+\mu\left(w>s\right)\cdot\left\|f\right\|_{{\rm
osc}}^{2}$
$\displaystyle=s\cdot\mathrm{var}_{\nu}\left(f\right)+\mu\left(w>s\right)\cdot\left\|f\right\|_{{\rm
osc}}^{2}$

where we have used the fact that since $c=\nu\left(f\right)$ and $\nu\ll\mu$,

$\mathrm{ess_{\mu}}\inf f\leqslant\mathrm{ess_{\nu}}\inf f\leqslant
c\leqslant\mathrm{ess_{\nu}}\sup f\leqslant\mathrm{ess_{\mu}}\sup f.$

∎

######  Lemma 32.

In Example 27, $P$ satisfies

$\sup_{f\in\mathrm{L}^{2}_{0}\left(\pi\right):\left\|f\right\|_{{\rm
osc}}=1}\left\|P^{n}f\right\|_{2}^{2}\leqslant C\cdot n^{-\frac{a-b-1}{b}},$

for some $C>0$.

###### Proof.

First observe that for $f\in\mathrm{L}^{2}_{0}(\mu)$,

$\displaystyle\left\langle Pf,f\right\rangle$ $\displaystyle=\int\mu\left({\rm
d}x\right)\cdot P\left(x,\mathrm{d}y\right)\cdot f\left(x\right)\cdot
f\left(y\right)$
$\displaystyle\geqslant\nu\left(w^{-1}\right)^{-1}\cdot\int\nu\left(\mathrm{d}x\right)\cdot
w\left(x\right)^{-1}\cdot w\left(x\right)\cdot\nu\left(\mathrm{d}y\right)\cdot
f\left(x\right)\cdot f\left(y\right)$
$\displaystyle=\nu\left(w^{-1}\right)^{-1}\cdot\nu\left(f\right)^{2}$
$\displaystyle\geqslant 0$

and so $P$ is positive and hence
$\mathcal{E}\left(P^{2},f\right)\geqslant\mathcal{E}\left(P,f\right)$.
Similarly, we find that

$\displaystyle\mathcal{E}\left(P,f\right)$
$\displaystyle=\frac{1}{2}\cdot\int\mu\left({\rm d}x\right)\cdot
P\left(x,\mathrm{d}y\right)\cdot\left(f\left(y\right)-f\left(x\right)\right)^{2}$
$\displaystyle=\nu\left(w^{-1}\right)^{-1}\cdot\frac{1}{2}\cdot\int\int\nu\left(\mathrm{d}x\right)\cdot\nu\left(\mathrm{d}y\right)\cdot\left(f\left(y\right)-f\left(x\right)\right)^{2}$
$\displaystyle=\nu\left(w^{-1}\right)^{-1}\cdot{\rm var}_{\nu}\left(f\right).$

Hence, we may obtain a WPI for $P$ by chaining [1, Section 3]. Applying Lemma
31 and taking $s=t\cdot\nu\left(w^{-1}\right)$, we have

$\displaystyle{\rm var}_{\mu}\left(f\right)$ $\displaystyle\leqslant
t\cdot{\rm var}_{\nu}\left(f\right)+\mu\left(\frac{{\rm d}\mu}{{\rm
d}\nu}>t\right)\cdot\left\|f\right\|_{{\rm osc}}^{2}$
$\displaystyle=s\cdot\mathcal{E}\left(P,f\right)+\mu\left(\frac{{\rm
d}\mu}{{\rm
d}\nu}>s\cdot\nu\left(w^{-1}\right)^{-1}\right)\cdot\left\|f\right\|_{{\rm
osc}}^{2}$
$\displaystyle=s\cdot\mathcal{E}\left(P,f\right)+\mathbb{P}_{\mu}\left(X^{b}>s\right)\cdot\left\|f\right\|_{{\rm
osc}}^{2},$

so we obtain that $P$, and hence $P^{2}$ by positivity, satisfies an
$\left\|\cdot\right\|_{{\rm osc}}^{2}$_–_ WPI with

$\beta\left(s\right)=\mathbb{P}_{\mu}\left(X^{b}>s\right)=s^{-\frac{a-b-1}{b}}.$

We conclude by combining Theorem 8 and Example 9. ∎

######  Lemma 33.

In Example 27, if $q>\frac{a-b-1}{b}$, then there exists
$f\in\mathrm{L}^{2}_{0}\left(\mu\right)$ with $\left\|f\right\|_{{\rm osc}}=1$
such that

$\left\|P^{n}f\right\|_{2}^{2}\not\in\mathcal{O}\left(n^{-q}\right).$

###### Proof.

Let

$\displaystyle A_{\varepsilon}$
$\displaystyle=\left\\{x\in\mathsf{E}:P\left(x,\left\\{x\right\\}\right)\geqslant
1-\varepsilon\right\\}$
$\displaystyle=\left\\{x\in\mathsf{E}:w\left(x\right)\leqslant\varepsilon\right\\}$
$\displaystyle=\left[t_{\varepsilon},\infty\right)$

where $t_{\varepsilon}=\varepsilon^{-1/b}$. Direct calculation gives that

$\mu\left(A_{\varepsilon}\right)=\frac{1}{t_{\varepsilon}^{a-b-1}}=\varepsilon^{\frac{a-b-1}{b}}.$

The result then follows from Example 20. ∎

##  3 Existence of WPIs

This section is concerned with the existence and practical establishment of
WPIs, and subsequent mixing time bounds, using arguments of a probabilistic
nature. More specifically, we extend in Section 3.1 the notion of Cheeger
inequalities [24] to Markov chains which do not admit a right spectral gap.
Namely, we show that establishing certain properties of Dirichlet forms on the
subset of indicator functions $f=\mathbf{1}_{A}$ (with $A\in\mathscr{E}$ and
$\mu(A)\leqslant 1/2$) is equivalent to a WPI. This is then employed in two
separate settings. Then in Section 3.2, we establish necessary and sufficient
conditions for the qualitative existence of WPIs for a Markov kernel in terms
of its probabilistic pathwise properties, namely irreducibility and
aperiodicity. In Section 3.3 we deduce quantitative bounds on WPIs and mixing
times from isoperimetric properties of the invariant distribution of the
Markov chain and a close coupling condition. These latter results will form
the theoretical backbone of our later application of studying the convergence
of Random Walk Metropolis on heavy-tailed targets in high dimensions; see
Section 5.

###  3.1 Weak Cheeger inequalities

We first introduce a generalisation of the notion of conductance of a Markov
kernel [24].

######  Definition 34 (Weak conductance profile).

The weak conductance profile (WCP) of a $\mu$-invariant Markov kernel $P$ is

$\Phi_{P}^{\mathrm{W}}\left(v\right):=\inf\left\\{\frac{\left(\mu\otimes
P\right)\left(A\times
A^{\complement}\right)}{\mu\left(A\right)}:v\leqslant\mu\left(A\right)\leqslant\frac{1}{2}\right\\},\qquad
v\in\left(0,\frac{1}{2}\right].$

We say that $P$ has a positive WCP if $\Phi_{P}^{\mathrm{W}}\left(v\right)>0$
for all $v\in\left(0,\frac{1}{2}\right]$.

The classical notion of conductance corresponds to $\lim_{v\rightarrow
0}\Phi_{P}^{\mathrm{W}}\left(v\right)$, and Lemma 102 outlines the link
between this quantity and Dirichlet forms, while the probabilistic
interpretation opens the door to practical uses. Note the distinction with the
‘strong’ conductance profile (defined in, say, Definition 3 of [2]), wherein
the infimum is taken over sufficiently small sets $A$, rather than
sufficiently large sets. The main results of this section are the following;
the proofs are at the end of the section.

######  Theorem 35 (Weak Cheeger inequalities).

Let $P$ be a $\mu$-invariant Markov kernel.

  1. a).

If $P$ has a positive WCP $\Phi_{P}^{\mathrm{W}}$, then $P$ satisfies a
$K^{*}$–WPI, for some
$K^{*}\colon\left[0,\frac{1}{4}\right]\rightarrow\left[0,\infty\right)$
satisfying for any $v\in\left(0,\frac{1}{4}\right]$,

$K^{*}\left(v\right)\geqslant 2^{-2}\cdot
v\cdot\Phi_{P}^{\mathrm{W}}\left(2^{-2}\cdot v\right)^{2}.$

  2. b).

If $P$ satisfies a $K^{*}$–WPI then $P$ has a positive WCP with conductance
profile $\Phi_{P}^{\mathrm{W}}$ satisfying for any
$v\in\left(0,\frac{1}{4}\right]$,

$\frac{1}{2}\cdot
v\cdot\Phi_{P}^{\mathrm{W}}\left(\frac{1-\sqrt{1-4v}}{2}\right)\geqslant
K^{*}\big{(}v\big{)}\,.$

######  Corollary 36.

Let $P$ be a $\mu$-reversible positive Markov kernel with a positive WCP. In
Theorem 8 we have for $x\in\left(0,\frac{1}{4}\right]$,

$2\cdot\int_{x}^{1/4}\frac{{\rm
d}v}{v\cdot\Phi_{P}^{\mathrm{W}}\left(\frac{1-\sqrt{1-4v}}{2}\right)}\leqslant
F\left(x\right)\leqslant 2^{2}\cdot\int_{x}^{1/4}\frac{{\rm
d}v}{v\cdot\Phi_{P}^{\mathrm{W}}\left(2^{-2}\cdot v\right)^{2}}$

from which upper and lower bounds on the rate of convergence
$\gamma\left(n\right)=F^{-1}\left(n\right)$ can be obtained. Further for any
$\epsilon>0$, it holds that

$\sup_{f\in\mathrm{L}^{2}_{0}(\mu)\setminus\\{0\\}}\left\|P^{n}f\right\|_{2}^{2}/\left\|f\right\|_{\mathrm{osc}}^{2}\leqslant\epsilon,$

whenever $n\geqslant
2^{2}\cdot\int_{2^{-2}\epsilon}^{2^{-4}}\frac{\mathrm{d}v}{v\cdot\Phi_{P}^{\mathrm{W}}\left(v\right)^{2}}$.

An expression for $K^{*}$ in the first statement of the theorem is obtained in
the proof, but may be of theoretical interest only. In an earlier technical
report we also established weak Cheeger inequalities [3, Theorem 38], but by
considering $\alpha$-WPIs instead of $K^{*}$-WPIs, therefore requiring
additional effort to lower bound $K^{*}$ in terms of $\Phi_{P}^{\mathrm{W}}$
when interested in mixing times or convergence rates. In addition to providing
a more direct route to applications, the results are sharper (see Appendix C).
We also point out that reversibility is not a requirement in Theorem 35 and
that in fact it is not either in [3, Theorem 38].

The proof of Theorem 35 differs from that of [3, Theorem 38] and relies on
“$\mathrm{L}^{1}\left(\mu\right)$– type” functional inequalities, which we
refer to as $\mathrm{L}^{1}$–WPI inequalities; [9] refers to these as Weak
Cheeger inequalities but we here prefer to reserve the term for the
inequalities in Theorem 35, by analogy with the use of “Cheeger inequalities”
in the (discrete-time) Markov chain literature. The proofs of Theorem 35 and
Corollary 36 can be found at the end of the section and rely on two
preparatory lemmas, Lemma 40 and Lemma 42, which link WCP,
$\mathrm{L}^{1}$–WPI and WPIs.

While various intermediate technical details of our proof are adapted from [9]
to the setting of discrete-time Markov chains, the overall objectives of the
two contributions are fairly distinct. In particular, [9] specifically focuses
on studying isoperimetric inequalities for probability measures on
$\mathbb{R}^{d}$ and their functional forms, with the connection to the
convergence of Markov semigroups being of secondary interest. By contrast, we
are primarily interested in the application of isoperimetric concepts to
convergence analysis of Markov chains, and so focus more attention on these
aspects of the problem. We also note that related functional inequalities of
the form developed here, and their connections to WPIs, have been discussed in
earlier work; for instance Sections 4 and 5 of [38] consider the links between
isoperimetry and WPIs in the specific scenarios of diffusion processes and
continuous-time Markov jump processes.

Recall that for $f\in\mathrm{L}^{1}\left(\mu\right)$, we defined
$\mathrm{var}_{\mu}^{\left(1\right)}\left(f\right):=\inf_{c\in\mathbb{R}}\left\|f-c\right\|_{1}$
and
$\mathcal{E}_{1}\left(P,f\right):=\frac{1}{2}\cdot\int\mu\left(\mathrm{d}x\right)\cdot
P\left(x,\mathrm{d}y\right)\cdot\left|f\left(y\right)-f\left(x\right)\right|$.

######  Definition 37 ($\mathrm{L}^{1}$–WPI).

We say that a $\mu$-invariant kernel $T$ satisfies a $\mathrm{L}^{1}$–weak
Poincaré inequality, abbreviated $\mathrm{L}^{1}$–WPI, if for a decreasing
function $\alpha:\left(0,\infty\right)\to\left[0,\infty\right)$ such that
$\alpha\left(r\right)>0$ for $r\in\left(0,\frac{1}{2}\right)$,

$\mathrm{var}_{\mu}^{\left(1\right)}\left(f\right)\leqslant\alpha\left(r\right)\cdot\mathcal{E}_{1}\left(T,f\right)+r\cdot\left\|f\right\|_{\mathrm{osc}},\quad\forall
r>0,f\in\mathrm{L}^{1}\left(\mu\right).$ (12)

######  Remark 38.

Note that with $A\in\mathscr{E}$ such that $\mu\left(A\right)=\frac{1}{2}$
then letting $f=\mathbf{1}_{A}$ and remarking that
$\mathrm{var}_{\mu}^{(1)}\left(\mathbf{1}_{A}\right)=\frac{1}{2}$ the
existence of an $\mathrm{L}^{1}$–WPI implies

$\frac{1}{2}\leqslant\alpha\left(r\right)\cdot\mathcal{E}_{1}\left(T,\mathbf{1}_{A}\right)+r\cdot
1,$

from which we see that the condition $\alpha\left(r\right)>0$ for
$r\in\left(0,\frac{1}{2}\right)$ is a requirement. Further, since
$\mathrm{var}_{\mu}^{\left(1\right)}\left(f\right)\leqslant\frac{1}{2}\cdot\left\|f\right\|_{\mathrm{osc}}$
by considering $c=\frac{1}{2}\cdot\left(\mathrm{ess_{\mu}}\sup
f-\mathrm{ess_{\mu}}\inf f\right)$, we see that (12) is non-trivial for
$r\in\left(0,\frac{1}{2}\right)$ only.

######  Remark 39.

As for standard WPIs, it is possible to parametrize $\mathrm{L}^{1}$–WPIs in
different ways; however the $\alpha-$parametrization will be sufficient for
our purposes.

The following is related to aspects of [9, Lemma 8.3].

######  Lemma 40.

Let $P$ be a $\mu$-invariant Markov kernel. Then,

  1. a).

if $P$ has a positive WCP $\Phi_{P}^{\mathrm{W}}$, then it satisfies an
$\mathrm{L}^{1}$–WPI with

$\alpha_{1}\left(r\right):=\sup_{r\leqslant
s\leqslant\frac{1}{2}}\left\\{\frac{s-r}{s\cdot\Phi_{P}^{\mathrm{W}}\left(s\right)}\right\\},\qquad
r\in\left(0,\frac{1}{2}\right);$ (13)

  2. b).

if $P$ satisfies an $\mathrm{L}^{1}$–WPI for some $\alpha_{1}$ then it has a
positive WCP and

$\Phi_{P}^{\mathrm{W}}\left(v\right)\geqslant\sup_{0<r\leqslant
v}\left\\{\frac{v-r}{\alpha_{1}\left(r\right)\cdot v}\right\\},\qquad
v\in\left(0,\frac{1}{2}\right].$ (14)

Additionally, if $\alpha_{1}$ in (14) is the optimal $\mathrm{L}^{1}$–WPI
function given in (13), then equality holds in (14).

###### Proof.

Since a $\mathrm{L}^{1}$–WPI depends only on $P$ through
$\mathcal{E}_{1}\left(P,\cdot\right)$ and a WCP depends only on $P$ through
$\mathcal{E}_{1}\left(P,\cdot\right)$ or
$\mathcal{E}_{2}\left(P,\cdot\right)$, we may assume without loss of
generality that $P$ is $\mu$-reversible; otherwise we may treat
$S=\frac{1}{2}\left(P+P^{*}\right)$.

Suppose first that $P$ has a positive WCP, $\Phi_{P}^{\mathrm{W}}.$ Let
$f\in\left[0,1\right]$ such that
$\mu\left(\left\\{f=0\right\\}\right)\geqslant\frac{1}{2}$ and
$\mathrm{ess\,sup}f=1$. Since $P$ may be assumed reversible, we mirror the
approaches of [24] to deduce that

$\displaystyle\mathcal{E}_{1}\left(P,f\right)$
$\displaystyle=\frac{1}{2}\cdot\int\mu\left(\mathrm{d}x\right)\cdot
P\left(x,\mathrm{d}y\right)\cdot\left|f\left(y\right)-f\left(x\right)\right|$
$\displaystyle=\int\mu\left(\mathrm{d}x\right)\cdot
P\left(x,\mathrm{d}y\right)\cdot\left\\{f\left(y\right)-f\left(x\right)\right\\}\cdot{\bf
1}\left[f\left(x\right)<f\left(y\right)\right]$
$\displaystyle=\int\mu\left(\mathrm{d}x\right)\cdot
P\left(x,\mathrm{d}y\right)\cdot\int{\bf 1}\left[f\left(x\right)\leqslant
t\right]\cdot\mathbf{1}\left[f\left(y\right)>t\right]\,\mathrm{d}t$
$\displaystyle=\int_{0}^{1}\left(\mu\otimes
P\right)\left(F_{t}^{\complement}\times F_{t}\right)\,\mathrm{d}t$
$\displaystyle=\int_{0}^{1}\left(\mu\otimes P\right)\left(F_{t}\times
F_{t}^{\complement}\right)\,\mathrm{d}t,$

with $F_{t}:=\left\\{x\in\mathsf{E}\colon f\left(x\right)>t\right\\}$. Note
also that $\mu\left(F_{t}\right)\leqslant\frac{1}{2}$ for $t>0$ since
$\mu\left(\left\\{f=0\right\\}\right)\geqslant\frac{1}{2}$. Further, by the
assumptions on $f$, for any $r>0$,

$\mu\left(f\right)-r\cdot\left\|f\right\|_{{\rm
osc}}=\int_{0}^{1}\left\\{\mu\left(F_{t}\right)-r\right\\}\,\mathrm{d}t.$

Thus by comparing integrands, we deduce that a sufficient condition for

$\mu\left(f\right)-r\cdot\left\|f\right\|_{{\rm
osc}}\leqslant\alpha_{1}\left(r\right)\cdot\mathcal{E}_{1}\left(P,f\right)$

to hold for a given $\alpha_{1}\left(r\right)\geqslant 0$ is that

$\displaystyle\mu\left(A\right)-r$
$\displaystyle\leqslant\alpha_{1}\left(r\right)\cdot\left(\mu\otimes
P\right)\left(A\times A^{\complement}\right),$

for all $A\in\mathscr{E}$ with $\mu\left(A\right)\leqslant\frac{1}{2}$. From
the assumed WCP, we have that

$\left(\mu\otimes P\right)\left(A\times
A^{\complement}\right)\geqslant\mu\left(A\right)\cdot\Phi_{P}^{\mathrm{W}}\left(\mu\left(A\right)\right)\,,$

implying that a sufficient condition is, with $s:=\mu\left(A\right)$ and for
$r\leqslant s$ to avoid a trivial bound,

$\displaystyle s-r$ $\displaystyle\leqslant\alpha_{1}\left(r\right)\cdot
s\cdot\Phi_{P}^{\mathrm{W}}\left(s\right),$
$\displaystyle\alpha_{1}\left(r\right)$ $\displaystyle:=\sup_{r\leqslant
s\leqslant\frac{1}{2}}\left\\{\frac{s-r}{s\cdot\Phi_{P}^{\mathrm{W}}\left(s\right)}\right\\}>0,$

which is a decreasing function on $\left(0,\frac{1}{2}\right)$. For
$f\in\mathrm{L}^{1}\left(\mu\right)$ satisfying $\mathrm{ess_{\mu}}\sup
f=\mathrm{ess_{\mu}}\sup\left(-f\right)=1$ and $\mu\left(\left\\{f\geqslant
0\right\\}\right)\geqslant\frac{1}{2},\mu\big{(}\left\\{f\leqslant
0\right\\}\big{)}\geqslant\frac{1}{2}$, consider the two functions
$f_{\pm}=\max\left\\{\pm f,0\right\\}$ and observe that
$\mu\left(\left\\{f_{\pm}=0\right\\}\right)\geqslant\frac{1}{2}$. The result
above yields

$\mu\left(f_{\pm}\right)\leqslant\alpha_{1}\left(r\right)\cdot\mathcal{E}_{1}\left(P,f_{\pm}\right)+r\cdot\left\|f_{\pm}\right\|_{{\rm
osc}}.$

Adding up these two inequalities and using that
$\mu\left(f_{+}\right)+\mu\left(f_{-}\right)=\mu\left(\left|f\right|\right)$,
$\mathcal{E}_{1}\left(P,f_{+}\right)+\mathcal{E}_{1}\left(P,f_{-}\right)=\mathcal{E}_{1}\left(P,f\right)$
and $\left\|f_{+}\right\|_{{\rm osc}}+\left\|f_{-}\right\|_{{\rm
osc}}=\left\|f\right\|_{{\rm osc}}$ gives that

$\mu\left(\left|f\right|\right)\leqslant\alpha_{1}\left(r\right)\cdot\mathcal{E}_{1}\left(P,f\right)+r\cdot\left\|f\right\|_{{\rm
osc}},$

and it holds in general that
$\mathrm{var}_{\mu}^{\left(1\right)}\left(f\right)\leqslant\mu\left(\left|f\right|\right)$.
For general bounded $f\in\mathrm{L}^{1}\left(\mu\right)$, one can always find
constants $c_{1},c_{2}\in\mathbb{R}$ such that $f=c_{1}+c_{2}\cdot\tilde{f}$
with
$\mathrm{ess_{\mu}}\sup\tilde{f}=\mathrm{ess_{\mu}}\sup\big{(}-\tilde{f}\big{)}=1$
and apply the above; we thus conclude.

Assume now that an $\mathrm{L}^{1}$–WPI holds. For $A\in\mathcal{E}$, let
$f=\mathbf{1}_{A}$ and note that
$\mathrm{var}_{\mu}^{\left(1\right)}\left(\mathbf{1}_{A}\right)=\min\left\\{\mu\left(A\right),\mu\left(A^{\complement}\right)\right\\}$.
Now, from Lemma 102 we have
$\mathcal{E}_{1}\left(P,\mathbf{1}_{A}\right)=\mu\otimes P\left(A\times
A^{\complement}\right)$ and the $\mathrm{L^{1}}$–WPI therefore implies for all
$r>0$ that

$\displaystyle\min\left\\{\mu\left(A\right),\mu\left(A^{\complement}\right)\right\\}$
$\displaystyle\leqslant\alpha_{1}\left(r\right)\cdot\mu\otimes P\left(A\times
A^{\complement}\right)+r\cdot 1.$

Notice again that for $\mu\left(A\right)=\frac{1}{2}$, we see that this
requires $\alpha_{1}\left(r\right)>0$ for $r\in\left(0,\frac{1}{2}\right)$.
Therefore for $r\in\left(0,\frac{1}{2}\right)$,

$\frac{\min\left\\{\mu\left(A\right),\mu\left(A^{\complement}\right)\right\\}-r}{\alpha_{1}\left(r\right)\cdot\min\left\\{\mu\left(A\right),\mu\left(A^{\complement}\right)\right\\}}\leqslant\frac{\mu\otimes
P\left(A\times
A^{\complement}\right)}{\min\left\\{\mu\left(A\right),\mu\left(A^{\complement}\right)\right\\}},$

and hence for $v\in\left(0,\frac{1}{2}\right]$,
$r\in\left(0,\frac{1}{2}\right)$ we have

$\displaystyle\Phi_{P}^{\mathrm{W}}\left(v\right)$
$\displaystyle\geqslant\inf\left\\{\frac{\min\left\\{\mu\left(A\right),\mu\left(A^{\complement}\right)\right\\}-r}{\alpha_{1}\left(r\right)\cdot\min\left\\{\mu\left(A\right),\mu\left(A^{\complement}\right)\right\\}}:\min\left\\{\mu\left(A\right),\mu\left(A^{\complement}\right)\right\\}\geqslant
v\right\\}$
$\displaystyle\geqslant\inf\left\\{\frac{s-r}{\alpha_{1}\left(r\right)\cdot
s}:s\in\left[v,\frac{1}{2}\right]\right\\}$
$\displaystyle=\frac{v-r}{\alpha_{1}\left(r\right)\cdot v}.$

Finally, for $v\in\left(0,\frac{1}{2}\right]$ take a supremum over
$r\in\left(0,v\right]$ to see that

$\Phi_{P}^{\mathrm{W}}\left(v\right)\geqslant\sup_{0<r\leqslant
v}\left\\{\frac{v-r}{\alpha_{1}\left(r\right)\cdot v}\right\\}>0,$

showing that $P$ has a positive WCP.

For sharpness, the preceding calculations demonstrate that the optimal
$\Phi_{P}^{\mathrm{W}}$ and $\alpha_{1}$ are in bijection with one another as
the sharp functions in the inequality

$\Phi_{P}^{\mathrm{W}}\left(v\right)\cdot\alpha_{1}\left(r\right)\geqslant\frac{v-r}{v},\qquad
v\in\left(0,\frac{1}{2}\right],\,r\in\left(0,\frac{1}{2}\right),$

from which the final claim follows. ∎

######  Remark 41.

From the existence of a positive WCP and the theorem above, one can obtain a
bound on the corresponding $\mathrm{L^{1}}$–WPI function, for
$r\in\left(0,\frac{1}{2}\right)$:

$\alpha_{1}\left(r\right)=\sup_{r\leqslant
s\leqslant\frac{1}{2}}\left\\{\frac{s-r}{s\cdot\Phi_{P}^{\mathrm{W}}\left(s\right)}\right\\}\leqslant\sup_{r\leqslant
s\leqslant\frac{1}{2}}\left\\{\frac{1}{\Phi_{P}^{\mathrm{W}}\left(s\right)}\right\\}\leqslant\Phi_{P}^{\mathrm{W}}\left(r\right)^{-1},$

since $\Phi_{P}^{\mathrm{W}}$ is non-decreasing.

We now show, by following arguments of [9], that an $\mathrm{L}^{1}$–WPI
implies an $\alpha$–WPI, with an explicit bound on the function $\alpha$.

######  Lemma 42.

Let $P$ be a $\mu$-invariant Markov kernel satisfying an $\mathrm{L}^{1}$–WPI
with function $\alpha_{1}$. Then $P$ also satisfies an $\alpha$–WPI with

$\alpha\left(r\right)=\inf_{\theta\in\left(0,1\right)}\left\\{\frac{\alpha_{1}\left(\left(1-\theta\right)\cdot
r\right)^{2}}{2\cdot\theta\cdot\left(1-\theta\right)}\right\\}\leqslant
2\cdot\alpha_{1}\left(\frac{1}{2}\cdot r\right)^{2},\quad r>0.$

###### Proof.

Assume that a $\mathrm{L^{1}}$–WPI holds. Let
$f\in\mathrm{L}^{2}\left(\mu\right)$ be nonnegative with
$\mu\left(\left\\{f=0\right\\}\right)=\mu\left(\left\\{f^{2}=0\right\\}\right)\geqslant\frac{1}{2}$.
We then apply the $\mathrm{L}^{1}$–WPI to $f^{2}$, followed by the
Cauchy–Schwarz and triangle inequalities, together with the inequality
$2ab\leqslant\lambda a^{2}+\lambda^{-1}b^{2}$, to see for $r>0$,
$\theta\in\left(0,1\right)$ that

$\displaystyle\left\|f\right\|_{2}^{2}=\mathrm{var}_{\mu}^{(1)}\left(f^{2}\right)$
$\displaystyle\leqslant\alpha_{1}\left(r\right)\cdot\mathcal{E}_{1}\left(P,f^{2}\right)+r\cdot\left\|f^{2}\right\|_{\mathrm{osc}}$
$\displaystyle\leqslant\sqrt{2}\cdot\alpha_{1}\left(r\right)\cdot\left\|f\right\|_{2}\cdot\mathcal{E}\left(P,f\right)^{1/2}+r\cdot\left\|f\right\|_{\mathrm{osc}}^{2}$
$\displaystyle\leqslant\theta\cdot\left\|f\right\|_{2}^{2}+\frac{\alpha_{1}\left(r\right)^{2}}{2\cdot\theta}\cdot\mathcal{E}\left(P,f\right)+r\cdot\left\|f\right\|_{\mathrm{osc}}^{2}.$

Rearranging, we obtain for $r>0$ that

$\left\|f\right\|_{2}^{2}\leqslant\frac{\alpha_{1}\left(r\right)^{2}}{2\cdot\theta\cdot\left(1-\theta\right)}\cdot\mathcal{E}\left(P,f\right)+\frac{r}{1-\theta}\cdot\left\|f\right\|_{\mathrm{osc}}^{2},$

that is, for $\tilde{r}>0$,

$\displaystyle\left\|f\right\|_{2}^{2}$
$\displaystyle\leqslant\frac{\alpha_{1}\left(\tilde{r}\cdot\left(1-\theta\right)\right)^{2}}{2\cdot\theta\cdot\left(1-\theta\right)}\cdot\mathcal{E}\left(P,f\right)+\tilde{r}\cdot\left\|f\right\|_{\mathrm{osc}}^{2}.$

Now, for general bounded $f\in\mathrm{L}^{2}\left(\mu\right)$ such that
$\mu\left(\left\\{f\geqslant
0\right\\}\right)\geqslant\frac{1}{2},\mu\big{(}\left\\{f\leqslant
0\right\\}\big{)}\geqslant\frac{1}{2}$ we apply the above to
$f_{\pm}=\max\left\\{\pm f,0\right\\}$ such that
$\mu\left(\left\\{f_{+}\geqslant
0\right\\}\right),\mu\big{(}\left\\{f_{-}\geqslant
0\right\\}\big{)}\geqslant\frac{1}{2}$. Recalling that
$\left\|f_{+}\right\|_{2}^{2}+\left\|f_{-}\right\|_{2}^{2}=\left\|f\right\|_{2}^{2}$,
$\mathcal{E}\left(P,f_{+}\right)+\mathcal{E}\left(P,f_{-}\right)\leqslant\mathcal{E}\left(P,f\right)$,
and
$\left\|f_{+}\right\|_{\mathrm{osc}}^{2}+\left\|f_{-}\right\|_{\mathrm{osc}}^{2}\leqslant\left\|f\right\|_{\mathrm{osc}}^{2}$,
it follows that for $r>0$ and $\theta\in\left(0,1\right)$,

$\left\|f\right\|_{2}^{2}\leqslant\frac{\alpha_{1}\left(\left(1-\theta\right)\cdot
r\right)^{2}}{2\cdot\theta\cdot\left(1-\theta\right)}\cdot\mathcal{E}\left(P,f\right)+r\cdot\left\|f\right\|_{\mathrm{osc}}^{2}.$

Defining $\alpha:\left(0,\infty\right)\to\left[0,\infty\right)$ by

$\displaystyle\alpha\left(r\right)$
$\displaystyle=\inf_{\theta\in\left(0,1\right)}\frac{\alpha_{1}\left(\left(1-\theta\right)\cdot
r\right)^{2}}{2\cdot\theta\cdot\left(1-\theta\right)}$ $\displaystyle\leqslant
2\cdot\alpha_{1}\left(\frac{1}{2}\cdot r\right)^{2},$

we thus have with $\bar{f}=f-\mu\left(f\right)$ that

$\left\|\bar{f}\right\|_{2}^{2}\leqslant\left\|f\right\|_{2}^{2}\leqslant\alpha\left(r\right)\cdot\mathcal{E}\left(P,\bar{f}\right)+r\cdot\left\|\bar{f}\right\|_{\mathrm{osc}}^{2}.$

Finally, for general $f\in\mathrm{L}^{2}\left(\mu\right)$, we apply the above
to $f-\mathrm{med}_{\mu}\left(f\right)$, where
$\mathrm{med}_{\mu}\left(f\right)$ is a median of $f$ under $\mu$, and
conclude that an $\alpha$–WPI holds since $\alpha$ is decreasing. ∎

We now present the proof of the main result, i.e. Theorem 35

###### Proof of Theorem 35.

For the first statement. By Lemmas 40, 42, we can define
$\alpha\left(r\right):=2\cdot\alpha_{1}\left(\frac{1}{2}\cdot r\right)^{2}$
and $\alpha_{1}\colon\left(0,\frac{1}{2}\right)\to\left(0,\infty\right)$ is as
in Lemma 40, so that for all $f\in\mathrm{L}^{2}_{0}\left(\pi\right)$ and
$r>0$, we have that

$\displaystyle\left\|f\right\|_{2}^{2}$
$\displaystyle\leqslant\alpha\left(r\right)\cdot\mathcal{E}\left(P,f\right)+r\cdot\left\|f\right\|_{\mathrm{osc}}^{2}.$

Furthermore for $r\in\left(0,\frac{1}{2}\right)$ we can write

$\frac{\mathcal{E}\left(P,f\right)}{\left\|f\right\|_{\mathrm{osc}}^{2}}\geqslant\frac{1}{\alpha\left(r\right)}\cdot\left\\{\frac{\left\|f\right\|_{2}^{2}}{\left\|f\right\|_{\mathrm{osc}}^{2}}-r\right\\}.$

Now defining

$K^{*}\left(v\right):=\sup_{r\in\left(0,\frac{1}{4}\right]}\left\\{\frac{v-r}{\alpha\left(r\right)}\right\\},$
(15)

and recalling that
$\left\|f\right\|_{2}^{2}/\left\|f\right\|_{\mathrm{osc}}^{2}\leqslant\frac{1}{4}$,
we see that this implies
$\frac{\mathcal{E}\left(P,f\right)}{\left\|f\right\|_{\mathrm{osc}}^{2}}\geqslant
K^{*}\left(\frac{\left\|f\right\|_{2}^{2}}{\left\|f\right\|_{\mathrm{osc}}^{2}}\right)$.
From the definition of $K^{*}$ it is clear that it is nondecreasing, and as a
supremum of linear functions it is convex. From Lemma 42 and Remark 41, we
have $\alpha\left(r\right)\leqslant
2\cdot\Phi_{P}^{\mathrm{W}}\left(\frac{1}{2}\cdot r\right)^{-2}$ for
$r\in\left[0,1\right]$, and hence taking $r=\frac{1}{2}\cdot v$ in the
definition of $K^{*}$, one obtains for $v\in\left[0,1\right]$ that

$K^{*}\left(v\right)\geqslant\frac{1}{2}\cdot
v\cdot\alpha\left(\frac{1}{2}\cdot v\right)^{-1}\geqslant\frac{1}{2}\cdot
v\cdot\left(2\cdot\Phi_{P}^{\mathrm{W}}\left(\frac{1}{4}\cdot
v\right)^{-2}\right)^{-1}=2^{-2}\cdot
v\cdot\Phi_{P}^{\mathrm{W}}\left(2^{-2}\cdot v\right)^{2};$ (16)

we are thus able to conclude. For the second statement, let $A\in\mathcal{E}$
such that $\mu\left(A\right)\in\left(0,1\right)$ and consider the function

$f=\frac{\mathbf{1}_{A}-\mu\left(A\right)}{\sqrt{\mu\left(A\right)\cdot\mu\left(A^{\complement}\right)}},$

so that from Lemma 102, it holds that $\left\|f\right\|_{2}^{2}=1$ and
$1/\left\|f\right\|_{\mathrm{osc}}^{2}=\mu\left(A\right)\cdot\mu\left(A^{\complement}\right)$.
Assumed existence of a $K^{*}$–WPI and Lemma 102 together imply that

$\frac{\mu\otimes P\left(A\times
A^{\complement}\right)}{\mu\left(A\right)}\geqslant\frac{K^{*}\left(\mu\otimes\mu\left(A\times
A^{\complement}\right)\right)}{\mu\left(A\right)\cdot\mu\otimes\mu\left(A\times
A^{\complement}\right)},$

from which we deduce for $v\in\left(0,\frac{1}{2}\right]$ that

$\displaystyle\inf\left\\{\frac{\left(\mu\otimes P\right)\left(A\times
A^{\complement}\right)}{\mu\left(A\right)}:v\leqslant\mu\left(A\right)\leqslant\frac{1}{2}\right\\}$
$\displaystyle\geqslant\inf\left\\{\frac{K^{*}\left(s\cdot\left(1-s\right)\right)}{s\cdot
s\cdot\left(1-s\right)}:v\leqslant s\leqslant\frac{1}{2}\right\\}$
$\displaystyle\geqslant
2\cdot\frac{K^{*}\left(v\cdot\left(1-v\right)\right)}{v\cdot\left(1-v\right)}\,,$

where the last inequality follows from noting that $v\mapsto v^{-1}\cdot
K^{*}\left(v\right)$ is increasing on $\left[0,\frac{1}{4}\right]$ from
convexity and $K^{*}\left(0\right)=0$. Now for
$v\in\left(0,\frac{1}{2}\right]$, observe that
$u=v\cdot\left(1-v\right)\in\left(0,\frac{1}{4}\right]$, and so taking
$v=\left(1-\sqrt{1-4u}\right)/2$ allows us to conclude. ∎

###### Proof of Corollary 36.

With $F$ as defined in Theorem 8, the upper and lower bounds follow
immediately from the upper and lower bounds on $K^{*}$ from Theorem 35. To
establish the subsequent claim that taking $n$ as written will guarantee
$\left\|P^{n}f\right\|_{2}^{2}/\left\|f\right\|_{\mathrm{osc}}^{2}\leqslant\epsilon$,
it remains to ensure that $K^{*}$ has a form which allows us to apply the
reasoning from [1, Proof of Theorem 8], namely that $K^{*}$ is the convex
conjugate of $K\left(u\right):=u\,\beta\left(1/u\right)$ for $u>0$ and
$K\left(0\right)=0$, where $\beta:=\alpha^{\mathrm{\shortminussymb}}$ with
$\alpha^{\mathrm{\shortminussymb}}\left(s\right):=\inf\left\\{u>0\colon\alpha\left(u\right)\leqslant
s\right\\}$ for $s>0$. One can check that
$\alpha_{1}\colon\left(0,\frac{1}{2}\right)\to\left(0,\infty\right)$ in Lemma
40 is strictly decreasing, implying that
$\alpha\colon\left(0,\frac{1}{4}\right]\to\left[0,\infty\right)$ is also
strictly decreasing. Setting $\alpha\left(r\right)=0$ for
$r>\mathfrak{a}=\frac{1}{4}$ (as we as free to do) and appealing to [3,
Proposition 5] yields that $\alpha^{-}\circ\alpha\left(r\right)=r$ for
$r\in\left(0,\frac{1}{4}\right]$. Consequently with
$u:=\left[\alpha\left(r\right)\right]^{-1}$, we have that
$\alpha^{-}\left(u^{-1}\right)=r$, and deduce the representation

$\sup_{r\in\left(0,\frac{1}{4}\right]}\left\\{\frac{v-r}{\alpha\left(r\right)}\right\\}=\sup_{u\in\left[0,\infty\right)}\left\\{v\cdot
u-u\cdot\alpha^{-}\left(u^{-1}\right)\right\\}.$

We can therefore use the properties of [1, Lemma 1, Supp. Material]. From the
assumed positivity and reversibility of $P$ and Theorem 35, we have for
$f\in\mathrm{L}^{2}\left(\mu\right)$,

$\frac{\mathcal{E}\left(P^{*}P,f\right)}{\left\|f\right\|_{\mathrm{osc}}^{2}}=\frac{\mathcal{E}\left(P^{2},f\right)}{\left\|f\right\|_{\mathrm{osc}}^{2}}\geqslant\frac{\mathcal{E}\left(P,f\right)}{\left\|f\right\|_{\mathrm{osc}}^{2}}\geqslant
K^{*}\left(\frac{\left\|f\right\|_{2}^{2}}{\left\|f\right\|_{\mathrm{osc}}^{2}}\right).$

Finally, following [1, Proof of Theorem 8, Supp. Material], with
$n\in\mathbb{N}$, $v_{0}=\mathfrak{a}=\frac{1}{4}$ and
$v_{n}=\left\|P^{n}f\right\|_{2}^{2}/\left\|f\right\|_{\mathrm{osc}}^{2}$,
there holds the bound

$\displaystyle n\leqslant\int_{v_{n}}^{v_{0}}\frac{{\rm
d}v}{K^{*}\left(v\right)}$ $\displaystyle\leqslant
2^{2}\cdot\int_{v_{n}}^{v_{0}}\frac{{\rm
d}v}{v\cdot\Phi_{P}^{\mathrm{W}}\left(2^{-2}\cdot
v\right)^{2}}=2^{2}\cdot\int_{2^{-2}\cdot v_{n}}^{2^{-4}}\frac{{\rm
d}v}{v\cdot\Phi_{P}^{\mathrm{W}}\left(v\right)^{2}}.$

As such, taking $n\geqslant
2^{2}\cdot\int_{2^{-2}\cdot\epsilon}^{2^{-4}}\frac{\mathrm{d}v}{v\cdot\Phi_{P}^{\mathrm{W}}\left(v\right)^{2}}$
guarantees that $v_{n}\leqslant\epsilon$, and so we conclude. ∎

###  3.2 $\mu$–irreducibility, aperiodicity and WPIs

Building on the concepts and results given in Section 3.1, this section
establishes necessary and sufficient conditions for the existence of WPIs, in
terms of probabilistic properties of the Markov chain. The results are
predominantly theoretical in nature; quantitative bounds are discussed in
Section 3.3. We consider here $\left\|\cdot\right\|_{\mathrm{osc}}^{2}$_–_
WPIs and $\left\|\cdot\right\|_{\mathrm{osc}}^{2}$-convergence. Throughout
this section, we are working with a $\mu$-invariant Markov kernel $P$, where
$\mu$ is a probability measure.

From earlier sections, we have seen that the existence of a WPI for the kernel
$T_{k}:=\left(P^{*}\right)^{k}P^{k}$ for some $k\in\mathbb{N}$ enables one to
deduce (subgeometric) convergence bounds for $\left\|P^{k\cdot
n}f\right\|_{2}$, and hence $\left\|P^{n}f\right\|_{2}$, for bounded $f$. This
section is concerned with

  1. a).

the necessity of the existence of a WPI for some $T_{k}$ to ensure convergence
of $P$;

  2. b).

whether simple conditions such as $\mu$-irreducibility and aperiodicity can
imply the existence of a WPI.

More specifically, we know that for a $\mu$-invariant, aperiodic and
$\mu$-irreducible Markov kernel, under mild regularity conditions, for
$\mu$-almost all $x\in\mathsf{E}$,

$\lim_{n\rightarrow\infty}\left\|\delta_{x}P^{n}-\mu\right\|_{\mathrm{TV}}=0;$

see [29, Theorem 13.3.5]. This implies that

$\left\|P^{n}f-\mu\left(f\right)\right\|_{2}\leqslant\mu\left(\left\|\delta_{x}P^{n}-\mu\right\|_{\mathrm{TV}}^{2}\right)^{1/2}\cdot\left\|f\right\|_{\mathrm{osc}},$
(17)

from the characterization
$\left\|\nu\right\|_{\mathrm{TV}}=\sup\left\\{\nu\left(f\right):\left\|f\right\|_{\mathrm{osc}}\leqslant
1\right\\}$ for signed measures with $\nu\left(\mathsf{E}\right)=0$, and the
dominated convergence theorem implies convergence. When $P$ is reversible, [1,
Proposition 23] establishes that this implies that a WPI holds for $P^{2}$,
therefore implying the necessity of the WPI condition. A natural question is
whether $\mu$-irreducibility and aperiodicity, or convergence, always imply
the existence of a WPI for $T_{k}$ for some $k\in\mathbb{N}$. The answer is
negative and we give a counterexample.

######  Definition 43.

We say that a Markov kernel $T$ on $\left(\mathsf{E},\mathscr{E}\right)$ is
$\nu$-irreducible for a measure $\nu$ on $\left(\mathsf{E},\mathscr{E}\right)$
if for any measurable set $A\in\mathscr{E}$ with $\nu\left(A\right)>0$, we
have

$\sum_{n=0}^{\infty}\lambda^{n}\cdot T^{n}\left(x,A\right)>0,\quad\forall
x\in\mathsf{E},$

for some (and hence all) $0<\lambda<1$.

######  Definition 44.

For a $\mu$-invariant Markov kernel $T$, and $\epsilon\in\left(0,1\right)$, we
denote by $T_{\epsilon}$ the $\mu$-invariant kernel
$T_{\epsilon}=\epsilon\cdot{\rm Id}+\left(1-\epsilon\right)\cdot T$.

######  Definition 45.

We say a Markov kernel $T$ has positive holding probabilities almost
everywhere, or satisfies (PH), if

$\mu\left(\left\\{x\in\mathsf{E}:T\left(x,\left\\{x\right\\}\right)=0\right\\}\right)=0.$

Throughout this section, we write $S:=\left(P+P^{*}\right)/2$ for the additive
reversibilization of $P$.

The following summarizes the most salient general results, i.e. without
assuming (PH), in this section:

######  Theorem 46.

Throughout, we write ‘convergent’ for
‘$\left\|\cdot\right\|_{\mathrm{osc}}^{2}$-convergent’. The following hold:

  1. a).

$P^{*}P$ satisfies a WPI $\Rightarrow$ $P$ convergent [Theorem 8]

  2. b).

$P$ $\mu$-irreducible $\Rightarrow$ $P$ satisfies a WPI [Corollary 51]

  3. c).

$P$ convergent $\Rightarrow$ $P^{k}$ satisfies a WPI for all $k\in\mathbb{N}$.
[Lemma 56]

  4. d).

$P$ convergent $\Rightarrow$ $S$ convergent [Proposition 60]

  5. e).

The following are equivalent: [Proposition 63]

    1. i

$P$ satisfies a WPI.

    2. ii

For some $\epsilon\in\left(0,1\right)$, $P_{\epsilon}^{*}P_{\epsilon}$
satisfies a WPI.

    3. iii

For all $\epsilon\in\left(0,1\right)$, $P_{\epsilon}^{*}P_{\epsilon}$
satisfies a WPI.

  6. f).

The following are equivalent: [Remark 6]

    1. i

$P$ satisfies a WPI

    2. ii

$P^{*}$ satisfies a WPI

    3. iii

$S$ satisfies a WPI

  7. g).

$P$ satisfying a WPI does not imply $P$ is convergent [Example 61].

  8. h).

$S$ convergent does not imply $P$ convergent [Example 61].

  9. i).

$P$ convergent does not imply $\left(P^{*}\right)^{k}P^{k}$ satisfies a WPI
for some $k\in\mathbb{N}$ [Proposition 66].

Item e) above can be connected with the notion of variance bounding: [34,
Theorem 6] shows that the variance bounding property is equivalent to
geometric $\mathrm{L}^{2}$-convergence of $P_{\epsilon}$ for any
(equivalently, all) $\epsilon\in\left(0,1\right)$, for reversible $P$. The
situation under (PH) is much simpler. The proof of Theorem 47 can be found in
Section 3.2.2.

######  Theorem 47.

Assume $P$ satisfies (PH). The following are equivalent:

  1. a).

$P$ satisfies a WPI;

  2. b).

$P^{*}P$ satisfies a WPI;

  3. c).

$P$ is convergent;

  4. d).

$PP^{*}$ satisfies a WPI;

  5. e).

$P^{*}$ is convergent.

We note that $P$ satisfies (PH) $\iff$ $P^{*}$ satisfies (PH) $\iff$ $S$
satisfies (PH) by Lemma 62.

These implications and equivalences are quite useful for showing several
derivative results. For example, we may deduce that reversible Markov kernels
$P$ satisfies $P$ convergent $\iff$ $P^{2}=P^{*}P$ satisfies a WPI by Lemma 56
and Theorem 8. Similarly, it is straightforward to deduce that under (PH), $S$
convergent $\iff$ $S$ satisfies a WPI $\iff$ $P$ satisfies a WPI $\iff$
$P^{*}P$ satisfies a WPI $\iff$ $P$ convergent. This can be contrasted with
the general case where one only has $P$ convergent $\Rightarrow$ $S$
convergent, and $S$ convergent $\Rightarrow$ for all
$\epsilon\in\left(0,1\right)$, $P_{\epsilon}^{*}P_{\epsilon}$ satisfies a WPI;
see Proposition 64.

####  3.2.1 Equivalence of $\left\|\cdot\right\|_{{\rm osc}}^{2}$_–_ WPI and
RUPI

We will see that for a Markov operator $T$, a necessary and sufficient
condition for a
$\left(\left\|\cdot\right\|_{\mathrm{osc}}^{2},\alpha\right)-$WPI to hold is
the abstract resolvent-uniform-positivity-improving (RUPI) property. This
property appeared in [20], and in [40] it was suggested that an equivalence
between the RUPI property and the existence of a WPI was already established
in an unpublished manuscript by L. Wu. However, we have not been able to
access this manuscript, and so in this section we provide a direct proof of
this equivalence. This abstract RUPI condition will provide the crucial link
between irreducibility and existence of a WPI in Corollary 51; it has been
established in [20, Corollary 4.5] that $\mu$-irreducibility implies the RUPI
property.

######  Definition 48 (UPI and RUPI).

A $\mu$-invariant kernel $T$ is uniform-positivity-improving (UPI) if for each
$\epsilon>0$,

$\inf\left\\{\left\langle\mathbf{1}_{A},T\mathbf{1}_{B}\right\rangle:\mu\left(A\right)\wedge\mu\left(B\right)\geqslant\epsilon\right\\}>0.$

A Markov kernel $T$ is said to be resolvent-uniform-positivity-improving
(RUPI) if for some (and hence all) $0<\lambda<1$, we have that the resolvent

$R\left(\lambda,T\right):=\sum_{n=0}^{\infty}\lambda^{n}\cdot
T^{n}=\left(\mathrm{Id}-\lambda\cdot T\right)^{-1}\,,$

is UPI.

Although the RUPI condition is abstract, in order to establish that a given
kernel $T$ is RUPI, it is sufficient to show that a simple irreducibility
condition holds. Recall that we are assuming in this section that $\mu$ is a
probability measure, and that $T$ is $\mu$-invariant.

######  Proposition 49 ([20, Corollary 4.5]).

Suppose that the $\mu$-invariant kernel $T$ is $\mu$-irreducible. Then $T$ is
RUPI.

The aim of this subsection is to establish the following abstract result,
which immediately implies the subsequent practical corollary.

######  Theorem 50.

Suppose that $T$ is a $\mu$–invariant Markov kernel. Then $T$ satisfies a WPI
if and only if $T$ is RUPI.

###### Proof.

This follows from Proposition 55 and Proposition 58 below. ∎

######  Corollary 51.

Suppose the Markov kernel $T$ is $\mu$-irreducible. Then $T$ possesses a WPI
by Proposition 49 and Theorem 50.

We follow [40] and give an equivalent condition for RUPI which will be
convenient to work with.

######  Lemma 52.

An equivalent condition for a Markov kernel $T$ to be RUPI is the following:
for any $\epsilon>0$, there exists $m\in\mathbb{N}$ such that

$\inf\left\\{\left\langle\mathbf{1}_{A},\sum_{n=0}^{m}T^{n}\mathbf{1}_{B}\right\rangle:\mu\left(A\right)\wedge\mu\left(B\right)\geqslant\epsilon\right\\}>0.$
(18)

###### Proof.

The condition in Lemma 52 directly implies RUPI. To see this, take
$\lambda\in\left(0,1\right)$, $\epsilon>0$ and let $m\in\mathbb{N}$ such that
$\inf\left\\{\left\langle\mathbf{1}_{A},\sum_{n=0}^{m}T^{n}\mathbf{1}_{B}\right\rangle:\mu\left(A\right)\wedge\mu\left(B\right)\geqslant\epsilon\right\\}>0$,
which exists by assumption. Write

$\displaystyle\left\langle\mathbf{1}_{A},R\left(\lambda,T\right)\mathbf{1}_{B}\right\rangle$
$\displaystyle=\left\langle\mathbf{1}_{A},\sum_{n=0}^{\infty}\lambda^{n}\cdot
T^{n}\mathbf{1}_{B}\right\rangle$
$\displaystyle\geqslant\left\langle\mathbf{1}_{A},\sum_{n=0}^{m}\lambda^{n}\cdot
T^{n}\mathbf{1}_{B}\right\rangle$
$\displaystyle\geqslant\lambda^{m}\cdot\left\langle\mathbf{1}_{A},\sum_{n=0}^{m}T^{n}\mathbf{1}_{B}\right\rangle,$

to deduce that

$\inf\\{\left\langle\mathbf{1}_{A},R\left(\lambda,T\right)\mathbf{1}_{B}\right\rangle:\mu\left(A\right)\wedge\mu\left(B\right)\geqslant\epsilon\\}\\\
\geqslant\lambda^{m}\cdot\inf\left\\{\left\langle\mathbf{1}_{A},\sum_{n=0}^{m}T^{n}\mathbf{1}_{B}\right\rangle:\mu\left(A\right)\wedge\mu\left(B\right)\geqslant\epsilon\right\\}>0,$

from which the RUPI condition follows.

Conversely, suppose that $T$ is RUPI, fix $\lambda\in\left(0,1\right)$ and
assume that for some $\epsilon>0$, (18) does not hold for any
$m\in\mathbb{N}$. We show that this leads to a contradiction. By the RUPI
assumption we have that

$\delta:=\inf\left\\{\left\langle\mathbf{1}_{A},\sum_{n=0}^{\infty}\lambda^{n}\cdot
T^{n}\mathbf{1}_{B}\right\rangle:\mu\left(A\right)\wedge\mu\left(B\right)\geqslant\epsilon\right\\}>0.$

Choose $m\in\mathbb{N}$ large enough so that

$\sum_{n=m+1}^{\infty}\lambda^{n}<\frac{\delta}{2}.$

Since we have assumed that (18) is violated for $\epsilon>0$ and
$m\in\mathbb{N}$ as chosen above, there exists a sequence
$\left\\{\left(A_{j},B_{j}\right)\right\\}_{j=1}^{\infty}$ of sets all with
mass at least $\epsilon$ such that
$\left\langle\mathbf{1}_{A_{j}},\sum_{n=0}^{m}T^{n}\mathbf{1}_{B_{j}}\right\rangle\to
0$, therefore implying for any $j\in\mathbb{N}$,

$\displaystyle\delta\leqslant\left\langle\mathbf{1}_{A_{j}},\sum_{n=0}^{\infty}\lambda^{n}T^{n}\mathbf{1}_{B_{j}}\right\rangle$
$\displaystyle=\left\langle\mathbf{1}_{A_{j}},\sum_{n=0}^{m}\lambda^{n}T^{n}\mathbf{1}_{B_{j}}\right\rangle+\left\langle\mathbf{1}_{A_{j}},\sum_{n=m+1}^{\infty}\lambda^{n}T^{n}\mathbf{1}_{B_{j}}\right\rangle$
$\displaystyle\leqslant\left\langle\mathbf{1}_{A_{j}},\sum_{n=0}^{m}T^{n}\mathbf{1}_{B_{j}}\right\rangle+\sum_{n=m+1}^{\infty}\lambda^{n}$
$\displaystyle\leqslant\left\langle\mathbf{1}_{A_{j}},\sum_{n=0}^{m}T^{n}\mathbf{1}_{B_{j}}\right\rangle+\frac{\delta}{2}$
$\displaystyle\overset{j\to\infty}{\rightarrow}\frac{\delta}{2}\,,$

therefore leading to a contradiction. The conclusion follows. ∎

We first establish that for reversible kernels, RUPI implies a WPI for the
resolvent.

######  Lemma 53.

Suppose that a $\mu$-reversible Markov kernel $T$ is RUPI. Then for any
$\lambda\in\left(0,1\right)$, the resolvent Markov kernel
$S_{\lambda}:=\left(1-\lambda\right)\cdot R\left(\lambda,T\right)$ is
$\mu$-reversible and has the following property: for any
$0<\epsilon\leqslant\frac{1}{2}$, for the quantity appearing in the WCP,

$\inf_{A:\epsilon\leqslant\mu\left(A\right)\leqslant\frac{1}{2}}\frac{\mathcal{E}\left(S_{\lambda},\mathbf{1}_{A}\right)}{\mu\left(A\right)}>0.$

Thus by Theorem 35, $S_{\lambda}$ satisfies a WPI.

###### Proof.

Fix $0<\epsilon\leqslant\frac{1}{2}$ and $\lambda\in\left(0,1\right)$. By the
RUPI condition, we know that
$\inf\left\\{\left\langle\mathbf{1}_{A},S_{\lambda}\mathbf{1}_{B}\right\rangle:\mu\left(A\right)\wedge\mu\left(B\right)\geqslant\epsilon\right\\}>0$.
In particular, if $A$ is such that
$\epsilon\leqslant\mu\left(A\right)\leqslant\frac{1}{2}$, we must have that
both $\mu\left(A\right)\geqslant\epsilon$ and
$\mu\left(A^{\complement}\right)\geqslant\frac{1}{2}\geqslant\epsilon$. Thus
since

$\mathcal{E}\left(S_{\lambda},\mathbf{1}_{A}\right)=\left\langle\mathbf{1}_{A},S_{\lambda}\mathbf{1}_{A^{\complement}}\right\rangle,$

by Lemma 102 we must have that

$\left\langle\mathbf{1}_{A},S_{\lambda}\mathbf{1}_{A^{\complement}}\right\rangle\geqslant\delta>0,$

for some $\delta>0$, whenever
$\epsilon<\mu\left(A\right)\leqslant\frac{1}{2}$. ∎

We now establish one direction of Theorem 50 through a sequence of lemmas: we
first consider the case when $T$ is reversible, and then deduce the case for
general $T$; see Remark 6.

######  Lemma 54.

Suppose $T$ is a $\mu$-reversible Markov kernel that is RUPI. Then $T$
satisfies a WPI.

###### Proof.

Since $T$ is RUPI, we have established above in Lemma 53 that the resolvent
$S_{\lambda}:=\left(1-\lambda\right)\cdot R\left(\lambda,T\right)$ satisfies a
WPI for any $\lambda\in\left(0,1\right)$. In other words, we can find some
$\alpha_{\lambda}:\left(0,\infty\right)\to\left[0,\infty\right)$ such that for
any $f\in\mathrm{L}_{0}^{2}\left(\mu\right)$ and $r>0$, it holds that

$\left\|f\right\|_{2}^{2}\leqslant\alpha_{\lambda}\left(r\right)\cdot\left\langle\left(\mathrm{Id}-S_{\lambda}\right)f,f\right\rangle+r\cdot\left\|f\right\|_{\mathrm{osc}}^{2}.$

Now, given a function $g\in\mathrm{L}_{0}^{2}\left(\mu\right)$, define
$f:=\left(1-\lambda\right)^{-1}\cdot\left(\mathrm{Id}-\lambda\cdot
T\right)\cdot g$, so that
$g=\left(1-\lambda\right)\cdot\left(\mathrm{Id}-\lambda\cdot
T\right)^{-1}\cdot f$. Note that since $0<\lambda<1$, the operator
$\left(\mathrm{Id}-\lambda\cdot T\right)$ is invertible.

Now since $g\in\mathrm{L}_{0}^{2}\left(\mu\right)$, we have that
$f\in\mathrm{L}_{0}^{2}\left(\mu\right)$ (by, for instance, considering the
power series representation of $R\left(\lambda,T\right)$). Furthermore, we
have that

$\displaystyle\left\|g\right\|_{2}^{2}$
$\displaystyle=\left(1-\lambda\right)^{2}\cdot\left\|\left(\mathrm{Id}-\lambda\cdot
T\right)^{-1}f\right\|_{2}^{2}$
$\displaystyle\leqslant\left(1-\lambda\right)^{2}\cdot\left\|\left(\mathrm{Id}-\lambda\cdot
T\right)^{-1}\right\|_{\mathrm{op}}\cdot\left\|f\right\|_{2}^{2}$
$\displaystyle\leqslant\left\|f\right\|_{2}^{2}$

since the operator norm can be bounded as
$\left\|\left(\mathrm{Id}-\lambda\cdot
T\right)^{-1}\right\|_{\mathrm{op}}\leqslant\frac{1}{1-\lambda}$, by standard
arguments.

Thus, we have that

$\displaystyle\left\|g\right\|_{2}^{2}\leqslant$
$\displaystyle\left\|f\right\|_{2}^{2}$ $\displaystyle\leqslant$
$\displaystyle\alpha_{\lambda}\left(r\right)\cdot\left\langle\left(\mathrm{Id}-S_{\lambda}\right)f,f\right\rangle+r\cdot\left\|f\right\|_{\mathrm{osc}}^{2}$
$\displaystyle=$
$\displaystyle\alpha_{\lambda}\left(r\right)\cdot\left\langle\left(\mathrm{Id}-\left(1-\lambda\right)\cdot\left(\mathrm{Id}-\lambda\cdot
T\right)^{-1}\right)f,f\right\rangle+r\cdot\left\|f\right\|_{\mathrm{osc}}^{2}$
$\displaystyle=$
$\displaystyle\alpha_{\lambda}\left(r\right)\cdot\left\langle\left(1-\lambda\right)^{-1}\cdot\left(\mathrm{Id}-\lambda\cdot
T\right)g-g,\left(1-\lambda\right)^{-1}\cdot\left(\mathrm{Id}-\lambda\cdot
T\right)g\right\rangle+r\cdot\left\|f\right\|_{\mathrm{osc}}^{2}$
$\displaystyle=$
$\displaystyle\alpha_{\lambda}\left(r\right)\cdot\left\langle\lambda\cdot\left(1-\lambda\right)^{-1}\cdot\left(\mathrm{Id}-T\right)g,g+\lambda\cdot\left(1-\lambda\right)^{-1}\cdot\left(\mathrm{Id}-T\right)g\right\rangle+r\cdot\left\|f\right\|_{\mathrm{osc}}^{2}$
$\displaystyle=$
$\displaystyle\alpha_{\lambda}\left(r\right)\cdot\left\\{\left\langle\lambda\cdot\left(1-\lambda\right)^{-1}\cdot\left(\mathrm{Id}-T\right)g,g\right\rangle+\lambda^{2}\cdot\left(1-\lambda\right)^{-2}\cdot\left\|\left(\mathrm{Id}-T\right)g\right\|_{2}^{2}\right\\}$
$\displaystyle\hskip 256.0748pt+r\cdot\left\|f\right\|_{\mathrm{osc}}^{2}.$

Note that

$\displaystyle\left\|\left(\mathrm{Id}-T\right)g\right\|_{2}^{2}$
$\displaystyle=\left\langle\left(\mathrm{Id}-T\right)g,\left(\mathrm{Id}-T\right)g\right\rangle$
$\displaystyle=\left\langle\left(\mathrm{Id}-T\right)g,g\right\rangle-\left\langle\left(\mathrm{Id}-T\right)g,Tg\right\rangle.$

It is enough to bound this final term by

$-\left\langle\left(\mathrm{Id}-T\right)g,Tg\right\rangle\leqslant\left\langle\left(\mathrm{Id}-T\right)g,g\right\rangle.$

To see why this inequality is true, note that it is equivalent to

$\displaystyle 0$
$\displaystyle\leqslant\left\langle\left(\mathrm{Id}-T\right)g,\left(\mathrm{Id}+T\right)g\right\rangle$
$\displaystyle=\left\langle\left(\mathrm{Id}+T\right)\cdot\left(\mathrm{Id}-T\right)g,g\right\rangle$
$\displaystyle=\left\langle\left(\mathrm{Id}-T^{2}\right)g,g\right\rangle$

where we have made use of reversibility of $T$, and we certainly have that
$0\leqslant\left\langle\left(\mathrm{Id}-T^{*}T\right)g,g\right\rangle=\left\langle\left(\mathrm{Id}-T^{2}\right)g,g\right\rangle$.

Overall, this gives us that

$\displaystyle\left\|g\right\|_{2}^{2}$
$\displaystyle\leqslant\alpha_{\lambda}\left(r\right)\cdot\left\\{\left\langle\lambda\cdot\left(1-\lambda\right)^{-1}\cdot\left(\mathrm{Id}-T\right)g,g\right\rangle+\lambda^{2}\cdot\left(1-\lambda\right)^{-2}\cdot\left\|\left(\mathrm{Id}-T\right)g\right\|_{2}^{2}\right\\}$
$\displaystyle\quad+r\cdot\left\|\left(1-\lambda\right)^{-1}\cdot\left(\mathrm{Id}-\lambda\cdot
T\right)g\right\|_{\mathrm{osc}}^{2}$
$\displaystyle\leqslant\alpha_{\lambda}\left(r\right)\cdot\left\\{\lambda\cdot\left(1-\lambda\right)^{-1}+2\cdot\lambda^{2}\cdot\left(1-\lambda\right)^{-2}\right\\}\cdot\left\langle\left(\mathrm{Id}-T\right)g,g\right\rangle$
$\displaystyle\quad+r\cdot\left(1+\lambda\right)^{2}\cdot\left(1-\lambda\right)^{-2}\cdot\left\|g\right\|_{\mathrm{osc}}^{2},$

and by reparameterizing with
$r^{\prime}=r\cdot\left(1+\lambda\right)^{2}\cdot\left(1-\lambda\right)^{-2}$,
one obtains a standard WPI for $T$. ∎

######  Proposition 55.

Suppose a $\mu$-invariant Markov kernel $T$ is RUPI. Then $T$ satisfies a WPI.

###### Proof.

It suffices to show that $\left(T+T^{*}\right)/2$ is RUPI, as then by Lemma
54, $\left(T+T^{*}\right)/2$ possesses a WPI, which is equivalent to $T$
possessing a WPI (see Remark 6). Since $T$ is RUPI, for any $\epsilon>0$, we
can find some $\delta>0$ and $N\in\mathbb{N}$ such that whenever
$\mu\left(A\right)\wedge\mu\left(B\right)\geqslant\epsilon$,

$\left\langle\mathbf{1}_{A},\sum_{n=0}^{N}T^{n}\mathbf{1}_{B}\right\rangle\geqslant\delta>0.$
(19)

We now wish to obtain such a statement for the kernel
$\left(T+T^{*}\right)/2$. To this end, fix $\epsilon>0$, and consider

$\left\langle\mathbf{1}_{A},\sum_{n=0}^{N}\left(\frac{T+T^{*}}{2}\right)^{n}\mathbf{1}_{B}\right\rangle=\left\langle\mathbf{1}_{A},\sum_{n=0}^{N}2^{-n}\cdot
T^{n}\mathbf{1}_{B}\right\rangle+\left\langle\mathbf{1}_{A},R\mathbf{1}_{B}\right\rangle,$

where $R$ is a sum of operators of the form $c\cdot
T^{a_{1}}\cdot\left(T^{*}\right)^{b_{1}}\cdot\dots\cdot
T^{a_{r}}\cdot\left(T^{*}\right)^{b_{r}}$ for some $r\in\mathbb{N}$,
$a_{i},b_{i}\in\mathbb{N}_{0}$ for all $i=1,\dots,r$ and $c\geqslant 0$. Thus
since $T$ and $T^{*}$ are Markov kernels, we have that
$\left\langle\mathbf{1}_{A},R\mathbf{1}_{B}\right\rangle\geqslant 0.$ So we
can continue and have, for any sets with
$\mu\left(A\right)\wedge\mu\left(B\right)\geqslant\epsilon$,

$\displaystyle\left\langle\mathbf{1}_{A},\sum_{n=0}^{N}\left(\frac{T+T^{*}}{2}\right)^{n}\mathbf{1}_{B}\right\rangle$
$\displaystyle\geqslant\left\langle\mathbf{1}_{A},\sum_{n=0}^{N}2^{-n}\cdot
T^{n}\mathbf{1}_{B}\right\rangle$ $\displaystyle\geqslant
2^{-N}\cdot\left\langle\mathbf{1}_{A},\sum_{n=0}^{N}T^{n}\mathbf{1}_{B}\right\rangle$
$\displaystyle\geqslant 2^{-N}\cdot\delta>0,$

since each summand is positive, and we have used the fact that $T$ is RUPI
(19). ∎

We now prove the other direction. The following is a useful implication of
convergence, meaning (4) holds, which we will rely on below and also in
Section 3.2.2.

######  Lemma 56.

Assume $T$ is convergent. Then for any $\epsilon>0$, there exists
$n_{0}\in\mathbb{N}$ such that for any $N\geqslant n_{0}$,

$\inf\left\\{\left\langle\mathbf{1}_{A},T^{N}\mathbf{1}_{B}\right\rangle:\mu\left(A\right)\wedge\mu\left(B\right)\geqslant\epsilon\right\\}>0.$

In particular, for all $k\in\mathbb{N}$, $T^{k}$ is RUPI and $T^{k}$ satisfies
a WPI.

###### Proof.

Let $\epsilon>0$ be arbitrary. Since $T$ is convergent, we may take
$n_{0}\in\mathbb{N}$ large enough such that
$\left\|T^{N}f\right\|_{2}\leqslant\left\|f\right\|_{{\rm
osc}}\cdot\epsilon^{2}/2$ for all $N\geqslant n_{0}$. Let $A,B\in\mathscr{E}$
be such that $\mu\left(A\right)\wedge\mu\left(B\right)\geqslant\epsilon$. For
any $N\geqslant n_{0}$ we have

$\displaystyle\left\langle\mathbf{1}_{A},T^{N}\mathbf{1}_{B}\right\rangle$
$\displaystyle=\left\langle\mathbf{1}_{A},\left(T^{N}-\mu\right)\mathbf{1}_{B}\right\rangle+\left\langle\mathbf{1}_{A},\mu\mathbf{1}_{B}\right\rangle$
$\displaystyle=\left\langle\mathbf{1}_{A},\left(T^{N}-\mu\right)\mathbf{1}_{B}\right\rangle+\mu\left(A\right)\cdot\mu\left(B\right).$

Let $f={\bf 1}_{B}-\mu\left(B\right)$, so that by Cauchy–Schwarz, we have that

$\left|\left\langle\mathbf{1}_{A},\left(T^{N}-\mu\right)\mathbf{1}_{B}\right\rangle\right|=\left|\left\langle\mathbf{1}_{A},T^{N}f\right\rangle\right|\leqslant\mu\left(A\right)^{1/2}\cdot\left\|f\right\|_{{\rm
osc}}\cdot\epsilon^{2}/2\leqslant\epsilon^{2}/2,$

and therefore

$\left\langle\mathbf{1}_{A},T^{N}\mathbf{1}_{B}\right\rangle\geqslant-\epsilon^{2}/2+\epsilon^{2}=\epsilon^{2}/2>0,$

from which we can conclude. Now let $k\in\left\\{1,2,\ldots\right\\}$ be
arbitrary. Since we may choose $N$ to be a multiple of $k$, it follows from
Lemma 52 that $T^{k}$ is RUPI. Hence, by Proposition 55, $T^{k}$ satisfies a
WPI. ∎

Lemma 57b) is an extension of the argument referenced by [30, Remark 2.16].

######  Lemma 57.

Assume $P$ satisfies (PH). Then

  1. a).

The following (chained) WPI holds: for $s>0$,
$f\in\mathrm{L}^{2}_{0}\left(\mu\right)$,

$\mathcal{\mathcal{E}}\left(P,f\right)\leqslant
s\cdot\mathcal{E}\left(P^{*}P,f\right)+\beta\left(s\right)\cdot\left\|f\right\|_{{\rm
osc}}^{2},$ (20)

for the function

$\beta\left(s\right):=\mu\otimes\left(\frac{P+P^{*}}{2}\right)\left(A\left(s\right)^{\complement}\cap\left\\{X\neq
Y\right\\}\right),$ (21)

where
$A\left(s\right):=\left\\{\left(x,y\right)\in\mathsf{E}\times\mathsf{E}:s\cdot\delta\left(x,y\right)>1\right\\}$,
where
$\delta\left(x,y\right):=2\cdot\left(\epsilon\left(x\right)\wedge\epsilon\left(y\right)\right)$,
and $\epsilon\left(x\right):=P\left(x,\left\\{x\right\\}\right)$.

  2. b).

If $P\left(x,\left\\{x\right\\}\right)\geqslant\varepsilon>0$ for $\mu$-almost
all $x\in E$, then for $f\in\mathrm{L}^{2}_{0}\left(\mu\right)$,
$\mathcal{E}\left(P^{*}P,f\right)\geqslant
2\cdot\varepsilon\cdot\mathcal{E}\left(P,f\right)$.

In particular, if $P$ satisfies a WPI then $P^{*}P$ satisfies a WPI.

###### Proof.

First, we represent the kernel $P$ as

$P\left(x,\mathrm{d}y\right)=\epsilon\left(x\right)\cdot\delta_{x}\left(\mathrm{d}y\right)+R\left(x,\mathrm{d}y\right),$

for a function $\epsilon:\mathsf{E}\to\left[0,1\right]$ given by
$\epsilon\left(x\right)=P\left(x,\left\\{x\right\\}\right)$ and some (possibly
non-Markov) kernel $R$. Note that since $f\mapsto Pf$ and the pointwise
product $f\mapsto\epsilon\cdot f$ are bounded linear operators on
$\mathrm{L}^{2}\left(\mu\right)$, it follows that $f\mapsto Rf$ is also a
bounded linear operator on $\mathrm{L}^{2}\left(\mu\right)$, and hence there
exists a bounded linear adjoint operator $R^{*}$.

Thus we have the representation

$P^{*}\left(x,\mathrm{d}y\right)=\epsilon\left(x\right)\cdot\delta_{x}\left(\mathrm{d}y\right)+R^{*}\left(x,\mathrm{d}y\right).$

In order to establish the desired WPI, we seek to apply [1, Theorem 38],
namely we seek to establish for any measurable set $A\subset\mathsf{E}$, and
$x\in\mathsf{E}$,

$P^{*}P\left(x,A\setminus\left\\{x\right\\}\right)\geqslant\int_{A\backslash\left\\{x\right\\}}2\cdot\min\left\\{\epsilon\left(x\right),\epsilon\left(y\right)\right\\}\cdot\left(\frac{P+P^{*}}{2}\right)\left(x,\mathrm{d}y\right),$

Note that this is sufficient since
$\mathcal{E}\left(P,f\right)=\mathcal{E}\left(\left(P+P^{*}\right)/2,f\right)$.

Through direct calculation, we find that

$\displaystyle P^{*}P\left(x,A\setminus\left\\{x\right\\}\right)$
$\displaystyle=\int\left(\epsilon\left(x\right)R\left(x,\mathrm{d}y\right)+R^{*}\left(x,\mathrm{d}y\right)\epsilon\left(y\right)\right)\mathbf{1}_{A\setminus\left\\{x\right\\}}\left(y\right)$
$\displaystyle\hskip 113.81102pt+\int
R^{*}\left(x,\mathrm{d}z\right)R\left(z,\mathrm{d}y\right)\mathbf{1}_{A\setminus\left\\{x\right\\}}\left(y\right)$
$\displaystyle\geqslant\int\left(\epsilon\left(x\right)R\left(x,\mathrm{d}y\right)+R^{*}\left(x,\mathrm{d}y\right)\epsilon\left(y\right)\right)\mathbf{1}_{A\setminus\left\\{x\right\\}}\left(y\right)$
$\displaystyle\geqslant\int\min\left\\{\epsilon\left(x\right),\epsilon\left(y\right)\right\\}\cdot\left[R\left(x,\mathrm{d}y\right)+R^{*}\left(x,\mathrm{d}y\right)\right]$
$\displaystyle=2\cdot\int\min\left\\{\epsilon\left(x\right),\epsilon\left(y\right)\right\\}\cdot\left(\frac{P+P^{*}}{2}\right)\left(x,\mathrm{d}y\right).$

Thus by [1, Theorem 36], we have that (20) holds with $\beta\left(s\right)$ as
in (21). Note that $\beta\left(s\right)\to 0$ as $s\to\infty$ since we have
assumed that
$\int\mu\left(\mathrm{d}x\right)\cdot\mathbf{1}\left[\left\\{x:P\left(x,\left\\{x\right\\}\right)=0\right\\}\right]=0$.
The last statement is immediate. ∎

######  Proposition 58.

Let $T$ be a $\mu$-invariant Markov kernel satisfying an $\alpha$_–_ WPI for
some $\alpha:\left(0,\infty\right)\to\left[0,\infty\right)$. Then $T$ is RUPI.

###### Proof.

Consider the Markov operator $T_{1/2}:=\frac{1}{2}\left(\mathrm{Id}+T\right)$,
which satisfies
$T_{1/2}\left(x,\left\\{x\right\\}\right)\geqslant\frac{1}{2}$ by
construction. Note that

$\displaystyle\mathcal{E}\left(T_{1/2},f\right)$
$\displaystyle=\left\langle\left(\mathrm{Id}-\frac{1}{2}\cdot\left(\mathrm{Id}+T\right)\right)f,f\right\rangle$
$\displaystyle=\frac{1}{2}\cdot\left\langle\left(\mathrm{Id}-T\right)f,f\right\rangle$
$\displaystyle=\frac{1}{2}\cdot\mathcal{E}\left(T,f\right).$

Therefore, since $T$ satisfies a $\alpha-$WPI, we have that $T_{1/2}$
satisfies a
$\left(\left\|\cdot\right\|_{\mathrm{osc}}^{2},2\cdot\alpha\right)$_–_ WPI:

$\displaystyle\left\|f\right\|_{2}^{2}$
$\displaystyle\leqslant\alpha\left(r\right)\cdot\mathcal{E}\left(T,f\right)+r\cdot\left\|f\right\|_{{\rm
osc}}^{2}$
$\displaystyle=2\cdot\alpha\left(r\right)\cdot\mathcal{E}\left(T_{1/2},f\right)+r\cdot\left\|f\right\|_{{\rm
osc}}^{2}.$

Since
$\mathrm{ess_{\mu}\inf}_{x}T_{1/2}\left(x,\left\\{x\right\\}\right)\geqslant\frac{1}{2}$,
by Lemma 57 we have the inequality
$\mathcal{E}\left(T_{1/2},f\right)\leqslant\mathcal{E}\left(T_{1/2}^{*}T_{1/2},f\right)$,
so we deduce a $(2\cdot\alpha)$_–_ WPI for $T_{1/2}^{*}T_{1/2}$. Hence,
$T_{1/2}$ is convergent by Theorem 8.

We will now verify the condition for RUPI in Lemma 52. Let
$\epsilon\in\left(0,1\right)$ be arbitrary. Since $T_{1/2}$ is convergent,
Lemma 56 implies that there exists $N\in\mathbb{N}$ such that

$\delta=\inf\left\\{\left\langle\mathbf{1}_{A},T_{1/2}^{N}\mathbf{1}_{B}\right\rangle:\mu\left(A\right)\wedge\mu\left(B\right)\geqslant\epsilon\right\\}>0.$

Now,

$T_{1/2}^{N}=\left(\frac{\mathrm{Id}+T}{2}\right)^{N}=\frac{1}{2^{N}}\cdot\sum_{k=0}^{N}a_{k}\cdot
T^{k},$

for binomial coefficients $\\{a_{i}\\}$. Since $\sum_{i=0}^{N}a_{i}=2^{N}$ and
${\bf 1}_{A}$, ${\bf 1}_{B}$ are non-negative, we have

$\left\langle\mathbf{1}_{A},\sum_{k=0}^{N}T^{k}\mathbf{1}_{B}\right\rangle\geqslant\left\langle\mathbf{1}_{A},\left(\frac{\mathrm{Id}+T}{2}\right)^{N}\mathbf{1}_{B}\right\rangle,\qquad
A,B\in\mathscr{E},$

and this implies that

$\inf\left\\{\left\langle\mathbf{1}_{A},\sum_{k=0}^{N}T^{k}\mathbf{1}_{B}\right\rangle:\mu\left(A\right)\wedge\mu\left(B\right)\geqslant\epsilon\right\\}\geqslant\delta>0,$

so $T$ is RUPI. ∎

####  3.2.2 WPIs and convergence under (PH)

Recall from Theorem 8 that if $P^{*}P$ satisfies a WPI, then we know that $P$
is convergent: