# Almost sure convergence rates of adaptive increasingly rare Markov chain
Monte Carlo

Julian Hofstadler1 Krzysztof Łatuszyński2 Gareth O. Roberts2 Daniel Rudolf1

(1University of Passau
2University of Warwick
)

###### Abstract

We consider adaptive increasingly rare Markov chain Monte Carlo (AIR MCMC),
which is an adaptive MCMC method, where the adaptation concerning the “past”
happens less and less frequently over time. Under a contraction assumption for
a Wasserstein-like function we deduce upper bounds of the convergence rate of
Monte Carlo sums taking a renormalisation factor into account that is close to
the one that appears in a law of the iterated logarithm. We demonstrate the
applicability of our results by considering different settings, among which
are those of simultaneous geometric and uniform ergodicity. All proofs are
carried out on an augmented state space, including the classical non-augmented
setting as a special case. In contrast to other adaptive MCMC limit theory,
some technical assumptions, like diminishing adaptation, are not needed.

Keywords: adaptive Markov chain Monte Carlo, almost sure convergence rate,
Wasserstein contraction

Classification: 65C05; 65C20; 60J22

##  1 Introduction

Given some probability space $(\mathcal{X},\mathcal{F}_{\mathcal{X}},\nu)$ an
omnipresent challenge in computational statistics is the approximation of
expectations

$\nu(f)=\int_{\mathcal{X}}f(x)\nu({\rm d}x).$ (1)

Here $\nu$ is a desired distribution and $f\colon\mathcal{X}\to\mathbb{R}$ a
suitable integrand of interest. It is a common situation that sampling from
$\nu$ directly is not possible or computationally infeasible, for instance,
since the density of $\nu$ might contain an unknown normalising constant.

Markov chain Monte Carlo (MCMC) methods provide a powerful tool for
approximating expectations as given in (1), especially if direct sampling is
no option. The main idea is to construct a Markov chain
$(X_{n})_{n\in\mathbb{N}_{0}}$ which admits $\nu$ as limit distribution and
under rather mild conditions one can show that the Monte Carlo sums111To ease
notation later we do not use the initial point $X_{0}$ in the Monte Carlo sum.
$S_{n}f=\frac{1}{n}\sum_{j=1}^{n}f(X_{j})$ converge to $\nu(f)$ (in some
sense), provided that $f\colon\mathcal{X}\to\mathbb{R}$ is measurable and
$\nu(f)$ is finite, cf. [MT12, DMPS18]. Furthermore, explicit error bounds
[Rud12, ŁMN13] as well as concentration inequalities [GO02, Mia14] are
available.

Adaptive MCMC allows to update the single step transition mechanism by taking
the whole history into account and therefore leads to non-Markovian processes
(in general). It is well known that if the adaptation is done right,
convergence to the correct limit distribution can be improved. However, a
careless implementation of the adjustment might prevent the algorithm from
achieving desirable convergence properties, we refer to [AR05, Section 2] as
well as [ŁR14].

The non-Markovian behaviour of adaptive MCMC algorithms leads to rather
complicated proof techniques in the literature. To ease the mathematical
analysis the authors of [CŁR18] suggest to use adaptive increasingly rare
(AIR) algorithms, where adaptation takes place less and less frequently.
Indeed, certain technical requirements can be dropped in the theory of those
methods. Moreover, numerical studies, cf. [CŁR18, Section 2.2], show that AIR
versions of algorithms can perform equally well compared to their purely
adaptive counterparts, while saving computational resources.

The goal of this paper is to study the convergence properties of renormalised
Monte Carlo sums corresponding to an AIR scheme
$(X_{n})_{n\in\mathbb{N}_{0}}$. Setting
$r(n)=\sqrt{n}(\log(n))^{1/2+\operatorname{\varepsilon}}$, or
$r(n)=n^{1/2+\operatorname{\varepsilon}}$, our main results, see Section 3,
show

$\limsup_{n\to\infty}\left|\frac{1}{r(n)}\sum_{j=1}^{n}\left(f(X_{j})-\nu(f)\right)\right|=0,$
(2)

almost surely, given suitable $\operatorname{\varepsilon}>0$ and
$f\colon\mathcal{X}\to\mathbb{R}$. The main assumption in Section 3 is a
contraction condition for a Wasserstein-like function. We apply our results to
different settings in Section 4 leading to convergence rates for $S_{n}f$
based on AIR MCMC methods.

The rate $r(n)$ in (2) involves some number $\operatorname{\varepsilon}>0$. As
is discussed in Section 4 this parameter depends on

  * 1)

the size of the windows between consecutive adaptations, and

  * 2)

the complexity of the integrand under consideration.

The main idea of the proofs is to use a martingale approximation based on
solving Poisson’s equation, which is known to be a powerful tool, we refer for
instance to [AM06, SV10]. See also [AF10, AF12] for a generalisation of
Poisson’s equation. While the martingale part is typically easy, there is also
a remainder part, which for purely adaptive schemes can be tricky to control.
Usually one requires technical assumptions, however, for some algorithms, e.g.
adaptive Metropolis [HST01], we know these conditions are satisfied and limit
theory is available [AM06, SV10].

In this paper we focus on AIR versions of algorithms. An important observation
is that for this class of methods most of the (difficult) terms in the
martingale approximation cancel. Hence some rather technical requirements,
e.g. diminishing adaptation, are not required to show that the limit in (2) is
$0$ almost surely. Note that without diminishing adaptation difficulties
concerning convergence of purely adaptive MCMC may occur [AR05].

Our analysis of AIR MCMC is done in an augmented state space setting,
including the classical (non augmented) regime as special case. This is
motivated by the recent work [PHŁ20], where it is shown that moving to an
augmented state space allows to tackle difficulties like multimodality of the
target $\nu$.

Let us comment on how our results fit into the existing literature. Different
results for the ergodicity of adaptive MCMC methods in general and the
corresponding Monte Carlo sums are available. There are laws of large numbers
[HST01, AR05, AM06, RR07, SV10, FMP11, PHŁ20], central limit theorems [AM06,
SV10] as well as results about convergence to the correct target measure
[RR07, FMP11]. In [CŁR18] the convergence of AIR MCMC schemes is provided and,
moreover, the authors of [AA07] studied the efficiency of adaptive MCMC
compared to an “optimal” MCMC method.

Limit theory is usually proven under uniform or geometric convergence
assumptions and we are not aware of results that hold in a Wasserstein
contraction setting. Furthermore, convergence of renormalised Monte Carlo sums
as in (2) (or also laws of iterated logarithm) are to the best of our
knowledge not known for adaptive MCMC methods.

The goal of this paper is to fill this gap: Under a contraction condition for
a Wasserstein-like function we provide results describing the convergence
behaviour of (2), see Theorems 3.5, 3.6, 3.11. These are applied to different
examples, including simultaneous geometric and uniform ergodicity, which are
of interest in adaptive MCMC literature.

An outline of the paper is as follows. In Section 2 we introduce some notation
and general concepts which are used throughout the paper. Theoretical results
are stated and proven in Section 3 and applied to several examples in Section
4. The appendix contains some (technical) auxiliary tools.

##  2 Preliminaries

In this section we provide all required objects to establish our limit
theorems later. We start with some general notation and assumptions.

Following [PHŁ20], we consider an augmented state space, denoted by
$\mathcal{Y}=\mathcal{X}\times\Phi$, where $(\Phi,\mathcal{F}_{\Phi})$ is some
“auxiliary” measurable space and
$\mathcal{F}_{\mathcal{Y}}=\mathcal{F}_{\mathcal{X}}\otimes\mathcal{F}_{\Phi}$.
To avoid problems concerning measurability, all involved spaces are assumed to
be Polish with countable generated $\sigma$-algebra. Besides that there are no
particular requirements on $(\Phi,\mathcal{F}_{\Phi})$ and, moreover,
different choices for $\mathcal{F}_{\mathcal{Y}}$ may be feasible as well.
Note that choosing $\Phi$ to be a singleton corresponds to the classical non-
augmented case.

Given some measurable space $\mathcal{I}$ we consider

  * 1)

a family of probability distributions
$\\{{\pi}_{\gamma}\\}_{\gamma\in\mathcal{I}}$ on
$(\mathcal{Y},\mathcal{F}_{\mathcal{Y}})$, and

  * 2)

a family of transition kernels $\\{{P}_{\gamma}\\}_{\gamma\in\mathcal{I}}$ on
$(\mathcal{Y},\mathcal{F}_{\mathcal{Y}})$.

To be more precise, by 2) we mean that for any $\gamma\in\mathcal{I}$ the
mapping ${P}_{\gamma}\colon\mathcal{Y}\times\mathcal{F}_{\mathcal{Y}}\to[0,1]$
is a transition kernel. We assume that ${\pi}_{\gamma}$ has marginal $\nu$ for
any $\gamma\in\mathcal{I}$, that is, ${\pi_{\gamma}}(B\times\Phi)=\nu(B)$ for
any $B\in\mathcal{F}_{\mathcal{X}}$ and is the invariant distribution of
${P}_{\gamma}({y},\cdot)$ (for any ${y}\in\mathcal{Y}$). Moreover, we assume
that any kernel $P_{\gamma}$ is $\psi$-irreducible and aperiodic, see [MT12]
for definitions.

###  2.1 Adaptive and AIR MCMC

We consider an adaptive MCMC chain, say
$(Y_{n},\Gamma_{n})_{n\in\mathbb{N}_{0}}$, evolving in the space
$(\mathcal{Y},\mathcal{I})$, where $(\Gamma_{n})_{n\in\mathbb{N}_{0}}$ denotes
the sequence of (possibly random) adaptation parameters. In other words,
$\mathcal{Y}$ is the state space of $(Y_{n})_{n\in\mathbb{N}_{0}}$ and
$\mathcal{I}$ that of $(\Gamma_{n})_{n\in\mathbb{N}_{0}}$. We assume that
there exists some underlying (sufficiently regular) probability space
$(\Omega,\mathcal{F},\mathbb{P})$ and set
$\mathcal{F}_{n}=\sigma\left(Y_{0},\dots,Y_{n},\Gamma_{0},\dots,\Gamma_{n}\right)$,
such that $(\mathcal{F}_{n})_{n\in\mathbb{N}}$ is the natural filtration of
$(Y_{n},\Gamma_{n})_{n\in\mathbb{N}_{0}}$. We assume that
$(Y_{0},\Gamma_{0})\equiv(y_{0},\gamma_{0})$ for some initial values
$y_{0}\in\mathcal{Y}$ and $\gamma_{0}\in\mathcal{I}$.

The one step transition probabilities of $(Y_{n})_{n\in\mathbb{N}_{0}}$ are
specified via

$\mathbb{P}\left[Y_{n+1}\in
B|Y_{n}={y},\Gamma_{n}=\gamma\right]={P}_{\gamma}(y,B),$ (3)

where $B\in\mathcal{F}_{\mathcal{Y}}$. Moreover, the adaptation parameter
$\Gamma_{n}$ is chosen according to some update rule222The functions $U_{n}$
are assumed to be measurable.
$U_{n}\colon\mathcal{Y}^{n+1}\times\mathcal{I}^{n}\to\mathcal{I}$, allowing to
“improve” the kernel of $Y_{n+1}$ based on the already known values of
$Y_{0},\dots,Y_{n}$ and $\Gamma_{0},\dots,\Gamma_{n-1}$. That is
$\Gamma_{n}=U_{n}(Y_{0},\dots,Y_{n},\Gamma_{0},\dots,\Gamma_{n-1})$.

Using the regularity of all involved spaces and standard techniques of measure
theory we have that $\mathbb{P}$-almost surely

$\mathbb{E}\left[g(Y_{n+1})|\mathcal{F}_{n}\right]=P_{\Gamma_{n}}g(Y_{n}),$
(4)

where $g\colon\mathcal{Y}\to\mathbb{R}$ is an arbitrary measurable function
such that $P_{\gamma}g(y)<\infty$ for any $\gamma,y$.

We assume that for any $n\in\mathbb{N}$ the mapping
$\omega\mapsto\pi_{\Gamma_{n}(\omega)}(g)$ is measurable for any (measurable)
function $g\colon\mathcal{Y}\to\mathbb{R}$ such that
$\pi_{\gamma}(|g|)<\infty$ for all $\gamma\in\mathcal{I}$.

Since $(Y_{n})_{n\in\mathbb{N}_{0}}$ evolves in the augmented space
$\mathcal{Y}=\mathcal{X}\times\Phi$, each state has the form
$Y_{n}=(X_{n},\phi_{n})$, with an $\mathcal{X}$-valued component $X_{n}$ and a
$\Phi$-valued component $\phi_{n}$. Given some measurable function of interest
$f\colon\mathcal{X}\to\mathbb{R}$ with $\nu(|f|)<\infty$, we set
$h=h_{f}=f\cdot\mathds{1}_{\Phi}$. Using the fact that $\pi_{\gamma}$ has
marginal $\nu$ we obtain the following

$S_{n}f-\nu(f)=\frac{1}{n}\sum_{j=1}^{n}f(X_{j})-\nu(f)=\frac{1}{n}\sum_{j=1}^{n}\left(h(Y_{j})-{\pi}_{\Gamma_{j-1}}(h)\right),$
(5)

holding $\mathbb{P}$-a.s. for any $n\in\mathbb{N}$.

In this paper we focus on AIR MCMC algorithms, as introduced in [CŁR18]. This
class of methods allows $\Gamma_{n}$ to be updated only at specific times, in
contrast to purely adaptive schemes where $\Gamma_{n}$ may be changed after
each step. Given $\beta>0$ we set $k_{j}=\lceil j^{\beta}\rceil$ and allow
adaptation only at the times

$T_{m}=\sum_{j=1}^{m}k_{j}.$

That is, for $i_{1},i_{2}\in\\{T_{j},\dots,T_{j+1}-1\\}$ (with
$j\in\mathbb{N}$) we have $\Gamma_{i_{1}}=\Gamma_{i_{2}}$.

A simple, yet important observation is that for any $\beta>0$ there exist
constants $c_{\beta},C_{\beta}\in\mathbb{R}$ such that

$c_{\beta}m^{1+\beta}\leq T_{m}\leq C_{\beta}m^{1+\beta}.$ (6)

In particular, this means that we adapt less and less frequently.

###  2.2 Martingale approximation

Our results shall be proven using a martingale approximation technique based
on solutions of Poisson’s equation. Providing a powerful tool for the analysis
of general state space Markov chains, see e.g. [MT12, DMPS18], Poisson’s
equation was later also used in the adaptive setting, e.g. in [AM06, SV10,
ALR13]. See also [AF10, AF12] for further generalisations.

For a function of interest $h\colon\mathcal{Y}\to\mathbb{R}$ as well as
$\gamma\in\mathcal{I}$ we say $u_{\gamma}$ is a solution of Poisson’s equation
if

$u_{\gamma}(y)-P_{\gamma}u_{\gamma}(y)=h(y)-{\pi}_{\gamma}(h)$

for any $y\in\mathcal{Y}$. Provided such an $u_{\gamma}$ exists for any
$\gamma\in\Gamma$ we obtain the following decomposition for the Monte Carlo
sum of an adaptive MCMC method driven by (3):

$\displaystyle\sum_{j=1}^{n}\left(h(Y_{j})-{\pi}_{\Gamma_{j-1}}(h)\right)$
$\displaystyle=\sum_{j=1}^{n}\left({u}_{\Gamma_{j-1}}(Y_{j})-{P}_{\Gamma_{j-1}}{u}_{\Gamma_{j-1}}(Y_{j-1})\right)$
$\displaystyle\qquad+\sum_{j=1}^{n}\left({u}_{\Gamma_{j}}(Y_{j})-{u}_{\Gamma_{j-1}}(Y_{j})\right)$
$\displaystyle\qquad+h(Y_{n})-h(Y_{0})+{u}_{\Gamma_{0}}(Y_{0})-{u}_{\Gamma_{n}}(Y_{n}).$
(7)

Recall that for an AIR scheme adaptation only happens at certain times
$\\{T_{j}\\}_{j\in\mathbb{N}_{0}}$ with $T_{0}:=0$. Assume $T_{m}\leq
n+1<T_{m+1}$, then, since
$u_{\Gamma_{i+1}}(Y_{i+1})-u_{\Gamma_{i}}(Y_{i+1})=0$ for any
$i\in\\{T_{j},\dots,T_{j+1}-2\\}$, equation (2.2) simplifies to

$\displaystyle\sum_{j=1}^{n}\left(h(Y_{j})-{\pi}_{\Gamma_{j-1}}(h)\right)$
$\displaystyle=\sum_{j=1}^{n}\left({u}_{\Gamma_{j-1}}(Y_{j})-{P}_{\Gamma_{j-1}}{u}_{\Gamma_{j-1}}(Y_{j-1})\right)$
$\displaystyle\qquad+\sum_{j=0}^{m-1}\left(u_{\Gamma_{T_{j+1}}}(Y_{T_{j+1}})-u_{\Gamma_{T_{j}}}(Y_{T_{j+1}})\right)$
$\displaystyle\qquad+h(Y_{n})-h(Y_{0})+{u}_{\Gamma_{0}}(Y_{0})-{u}_{\Gamma_{n}}(Y_{n}).$
(8)

Here “simplification” refers to the fact that (2.2) contains less summands of
the form ${u}_{\Gamma_{j}}(Y_{j})-{u}_{\Gamma_{j-1}}(Y_{j})$ compared to
(2.2).

Let us define the following quantities:

$\displaystyle M_{n}$
$\displaystyle=\sum_{j=1}^{n}\left({u}_{\Gamma_{j-1}}(Y_{j})-{P}_{\Gamma_{j-1}}{u}_{\Gamma_{j-1}}(Y_{j-1})\right);$
$\displaystyle\Delta_{j}$
$\displaystyle:={u}_{\Gamma_{j-1}}(Y_{j})-{P}_{\Gamma_{j-1}}{u}_{\Gamma_{j-1}}(Y_{j-1});$
$\displaystyle\text{and}\qquad R_{m}$
$\displaystyle:=\sum_{j=0}^{m-1}\left(u_{\Gamma_{T_{j+1}}}(Y_{T_{j+1}})-u_{\Gamma_{T_{j}}}(Y_{T_{j+1}})\right).$

Note that we have $\mathbb{E}[M_{n}|\mathcal{F}_{n-1}]=M_{n-1}$ almost surely,
see (4).

###  2.3 Distance-like and Wasserstein-like functions

We consider a measurable space $(\mathcal{A},\mathcal{F}_{\mathcal{A}})$,
which is Polish w.r.t. some metric $d_{\mathcal{A}}$ and has a countably
generated $\sigma$-algebra.

In the spirit of [HMS11], we call a lower semi-continuous function
$d\colon\mathcal{A}\times\mathcal{A}\to\mathbb{R}^{+}$ distance-like if

  * a)

For any $x,y\in\mathcal{A}$ holds $d(x,y)=d(y,x)$.

  * b)

We have $d(x,y)=0\;\Longleftrightarrow\;x=y$.

Roughly speaking, a distance-like function is a metric without triangle
inequality and clearly, any metric is distance-like. As for the case where $d$
is a metric, we call a function $\psi\colon\mathcal{A}\to\mathbb{R}$ Lipschitz
if

$|\psi(x)-\psi(y)|\leq L\cdot d(x,y),$

for any $x,y\in\mathcal{A}$ with some real constant $L>0$.

Given some distance-like function $d$ on $\mathcal{A}$ and probability
measures $\mu_{1},\mu_{2}$ on $(\mathcal{A},\mathcal{F}_{\mathcal{A}})$ we
define

$\mathcal{W}(\mu_{1},\mu_{2}):=\inf_{\xi\in\mathcal{C}(\mu_{1},\mu_{2})}\int_{\mathcal{A}^{2}}d(x,y)\xi({\rm
d}x,{\rm d}y),$

where $\mathcal{C}(\mu_{1},\mu_{2})$ is the set of all couplings of
$\mu_{1},\mu_{2}$. As is argued in [HMS11] below Definition 4.3, $\mathcal{W}$
is a reasonable tool for quantifying the distance between probability
measures. Following [HJ23] we call $\mathcal{W}$ a Wasserstein-like function
(w.r.t. $d$). However, if $d$ is a metric, then $\mathcal{W}$ corresponds to
the usual $1$-Wasserstein distance.

We write $\text{Lip}_{1}$ for the set of all functions that are Lipschitz
(w.r.t. $d$) with constant $1$.

######  Lemma 2.1.

For $\mathcal{W}$ w.r.t. distance-like function $d$ holds

$\mathcal{W}(\mu_{1},\mu_{2})\geq\sup_{\psi\in\text{Lip}_{1}}\left|\int_{\mathcal{A}}\psi(x)\mu_{1}({\rm
d}x)-\int_{\mathcal{A}}\psi(y)\mu_{2}({\rm d}y)\right|.$

###### Proof.

Let $\psi\in\text{Lip}_{1}$. By lower semi-continuity we have that there
exists an optimal coupling $\xi$, i.e. $\mathcal{W}(\mu_{1},\mu_{2})=\int
d(x,y)\xi({\rm d}x,{\rm d}y)$. Thus

$\displaystyle\left|\int_{\mathcal{A}}\psi(x)\mu_{1}({\rm
d}x)-\int_{\mathcal{A}}\psi(y)\mu_{2}({\rm
d}y)\right|=\left|\int_{\mathcal{A}}\int_{\mathcal{A}}\psi(x)-\psi(y)\xi({\rm
d}x,{\rm d}y)\right|$
$\displaystyle\leq\int_{\mathcal{A}}\int_{\mathcal{A}}d(x,y)\xi({\rm d}x,{\rm
d}y)=\mathcal{W}(\mu_{1},\mu_{2}).$

∎

The well known Kantorovich-Rubinstein duality states that if $d$ is a metric,
then one would have equality in Lemma 2.1.

##  3 Convergence rates via Wasserstein contraction

In this section we provide limit theorems based on a contraction assumption
for a Wasserstein-like function.

Recall that $\mathcal{Y}$ is Polish w.r.t. to some metric $d_{\mathcal{Y}}$
and let $d\colon\mathcal{Y}\times\mathcal{Y}\to\mathbb{R}^{+}$ be a distance-
like function. For $\gamma\in\mathcal{I}$ we define the following quantities:

  * 1)

The eccentricity (see [JO10]) is defined as

$E_{\gamma}(y)=\int_{\mathcal{Y}}d(y,y^{\prime})\pi_{\gamma}({\rm
d}y^{\prime}).$

  * 2)

Given $\ell\in\mathbb{N}$, the Wasserstein contraction coefficient333For the
sake of brevity we refer to this as Wasserstein contraction coefficient, even
if $\mathcal{W}$ is “only” a Wasserstein-like function. for
$P_{\gamma}^{\ell}$ is

$\tau(P_{\gamma}^{\ell})=\sup_{\begin{subarray}{c}y_{1},y_{2}\in\mathcal{Y}\\\
y_{1}\neq
y_{2}\end{subarray}}\frac{\mathcal{W}(P_{\gamma}^{\ell}(y_{1},\cdot),P_{\gamma}^{\ell}(y_{2},\cdot))}{d(y_{1},y_{2})}.$

Reviewing the proofs of [Dob96, Proposition 14.3 and Proposition 14.4] we see
that also for $\mathcal{W}$ based on a distance-like function one has

  * i)

submultiplicativity, that is
$\tau(P_{\gamma}P_{\gamma^{\prime}})\leq\tau(P_{\gamma})\tau(P_{\gamma^{\prime}})$,
and,

  * ii)

contractivity, that is
$\mathcal{W}(\mu_{1}P_{\gamma},\mu_{2}P_{\gamma})\leq\tau(P_{\gamma})\mathcal{W}(\mu_{1},\mu_{2})$,

for any $\gamma,\gamma^{\prime}\in\mathcal{I}$ and any probability measures
$\mu_{1},\mu_{2}$ on $(\mathcal{Y},\mathcal{F}_{\mathcal{Y}}$).

The following contraction condition is our main assumption.

######  Assumption 3.1.

For each $\gamma\in\mathcal{I}$ we assume that $\pi_{\gamma}$ is the invariant
distribution of $P_{\gamma}$ and that

$\tau(P_{\gamma})\leq
M\qquad\text{and}\qquad\tau(P_{\gamma}^{k_{0}})\leq\tau,$

with $M\in(0,\infty)$, $\tau\in[0,1)$ and $k_{0}\in\mathbb{N}$ independent of
$\gamma$.

######  Remark 3.2.

In what follows we have to work with expectations of the form
$\mathbb{E}[E_{\Gamma_{j}}(Y_{k})]$, for some $j,k\in\mathbb{N}_{0}$. Let us
emphasize that for any $j,k\in\mathbb{N}_{0}$ the quantity
$\mathbb{E}_{\Gamma_{j}}(Y_{k})$ is a well defined random variable, see
Appendix B.

###  3.1 Solving Poisson’s equation

The upcoming result shows that under Assumption 3.1 Poisson’s equation has a
solution which is bounded by the eccentricity.

######  Proposition 3.3.

Let $\gamma\in\mathcal{I}$, $f\colon\mathcal{Y}\to\mathbb{R}$ be
$\pi_{\gamma}$ integrable and Lipschitz with constant $L\in(0,\infty)$ and let
$E_{\gamma}(y)<\infty$ for any $y\in\mathcal{Y}$. If Assumption 3.1 is true,
then the function $u_{\gamma}\colon\mathcal{Y}\to\mathbb{R}$ given via

$u_{\gamma}(y)=\sum_{\ell=0}^{\infty}P_{\gamma}^{\ell}f(y)-\pi_{\gamma}(f)$

is well defined and we have

  * i)

$|u_{\gamma}(y)|\leq\frac{k_{0}LM^{k_{0}}E_{\gamma}(y)}{1-\tau}$ for any
$y\in\mathcal{Y}$,

  * ii)

$u_{\gamma}(y)-P_{\gamma}u(y)=f(y)-\pi_{\gamma}(f)$ for any $y\in\mathcal{Y}$.

###### Proof.

Let $\ell=tk_{0}+r\in\mathbb{N}$ with $t\in\mathbb{N}_{0}$ and $1\leq r<k_{0}$
as well as $y\in\mathcal{Y}$. By Lemma 2.1 and the properties of contraction
coefficients

$\displaystyle\left|P_{\gamma}^{\ell}f(y)-\pi_{\gamma}(f)\right|$
$\displaystyle\leq
L\sup_{\psi\in\text{Lip}_{1}}\left|P^{\ell}\psi(y)-\pi_{\gamma}(\psi)\right|\leq
L\mathcal{W}(P_{\gamma}^{\ell}(y,\cdot),\pi_{\gamma})$ $\displaystyle\leq
L\tau(P_{\gamma}^{r})\tau(P_{\gamma}^{tk_{0}})\mathcal{W}(\delta_{y},\pi_{\gamma})\leq
LM^{r}\tau^{t}\mathcal{W}(\delta_{y},\pi_{\gamma})$ $\displaystyle\leq
LM^{k_{0}}\tau^{t}E_{\gamma}(y)<\infty.$

Using this bound we obtain

$\displaystyle|u_{\gamma}(y)|$
$\displaystyle\leq\sum_{\ell=0}^{\infty}|P_{\gamma}^{\ell}f(y)-\pi_{\gamma}(f)|=\sum_{t=0}^{\infty}\sum_{j=tk_{0}+1}^{(t+1)k_{0}}|P_{\gamma}^{j}f(y)-\pi_{\gamma}(f)|$
$\displaystyle\leq\sum_{t=0}^{\infty}\sum_{j=tk_{0}+1}^{(t+1)k_{0}}LM^{k_{0}}\tau^{t}E_{\gamma}(y)=\frac{k_{0}LM^{k_{0}}E_{\gamma}(y)}{1-\tau}$

This shows that $u_{\gamma}$ is well defined and also that i) holds.

It remains to show ii): It is no restriction to assume that
$\pi_{\gamma}(f)=0$ (otherwise consider $g=f-\pi_{\gamma}(f)$). For any
$n\in\mathbb{N}$ we set

$\psi_{n}^{+}=\sum_{\ell=0}^{n}P^{\ell}_{\gamma}f^{+}\qquad\text{and}\qquad\psi_{n}^{-}=\sum_{\ell=0}^{n}P_{\gamma}^{\ell}f^{-},$

where $f^{+}$ and $f^{-}$ are the positive and negative part of $f$. For any
$n\in\mathbb{N}$ holds

$P_{\gamma}\psi_{n}^{+}=\psi_{n+1}^{+}-f^{+}\qquad\text{and}\qquad
P_{\gamma}\psi_{n}^{-}=\psi_{n+1}^{-}-f^{-}.$

By monotone convergence for any $y_{0}\in\mathcal{Y}$ holds

$\lim_{n\to\infty}\left(P_{\gamma}\psi_{n}^{+}(y_{0})\right)=P_{\gamma}\left(\lim_{n\to\infty}\psi_{n}^{+}\right)(y_{0}).$

By the same arguments a similar statement for $\psi_{n}^{-}$ is deduced.
Clearly, $u_{\gamma}=\lim_{n\to\infty}(\psi_{n}^{+}-\psi_{n}^{-})$ and hence
by linearity

$\displaystyle
P_{\gamma}u_{\gamma}=P_{\gamma}\left(\lim_{n\to\infty}(\psi_{n}^{+}-\psi_{n}^{-})\right)=\lim_{n\to\infty}P_{\gamma}\psi_{n}^{+}-\lim_{n\to\infty}P_{\gamma}\psi_{n}^{-}$
$\displaystyle=\lim_{n\to\infty}\left(\psi_{n+1}^{+}-\psi_{n+1}^{-}\right)-f=u_{\gamma}-f.$

This shows ii) and therefore completes the proof. ∎

###  3.2 Limit theorems

This section contains our main results based on Assumption 3.1 together with
either, a bounded eccentricity condition, or, the existence of a Lyapunov
function.

####  3.2.1 Uniformly bounded eccentricity

The first two of our main results are for the setting where the eccentricities
$E_{\gamma}$ are uniformly bounded.

######  Assumption 3.4.

There is some $K\in(0,\infty)$ such that for any $\gamma\in\mathcal{I}$ and
$y\in\mathcal{Y}$ holds

$E_{\gamma}(y)\leq K.$

Our first main result is for the case $\beta\geq 1$ in the AIR scheme.

######  Theorem 3.5.

Let $f\colon\mathcal{X}\to\mathbb{R}$ be a function such that
$h=h_{f}=f\cdot\mathds{1}_{\Phi}$ is Lipschitz with constant $L$ (w.r.t. $d$)
and bounded. Let Assumptions 3.1 and 3.4 be true. Then, for any $\beta\geq 1$
we have

$\lim_{n\to\infty}\frac{1}{\sqrt{n}(\log
n)^{1/2+\operatorname{\varepsilon}}}\left(\sum_{j=1}^{n}f(X_{j})-\nu(f)\right)=0,$

almost surely for arbitrary $\operatorname{\varepsilon}>0$.

Before we turn to the proof we state our result for $\beta\in(0,1)$, where the
rate gets worse as $\beta$ approaches $0$. This does not mean that small
values of $\beta$ or purely adaptive algorithms are worse, rather this
phenomena is due to our proof technique, which collapses as $\beta\to 0$.

######  Theorem 3.6.

Let $f\colon\mathcal{X}\to\mathbb{R}$ be a function such that
$h=h_{f}=f\cdot\mathds{1}_{\Phi}$ is Lipschitz with constant $L$ (w.r.t. $d$)
and bounded. Let Assumptions 3.1 and 3.4 be true. Then, for any
$\beta\in(0,1)$ and $\operatorname{\varepsilon}>\frac{1}{1+\beta}-\frac{1}{2}$
we have

$\lim_{n\to\infty}\frac{1}{n^{1/2+\operatorname{\varepsilon}}}\left(\sum_{j=1}^{n}f(X_{j})-\nu(f)\right)=0,$

almost surely.

To prove Theorems 3.5 and 3.6 we start with two auxiliary results.

######  Lemma 3.7.

Under the assumptions of Theorem 3.5 (or Theorem 3.6) we have that
$(M_{n})_{n\in\mathbb{N}}$ is a $(\mathcal{F}_{n})_{n\in\mathbb{N}}$
martingale and for its differences holds

$|\Delta_{j}|\leq 2\frac{k_{0}LM^{k_{0}}K}{1-\tau},$

almost surely, with $k_{0},L,M,\tau$ and $K$ as in Assumptions 3.1 and 3.4.

###### Proof.

We know from Section 2 that $\mathbb{E}[M_{n}|\mathcal{F}_{n-1}]=M_{n-1}$
almost surely. Now we establish the bound for the differences which also
implies $\mathbb{E}|M_{n}|<\infty$ for any $n\in\mathbb{N}$, showing that
$M_{n}$ is a martingale.

Thereto we observe that

$\sup_{y\in\mathcal{Y}\,;\,\gamma\in\mathcal{I}}|u_{\gamma}(y)|\leq\frac{k_{0}LM^{k_{0}}K}{1-\tau},$

where we used Proposition 3.3 as well as Assumption 3.4. With the same
arguments we deduce that

$\sup_{y\in\mathcal{Y}\,;\,\gamma\in\mathcal{I}}|P_{\gamma}u_{\gamma}(y)|\leq\frac{k_{0}LM^{k_{0}}K}{1-\tau},$

and thus $|\Delta_{j}|\leq 2\frac{k_{0}LM^{k_{0}}K}{1-\tau}$ almost surely. ∎

Next we bound the remainder term $R_{m}$. Recall that the times of adaptation
are $T_{\ell}=\sum_{j=1}^{\ell}k_{j}$, where $k_{j}$ are the “windows” between
adaptation times $T_{0}:=0$. Moreover, given $n\in\mathbb{N}$ we assume that
$m=m(n)$ is such that $T_{m}\leq n+1<T_{m+1}$.

######  Lemma 3.8.

Let $h\colon\mathcal{X}\to\mathbb{R}$ be a function such that
$f=f_{h}=h\cdot\mathds{1}_{\Phi}$ is Lipschitz with constant $L$ (w.r.t. $d$).
Let Assumptions 3.1 and 3.4 be true. Then, for any $\beta>0$ holds

$|R_{m}|\leq
n^{1/(1+\beta)}\cdot(2c_{\beta})^{1/(1+\beta)}\frac{k_{0}LM^{k_{0}}K}{1-\tau},$

almost surely.

###### Proof.

By Proposition 3.3 and Assumption 3.4 we have

$\sup_{y\in\mathcal{Y}\,;\,\gamma\in\mathcal{I}}|u_{\gamma}(y)|\leq\frac{k_{0}LM^{k_{0}}K}{1-\tau},$

such that $|R_{m}|\leq m\cdot\frac{LM^{k_{0}}K}{1-\tau^{k_{0}}}$ almost
surely. According to (6) and since $T_{m}\leq n+1$ it follows

$m\leq c_{\beta}^{1/(1+\beta)}T_{m}^{1/(1+\beta)}\leq
c_{\beta}^{1/(1+\beta)}(n+1)^{1/(1+\beta)}\leq(2c_{\beta})^{1/(1+\beta)}n^{1/(1+\beta)},$

which finishes the proof. ∎

Now we are ready to show our main results of this section.

###### Proof of Theorem 3.5.

We set $r(n)=\sqrt{n}(\log n)^{1/2+\operatorname{\varepsilon}}$. By Lemma 3.7
$M_{n}$ is a martingale with uniformly bounded increments. Hence an
application of Lemma A.1 shows

$\lim_{n\to\infty}\frac{1}{r(n)}M_{n}=0$

almost surely.

Since $\beta\geq 1$ and by Lemma 3.8 we moreover deduce that

$\frac{1}{r(n)}|R_{m}|\leq\frac{n^{1/(1+\beta)}}{r(n)}\widetilde{C}\leq\frac{n^{1/2}}{r(n)}\widetilde{C},$

with
$\widetilde{C}=(2c_{\beta})^{1/(1+\beta)}\frac{k_{0}LM^{k_{0}}K}{1-\tau}$.
Thus $\lim_{n\to\infty}\frac{1}{r(n)}|R_{m}|=0$ a.s.

Finally, by setting

$g_{n}=h(Y_{n})-h(Y_{0})+{u}_{\Gamma_{0}}(Y_{0})-{u}_{\Gamma_{n}}(Y_{n})$

we have a sequence of uniformly (in $n$) bounded random variables. Thus
$\frac{1}{r(n)}g_{n}\to 0$ almost surely.

Now the result is a consequence of equations (5) and (2.2). ∎

###### Proof of Theorem 3.6.

We set $r(n)=n^{1/2+\operatorname{\varepsilon}}$. Note that our choice of
$\operatorname{\varepsilon}$ implies that

$\lim_{n\to\infty}\frac{n^{1/(1+\beta)}}{r(n)}=0.$

The proof is finished similar as in the proof of Theorem 3.5. ∎

####  3.2.2 Lyapunov condition

In this section derive a limit result based on the existence of a Lyapunov
function, which satisfies an inequality w.r.t. $d$.

Recall that a function $V\colon\mathcal{Y}\to[1,\infty)$ is called a Lyapunov
function (for some transition kernel $P$ on
$(\mathcal{Y},\mathcal{F}_{\mathcal{Y}})$) if there exist constants
$\kappa\in(0,1)$ and $b\geq 0$ such that

$PV(y)\leq\kappa V(y)+b.$

Note that if $\pi$ is the invariant distribution of $P$, then
$\pi(V)\leq\frac{b}{1-\tau}$, see e.g. [Hai18, Proposition 4.24].

######  Assumption 3.9.

There exists a function $V\colon\mathcal{Y}\to[1,\infty)$ and constants
$\kappa\in(0,1)$ and $b\in(0,\infty)$ such that for any $\gamma\in\mathcal{I}$
and $y\in\mathcal{Y}$ holds

$P_{\gamma}V(y)\leq\kappa V(y)+b.$

If $V$ is a Lyapunov function then $V^{q}$ is also a Lyapunov function for any
value of $q\in(0,1)$. This is well known from the literature (see e.g. [MT12,
Lemma 15.2.9] for the case $q=1/2$). For the sake of completeness we included
a proof in the appendix, cf. Lemma A.2.

The second assumption relates the function $V^{1/p}$ to the distance-like
function $d$, where $V$ is from Assumption 3.9 and $p>2$.

######  Assumption 3.10.

There are constants $p>2$ and $K\in(0,\infty)$ such that for the function $V$
from Assumption 3.9 and any $y_{1},y_{2}\in\mathcal{Y}$ we have

$d(y_{1},y_{2})\leq K(V^{1/p}(y_{1})+V^{1/p}(y_{2})).$

######  Theorem 3.11.

Let $f\colon\mathcal{X}\to\mathbb{R}$ be a function such that
$h=h_{f}=f\cdot\mathds{1}_{\Phi}$ is Lipschitz with constant $L$ (w.r.t. $d$)
and let Assumptions 3.1, 3.9 and 3.10 hold true. If $|h|\leq V^{1/p}$, then
for any $\beta\in(0,\infty)$ and
$\operatorname{\varepsilon}>\max\left\\{0,\frac{1}{1+\beta}+\frac{1}{p}-\frac{1}{2}\right\\}$
we have

$\lim_{n\to\infty}\frac{1}{n^{1/2+\operatorname{\varepsilon}}}\left(\sum_{j=1}^{n}f(X_{j})-\nu(f)\right)=0,$

almost surely.

We establish two auxiliary lemmas before proving the main result.

######  Lemma 3.12.

Under the assumptions of Theorem 3.11 $(M_{n})_{n\in\mathbb{N}}$ is a
$(\mathcal{F}_{n})_{n\in\mathbb{N}}$ martingale and there exists some constant
$C_{1}\in(0,\infty)$ such that for any $j\in\mathbb{N}$ holds

$\mathbb{E}|\Delta_{j}|^{2}\leq C_{1}.$

###### Proof.

From Section 2 we know $\mathbb{E}[M_{n}|\mathcal{F}_{n-1}]=M_{n-1}$ almost
surely. Next we establish the claimed bound on the differences which implies
$\mathbb{E}|M_{n}|<\infty$ for any $n\in\mathbb{N}$, showing that
$(M_{n})_{n\in\mathbb{N}}$ is a martingale.

For arbitrary $j,k\in\mathbb{N}$ Proposition 3.3, our assumptions about the
Lyapunov functions $V$ and $V^{1/p}$ as well as the bound
$\pi_{\gamma}(V)\leq\frac{b}{1-\kappa}$, valid for any $\gamma\in\mathcal{I}$
(compare [Hai18, Proposition 4.24]), yield

$\displaystyle\mathbb{E}[|u_{\Gamma_{j}}(Y_{k})|^{p}]$
$\displaystyle\leq\left(\frac{k_{0}LM^{k_{0}}}{1-\tau}\right)^{p}\mathbb{E}[E_{\Gamma_{j}}(Y_{k})^{p}]$
$\displaystyle\leq\left(\frac{k_{0}LM^{k_{0}}}{1-\tau}\right)^{p}\mathbb{E}\left[\int_{\mathcal{X}}d(Y_{k},y)^{p}\pi_{\Gamma_{j}}({\rm
d}y)\right]$
$\displaystyle\leq\left(\frac{Kk_{0}LM^{k_{0}}}{1-\tau}\right)^{p}\mathbb{E}\left[\int_{\mathcal{X}}\left(V^{1/p}(Y_{k})+V^{1/p}(y)\right)^{p}\pi_{\Gamma_{j}}({\rm
d}y)\right]$
$\displaystyle\leq\left(\frac{2Kk_{0}LM^{k_{0}}}{1-\tau}\right)^{p}\mathbb{E}\left[\int_{\mathcal{X}}V(Y_{k})+V(y)\pi_{\Gamma_{j}}({\rm
d}y)\right]$
$\displaystyle\leq\left(\frac{2Kk_{0}LM^{k_{0}}}{1-\tau}\right)^{p}\left(\mathbb{E}\left[V(Y_{k})\right]+\frac{b}{1-\kappa}\right).$

Applying the Lyapunov property multiple times we see

$\displaystyle\mathbb{E}\left[V(Y_{k})\right]=\mathbb{E}[P_{\Gamma_{k_{j}}}V(Y_{k-1})]\leq\kappa\mathbb{E}[V(Y_{k-1})+b]\leq\cdots$
$\displaystyle\leq\kappa^{k}\mathbb{E}[V(Y_{0})]+\frac{b}{1-\kappa}\leq\mathbb{E}[V(Y_{0})]+\frac{b}{1-\kappa}=V(y_{0})+\frac{b}{1-\kappa}.$

Thus by an application of Jensen’s inequality we deduce that

$\displaystyle\mathbb{E}[$
$\displaystyle|u_{\Gamma_{j}}(Y_{k})|^{2}]\leq\mathbb{E}[|u_{\Gamma_{j}}(Y_{k})|^{p}]^{2/p}$
$\displaystyle\leq\left(\frac{2Kk_{0}LM^{k_{0}}}{1-\tau}\right)^{2}\left(V(y_{0})+\frac{2b}{1-\kappa}\right)^{2/p}<\infty.$

Employing Proposition 3.3 ii) we deduce

$|P_{\Gamma_{j}}u_{\Gamma_{j}}(Y_{k})|\leq|u_{\Gamma_{j}}(Y_{k})|+|V^{1/p}(Y_{k})|+\pi_{\Gamma_{j}}(|V^{1/p}|).$

Using the same arguments as above, one obtains a bound for
$|P_{\Gamma_{j}}u_{\Gamma_{j}}(Y_{k})|^{p}$ from which the desired result
follows. ∎

Next we bound the remainder term $R_{m}$ in a suitable way.

######  Lemma 3.13.

Under the assumptions of Theorem 3.11 there exists some constant
$C_{2}\in(0,\infty)$ such that for any $n\in\mathbb{N}$ and $m=m(n)$ holds

$\mathbb{E}|R_{m}|^{p}\leq m^{p}\cdot C_{2}.$

###### Proof.

From the proof of Lemma 3.12 we know

$\mathbb{E}[|u_{\Gamma_{j}}(Y_{k})|^{p}]\leq\left(\frac{2Kk_{0}LM^{k_{0}}}{1-\tau}\right)^{p}\left(V(y_{0})+\frac{2b}{1-\kappa}\right),$

for any $j,k\in\mathbb{N}$. Using this we can bound each summand of $R_{m}$
and the desired result follows. ∎

Now we turn to the proof of Theorem 3.11.

###### Proof of Theorem 3.11.

We show that all terms in the martingale decomposition (2.2) go to $0$ after
being normalised with $r(n)=n^{1/2+\operatorname{\varepsilon}}$.

We begin with the martingale part $M_{n}$ which has differences uniformly
bounded in $L^{2}$, see Lemma 3.12. Thus Lemma A.1 yields

$\lim_{n\to\infty}\frac{1}{r(n)}M_{n}=0$

almost surely.

To $R_{m}$: For any $\delta>0$ we have by Lemma 3.13 and (6) that

$\mathbb{P}\left[\left|R_{m}\right|>\delta
r(n)\right]\leq\frac{m^{p}}{r(n)^{p}}C_{2}\delta^{-p}\leq\frac{n^{p/(1+\beta)}}{r(n)^{p}}C_{2}(2c_{\beta})^{1/(1+\beta)}\delta^{-p}.$

By our assumptions on $\operatorname{\varepsilon}>0$ follows

$\sum_{n=1}^{\infty}\frac{n^{p/(1+\beta)}}{r(n)^{p}}=\sum_{n=1}^{\infty}\frac{n^{p/(1+\beta)}}{n^{p(1/2+\operatorname{\varepsilon})}}<\infty,$

such that $\lim_{n\to\infty}\frac{1}{r(n)}R_{m}=0$ almost surely by the Borel-
Cantelli lemma.

The remaining part that needs to be controlled is:

$\frac{1}{r(n)}\left(h(Y_{n})-h(Y_{0})+{u}_{\Gamma_{0}}(Y_{0})-{u}_{\Gamma_{n}}(Y_{n})\right).$

Using the estimate $\mathbb{E}|u_{\Gamma_{j}}(Y_{k})|^{p}\leq C$, which was
shown in Lemma 3.12 with some finite constant $C$, it follows by Markov’s
inequality and Borel-Cantelli lemma that

$\lim_{n\to\infty}\frac{1}{r(n)}{u}_{\Gamma_{0}}(Y_{0})=\lim_{n\to\infty}\frac{1}{r(n)}{u}_{\Gamma_{n}}(Y_{n})=0$

almost surely. Finally using the fact that $|h|\leq V^{1/p}\leq V$ and by
multiple applications of the Lyapunov condition we deduce the existence of
some $\widetilde{C}<\infty$ such that
$\mathbb{E}|h(Y_{k})|^{p}\leq\widetilde{C}$ for any $k\in\mathbb{N}_{0}$.
Another Borel-Cantelli argument finishes the proof. ∎

##  4 Examples

In this section we apply our theoretical results to three different settings,
depending on the ergodicity properties of the family
$\\{P_{\gamma}\\}_{\gamma\in\mathcal{I}}$. We start with the case where all
kernels are uniformly ergodic, then, we investigate an AIR scheme relying on
geometric ergodic kernels. Finally, we study algorithms where all the kernels
satisfy conditions similar to those shown in [HMS11].

###  4.1 Uniform Ergodicity

Here we use the trivial metric on $\mathcal{Y}$, i.e., we set
$d(y_{1},y_{2})=\mathds{1}_{\\{y_{1}\neq y_{2}\\}}$. The corresponding
$1$-Wasserstein distance $\mathcal{W}$ coincides with the total variation
distance, that is,

$\mathcal{W}(\mu_{1},\mu_{2})=\left\|\mu_{1}-\mu_{2}\right\|_{tv},$

for any two probability measures $\mu_{1},\mu_{2}$ on
$(\mathcal{Y},\mathcal{F}_{\mathcal{Y}})$.

To show our results we assume the following simultaneous uniform ergodicity
condition.

######  Assumption 4.1.

There exists some $\tau\in[0,1)$ such that for any $\gamma\in\mathcal{I}$,
$y\in\mathcal{Y}$ and $\ell\in\mathbb{N}$ holds

$\left\|P_{\gamma}^{\ell}(y,\cdot)-\pi_{\gamma}\right\|_{tv}\leq\tau^{\ell}.$

######  Remark 4.2.

It is well known from (adaptive) MCMC literature that one usually requires
compact state spaces to have uniform ergodicity. However, this still includes
different interesting examples, see e.g. [RR07] for a discussion.

######  Theorem 4.3.

Let $f\colon\mathcal{X}\to\mathbb{R}$ be bounded and Assumption 4.1 be true.
If $\beta\geq 1$, then for any $\operatorname{\varepsilon}>0$ we have that
almost surely

$\lim_{n\to\infty}\frac{1}{\sqrt{n}(\log
n)^{1/2+\operatorname{\varepsilon}}}\sum_{j=1}^{n}\left(f(X_{j})-\nu(f)\right)=0.$

###### Proof.

Assumption 3.1 is true since total variation distance and $\mathcal{W}$
coincide under the assumptions in this section. Moreover, by definition of
$E_{\gamma}(y)$ we have

$E_{\gamma}(y)=\int_{\mathcal{Y}}d(y,y^{\prime})\pi_{\gamma}({\rm
d}y^{\prime})\leq\int_{\mathcal{Y}}1\;\pi_{\gamma}({\rm d}y^{\prime})=1,$

for any $\gamma\in\mathcal{I}$. Hence also Assumption 3.4 is true. Since $f$
is bounded it is also Lipschitz w.r.t. $d$, so all requirements to apply
Theorem 3.5 are fulfilled. ∎

######  Corollary 4.4.

Let $f\colon\mathcal{X}\to\mathbb{R}$ be bounded, Assumption 4.1 be true,
$\beta\geq 1$ and $\operatorname{\varepsilon}>0$. Then, there exists some
(measurable) $C\colon\Omega\to\mathbb{R}$ such that for almost all
$\omega\in\Omega$ and any $n\in\mathbb{N}$ holds

$\left|\frac{1}{{n}}\sum_{j=1}^{n}\left(f(X_{j}(\omega))-\nu(f)\right)\right|\leq\frac{(\log
n)^{1/2+\operatorname{\varepsilon}}}{\sqrt{n}}C(\omega)$

We end this example by providing a result for $\beta\in(0,1)$. The proof is
along the lines of Theorem 4.3 and is therefore omitted. Moreover, a similar
statement as in Corollary 4.4 could be deduced.

######  Theorem 4.5.

Let $f\colon\mathcal{X}\to\mathbb{R}$ be bounded and Assumption 4.1 be true.
If $\beta\in(0,1)$, then for any
$\operatorname{\varepsilon}>\max\left\\{0,\frac{1}{1+\beta}-\frac{1}{2}\right\\}$
we have that almost surely

$\lim_{n\to\infty}\frac{1}{n^{1/2+\operatorname{\varepsilon}}}\sum_{j=1}^{n}\left(f(X_{j})-\nu(f)\right)=0.$

###  4.2 Geometric ergodicity

Next we study the case where we have a (simultaneous) drift and minorisation
condition. Assumptions of this form are standard in adaptive MCMC theory, e.g.
[AM06, RR07, FMP11, CŁR18]. Recall that there is some metric $d_{\mathcal{Y}}$
on $\mathcal{Y}$ such that the space is Polish. If we speak of (semi)
continuity we always refer to that metric.

######  Assumption 4.6.

There exists $C\in\mathcal{F}_{\mathcal{Y}}$, a lower semi-continuous
measurable function $V\colon\mathcal{Y}\to[1,\infty)$ as well as numbers
$\kappa\in(0,1)$, $\delta>0$, $b<\infty$ such that $\sup_{C}V=v<\infty$ and
for any $\gamma\in\mathcal{I}$ the following holds:

  * i)

$P_{\gamma}V\leq\kappa V+b$

  * ii)

There exists a probability measure $\eta_{\gamma}$ on $C$ such that for any
$x\in C$ we have $P_{\gamma}(x,\cdot)\geq\delta\eta_{\gamma}$.

With $V$ from Assumption 4.6 let us introduce the following family of lower
semi-continuous metrics

$d_{q}(y_{1},y_{2})=\mathds{1}_{\\{y_{1}\neq
y_{2}\\}}\left(V^{q}(y_{1})+V^{q}(y_{2})\right),$

where $q\in(0,1]$. The set of functions which are Lipschitz with constant $1$
w.r.t. $d_{q}$ is denoted by $\text{Lip}_{q,1}$ and the $1$-Wasserstein
distance corresponding to $d_{q}$ by $\mathcal{W}_{q}$.

######  Theorem 4.7.

Let Assumption 4.6 be true and $f\colon\mathcal{X}\to\mathbb{R}$ be such that
$h=h_{f}=\mathds{1}_{\Phi}f$ satisfies $|h|\leq V^{1/p}$, with $V$ from
Assumption 4.6 and some $p>2$. Then, for any $\beta>0$ and
$\operatorname{\varepsilon}>\max\left\\{0,\frac{1}{1+\beta}+\frac{1}{p}-\frac{1}{2}\right\\}$
holds almost surely

$\lim_{n\to\infty}\frac{1}{n^{1/2+\operatorname{\varepsilon}}}\sum_{j=1}^{n}\left(f(X_{i})-\nu(f)\right)=0.$

###### Proof.

Set $q=1/p\in(0,1]$. The goal is to apply Theorem 3.11 for the metric $d_{q}$.
In the same fashion as [HM11, Lemma 2.1] one can prove

$\mathcal{W}_{q}(\mu_{1},\mu_{2})=\sup_{|\psi|\leq
V^{q}}\left|\int_{\mathcal{Y}}\psi\;{\rm
d}\mu_{1}-\int_{\mathcal{Y}}\psi\;{\rm d}\mu_{2}\right|,$

where $V$ is the function from Assumption 4.6 and $\mu_{1},\mu_{2}$ are
arbitrary probability measures on $(\mathcal{Y},\mathcal{F}_{\mathcal{Y}})$.
Furthermore, Assumption 4.6 implies simultaneous geometric ergodicity (note
that also for $V^{q}$ we have drift and minorisation): There exist
$\alpha_{q}\in\mathbb{R}$ and $\tau_{q}\in(0,1)$ only depending on
$\delta,\kappa,b,v,q$ such that for any $\gamma\in\mathcal{I}$,
$y\in\mathcal{Y}$ holds

$\sup_{|\psi|\leq
V^{q}}\left|\int_{\mathcal{Y}}\psi(y^{\prime})P_{\gamma}^{\ell}(y,{\rm
d}y^{\prime})-\pi_{\gamma}(\psi)\right|\leq\alpha_{q}V^{q}(y)\tau_{q}^{\ell},$

for any $\ell\in\mathbb{N}$. This follows from [Bax05], see also [AM06,
Theorem 2] [RR07, Section 7].

Hence (by an application of triangle inequality) for any
$y_{1},y_{2}\in\mathcal{Y}$ and any $\ell\in\mathbb{N}$ holds

$\mathcal{W}_{q}(P^{\ell}_{\gamma}(y_{1},\cdot),P^{\ell}_{\gamma}(y_{2},\cdot))\leq\alpha_{q}\tau_{q}^{\ell}(V^{q}(y_{1})+V^{q}(y_{2})).$

This shows that Assumption 3.1 holds (for the metric $d_{q}$). Moreover,
Assumptions 3.9 and 3.10 (both w.r.t. the metric $d_{q}$) are immediate
consequences of Assumption 4.6.

Finally we note that since $|h|\leq V^{q}$ it follows that $h$ is Lipschitz
(with constant $1$) w.r.t. the metric $d_{q}$. Thus all requirements to apply
Theorem 3.11 are met and the claimed result follows. ∎

######  Corollary 4.8.

Under the assumptions of Theorem 3.11 for any $\beta\in(0,\infty)$ and
$\operatorname{\varepsilon}>\max\left\\{0,\frac{1}{1+\beta}+\frac{1}{p}-\frac{1}{2}\right\\}$
there exists some (measurable) $C\colon\Omega\to\mathbb{R}$ such that for
almost all $\omega\in\Omega$ and any $n\in\mathbb{N}$ holds

$\left|\frac{1}{{n}}\sum_{j=1}^{n}\left(f(X_{j}(\omega))-\nu(f)\right)\right|\leq\frac{1}{n^{1/2-\operatorname{\varepsilon}}}C(\omega).$

Clearly, we want to have $\operatorname{\varepsilon}>0$ as small as possible
in Corollary 4.8 to obtain a good rate of convergence. For fixed $\beta$ we
may choose $\operatorname{\varepsilon}$ small if $p$ is large. This can be
interpreted as follows: For large $p$ the requirement $|h|\leq V^{1/p}$ gets
more restrictive, such that “easier” functions are considered, leading to a
better rate.

###  4.3 Weak Harris ergodicity

In this section we investigate kernels which satisfy a contraction similar to
the one which one obtains from the weak Harris theorem of [HMS11]. For certain
(classical) MCMC algorithms contraction results, similar to those required in
this section, were shown in [HSV14, HJ23]. In the adaptive setting, however,
we are not aware of convergence results which are based on this kind of
contraction assumptions.

Throughout the section we assume that
$d\colon\mathcal{Y}\times\mathcal{Y}\to[0,1]$ is a distance-like function.

######  Assumption 4.9.

There exists a lower semi-continuous function
$V\colon\mathcal{Y}\to[1,\infty)$ which is a Lyapunov function for all
$P_{\gamma}$ simultaneously.

Following [HMS11] we define the following distance-like function(s)

$\widetilde{d}_{q}(y_{1},y_{2})=\sqrt{d(y_{1},y_{2})(1+V^{q}(y_{1})+V^{q}(y_{2}))},$

where $q\in(0,1)$ and $V$ is from Assumption 4.9. The corresponding
Wasserstein-like function is denoted by $\widetilde{\mathcal{W}}_{q}$ and the
contraction coefficient by $\widetilde{\tau}_{q}$. The second assumption we
need in this section is that one has a contraction for one of these distances.

######  Assumption 4.10.

There exist numbers $k_{0}\in\mathbb{N}$, $\tau\in[0,1)$, $M\in(0,\infty)$ and
$p>2$ such that for any $\gamma\in\mathcal{I}$ holds

$\widetilde{\tau}_{1/p}(P_{\gamma}^{k_{0}})\leq\tau\qquad\text{ and
}\qquad\widetilde{\tau}_{1/p}(P_{\gamma})\leq M.$

######  Remark 4.11.

A contraction w.r.t. $\widetilde{\mathcal{W}}_{q}$ can be checked by using the
weak Harris theorem of [HMS11]. It is worth noting that the mentioned result
is suitable for infinite dimensional settings, see also [HSV14, HJ23].

######  Theorem 4.12.

Let Assumptions 4.9 and 4.10 be true and $f\colon\mathcal{X}\to\mathbb{R}$ be
such that $h=f\cdot\mathds{1}_{\Phi}$ is Lipschitz w.r.t.
$\widetilde{d}_{1/p}$ and $|h|\leq V^{1/p}$, where $V,p$ are as in Assumption
4.10. Then, for any $\beta>0$ and
$\operatorname{\varepsilon}>\max\left\\{0,\frac{1}{1+\beta}+\frac{1}{p}-\frac{1}{2}\right\\}$
holds almost surely

$\lim_{n\to\infty}\frac{1}{n^{1/2+\operatorname{\varepsilon}}}\sum_{j=1}^{n}\left(f(X_{i})-\nu(f)\right)=0.$

###### Proof.

Clearly, Assumptions 3.1 and 3.9 are true. Moreover, by definition of
$\widetilde{d}_{1/p}$ also Assumption 3.10 (w.r.t. $\widetilde{d}_{1/p}$)
holds.

Since $h$ is Lipschitz and bounded by $V^{1/p}$ all requirements to apply
Theorem 3.11 are met and the result follows. ∎

Clearly, also a result similar to Corollary 4.8 can be deduced from the
previous result.

######  Remark 4.13.

A contraction for the pCN kernel w.r.t. $\widetilde{\mathcal{W}}_{q}$ was
shown in [HSV14], being valid for a tuning parameter between
$(0,\frac{1}{2}]$. Hence Theorem 4.12 may be useful to analyse the convergence
of an AIR version of that method. Ideas about the adaptation of a pCN
algorithm can be found in [Law14, HYL17], a detailed discussion, however, is
beyond the scope of this paper and is left for possible future research.

## Acknowledgements

The authors thank Mareike Hasenpflug for helpful discussions, in particular
related to questions of measurability.

JH and DR gratefully acknowledges support of the DFG within project 432680300
– SFB 1456 subproject B02. KL was supported by the Royal Society through the
Royal Society University Research Fellowship. GR is partly supported by the
following grants: EP/X028119/1, EP/R034710/1, EP/R018561 and EP/V009478/1
(EPSRC) and the OCEAN ERC synergy grant (UKRI). Major parts of this paper were
proven while JH was a visiting research fellow at the University of Warwick
supported by the Royal Society and SFB 1456.

##  Appendix A Auxiliary results

The upcoming limit results is shown in the iid case in [Kle14]. However, to
use it in this paper we adapted it to the martingale setting.

######  Lemma A.1.

Let $(\xi_{n})_{n\in\mathbb{N}}$ be a martingale (w.r.t. to some filtration
$(\mathcal{G}_{n})_{n\in\mathbb{N}}$) with differences uniformly bounded in
$L^{2}$. Then, for any $\operatorname{\varepsilon}>0$ we have
$\mathbb{P}$-almost surely

$\lim_{n\to\infty}\frac{1}{\sqrt{n}(\log
n)^{1/2+\operatorname{\varepsilon}}}\xi_{n}=0.$

###### Proof.

For any $n\in\mathbb{N}$ we set $k_{n}=2^{n}$ and
$v(n)=\sqrt{n}(\log(n))^{1/2+\delta}$. Clearly,
$\lim_{n\to\infty}v(k_{n+1})/v(k_{n})=\sqrt{2}$. For sufficiently large
$k\in\mathbb{N}$ and $n=n(k)\in\mathbb{N}$ with $k_{n-1}\leq k<k_{n}$ we have
that

$\frac{|\xi_{k}|}{v(k)}\leq\frac{2|\xi_{k}|}{v(k_{n})}\leq\frac{2\max_{1\leq
k\leq k_{n}}|\xi_{k}|}{v(k_{n})}.$ (9)

Denote by $\zeta_{j}$ the differences of $(\xi_{n})_{n\in\mathbb{N}}$, i.e.,
$\zeta_{j}=\xi_{j}-\xi_{j-1}$ and observe that for $j<m$ holds

$\mathbb{E}[\zeta_{j}\zeta_{m}]=\mathbb{E}\left[\mathbb{E}\left[\zeta_{j}\zeta_{m}|\mathcal{G}_{m-1}\right]\right]=\mathbb{E}\left[\zeta_{j}\mathbb{E}\left[\zeta_{m}|\mathcal{G}_{m-1}\right]\right]=0.$

Consequently, since the differences are uniformly bounded in $L^{2}$, there is
some $c\in\mathbb{R}$ such that

$\mathbb{E}\left[\xi_{k_{n}}^{2}\right]=\mathbb{E}\left[\sum_{j=1}^{k_{n}}\zeta_{j}^{2}+2\sum_{j<m}\zeta_{j}\zeta_{m}\right]=\mathbb{E}\left[\sum_{j=1}^{k_{n}}\zeta_{j}^{2}\right]\leq
k_{n}\cdot c.$

Let $\operatorname{\varepsilon}>0$ and define the sets

$A_{n}^{\operatorname{\varepsilon}}=\left\\{\max_{1\leq k\leq
k_{n}}|\xi_{k}|>\operatorname{\varepsilon}v(k_{n})\right\\}.$

An application of Doob’s maximal inequality yields that

$\mathbb{P}\left[A_{n}^{\operatorname{\varepsilon}}\right]\leq\frac{1}{\operatorname{\varepsilon}^{2}v(k_{n})^{2}}\mathbb{E}[\xi_{k_{n}}^{2}]\leq\frac{c}{\operatorname{\varepsilon}^{2}\log(k_{n})^{1+2\delta}}=\frac{c}{\operatorname{\varepsilon}^{2}\log(2)^{1+2\delta}n^{1+2\delta}}.$

Since $\delta>0$ the lemma of Borel-Cantelli yields that the right hand side
in (9) goes to zero which implies the result. ∎

The next result shows that if $V$ is a Lyapunov function also $V^{q}$ is such
a function whenever $q\in(0,1)$. This fact is well known and we include a
proof only to make the paper self contained.

######  Lemma A.2.

Under Assumption 3.9 for any $y\in\mathcal{Y}$, $\gamma\in\mathcal{I}$ and
$q\in(0,1]$ we have

$P_{\gamma}V^{q}(y)\leq\kappa^{q}V^{q}(y)+b^{q},$

with $\kappa,b$ as in Assumption 3.9.

###### Proof.

By Jensen’s inequality holds $(P_{\gamma}V^{q}(y))\leq(\kappa V(y)+b)^{q}$.
Since $q\leq 1$, subadditivity444A function
$\psi\colon\mathbb{R}\to\mathbb{R}$ is called subadditive if
$\psi(x+y)\leq\psi(x)+\psi(y)$ for any $x,y\in\mathbb{R}$. In particular,
subadditivity of $z\mapsto z^{q}$, with $q<1$ follows from §103 on p. 83 of
[HLP52]. of $z\mapsto z^{q}$ together with the above estimate implies

$P_{\gamma}V^{q}(y)\leq\left(\kappa
V(y)+b\right)^{q}\leq\kappa^{q}V^{q}(y)+b^{q}.$

∎

##  Appendix B Measurability of eccentricities

Here we provide a proof for the measurability of the
$E_{\Gamma_{j}}(Y_{k})$’s, with $j,k\in\mathbb{N}$. For the sake of brevity
fix $j,k\in\mathbb{N}$ and set $Y_{k}=Y$ as well as $\Gamma_{j}=\Gamma$.

We begin with showing that

$(\omega,\omega^{\prime})\mapsto\int_{\mathcal{Y}}\mathds{1}_{C}(\omega^{\prime},y)\pi_{\Gamma(\omega)}({\rm
d}y)$ (10)

is measurable for any
$C\in\mathcal{F}\times\mathcal{F}_{\mathcal{Y}}=:\mathcal{G}$, considered as
mapping from $(\Omega\times\Omega,\mathcal{F}\otimes\mathcal{F})$ to the
reals. Thereto we define the following set

$\mathscr{D}=\left\\{C\in\mathcal{G}\colon(\omega,\omega^{\prime})\mapsto\int_{\mathcal{Y}}\mathds{1}_{C}(\omega^{\prime},y)\pi_{\Gamma(\omega)}({\rm
d}y)\quad\text{is measurable}\right\\}.$

We note that $\mathscr{D}$ is a Dynkin system and sets of the form $A\times
B$, with $A\in\mathcal{F}$ and $B\in\mathcal{F}_{\mathcal{Y}}$, are contained
in $\mathscr{D}$. Now the claimed measurability relation of (10) follows by
the Dynkin system theorem.

The mapping $\omega\mapsto(\omega,Y(\omega))$ is measurable555Considered as a
mapping from $(\Omega,\mathcal{F})$ to
$(\Omega\times\Omega,\mathcal{F}\otimes\mathcal{F})$. so that for any
$C\in\mathcal{G}$ we have measurability of

$\omega\mapsto\int_{\mathcal{Y}}\mathds{1}_{C}(Y(\omega),\omega)\pi_{\Gamma(\omega)}({\rm
d}y).$

Since the distance-like function $d$ is measurable (recall that it is lower
semi-continuous) we deduce the measurability of $E_{\Gamma}(Y)$ by the usual
approximation arguments with simple functions.

## References

  * [AA07] Christophe Andrieu and Yves F. Atchadé, _On the efficiency of adaptive MCMC algorithms_ , Electronic Communications in Probability 12 (2007), 336 – 349.
  * [AF10] Yves F. Atchadé and Gersende Fort, _Limit theorems for some adaptive MCMC algorithms with subgeometric kernels_ , Bernoulli 16 (2010), no. 1, 116–154.
  * [AF12] , _Limit theorems for some adaptive MCMC algorithms with subgeometric kernels: Part II_ , Bernoulli 18 (2012), no. 3, 975–1001.
  * [ALR13] Yves F. Atchadé, Nicolas Lartillot, and Christian Robert, _Bayesian computation for statistical models with intractable normalizing constants_ , Brazilian Journal of Probability and Statistics 27 (2013), no. 4, 416–436.
  * [AM06] Christophe Andrieu and Éric Moulines, _On the ergodicity properties of some adaptive MCMC algorithms_ , The Annals of Applied Probability 16 (2006), no. 3, 1462–1505.
  * [AR05] Yves F. Atchadé and Jeffrey S. Rosenthal, _On adaptive Markov chain Monte Carlo algorithms_ , Bernoulli 11 (2005), no. 5, 815–828.
  * [Bax05] Peter H. Baxendale, _Renewal theory and computable convergence rates for geometrically ergodic Markov chains_ , The Annals of Applied Probability 15 (2005), no. 1B, 700 – 738.
  * [CŁR18] Cyril Chimisov, Krzysztof Łatuszyński, and Gareth O. Roberts, _Air Markov chain Monte Carlo_ , arXiv preprint arXiv:1801.09309, 2018.
  * [DMPS18] Randal Douc, Éric Moulines, Pierre Priouret, and Philippe Soulier, _Markov chains_ , Springer, 2018.
  * [Dob96] Roland L. Dobrushin, _Lectures on probability theory and statistics: Ecole d’eté de probabilités de Saint-Flour XXIV—1994_ , ch. Perturbation methods of the theory of Gibbsian fields, pp. 1–66, Springer Berlin Heidelberg, Berlin, Heidelberg, 1996.
  * [FMP11] Gersende Fort, Éric Moulines, and Pierre Priouret, _Convergence of adaptive and interacting Markov chain Monte Carlo algorithms_ , The Annals of Statistics 39 (2011), no. 6, 3262–3289.
  * [GO02] Peter W. Glynn and Dirk Ormoneit, _Hoeffding’s inequality for uniformly ergodic Markov chains_ , Statistics & Probability Letters 56 (2002), no. 2, 143–146.
  * [Hai18] Martin Hairer, _Ergodic properties of Markov processes_ , Lecture notes (2018).
  * [HJ23] Bamdad Hosseini and James E. Johndrow, _Spectral gaps and error estimates for infinite-dimensional Metropolis–Hastings with non-Gaussian priors_ , The Annals of Applied Probability 33 (2023), no. 3, 1827 – 1873.
  * [HLP52] Godfrey H. Hardy, John E. Littlewood, and George Pólya, _Inequalities_ , second ed., Cambridge University Press, 1952.
  * [HM11] Martin Hairer and Jonathan C. Mattingly, _Yet another look at Harris’ ergodic theorem for Markov Chains_ , Seminar on Stochastic Analysis, Random Fields and Applications VI (Basel) (Robert Dalang, Marco Dozzi, and Francesco Russo, eds.), Springer Basel, 2011, pp. 109–117.
  * [HMS11] Martin Hairer, Jonathan C Mattingly, and Michael Scheutzow, _Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations_ , Probability Theory and Related Fields 149 (2011), 223–259.
  * [HST01] Heikki Haario, Eero Saksman, and Johanna Tamminen, _An adaptive Metropolis algorithm_ , Bernoulli 7 (2001), no. 2, 223–242.
  * [HSV14] Martin Hairer, Andrew M. Stuart, and Sebastian J. Vollmer, _Spectral gaps for a Metropolis–Hastings algorithm in infinite dimensions_ , The Annals of Applied Probability 24 (2014), no. 6, 2455 – 2490.
  * [HYL17] Zixi Hu, Zhewei Yao, and Jinglai Li, _On an adaptive preconditioned Crank–Nicolson MCMC algorithm for infinite dimensional Bayesian inference_ , Journal of Computational Physics 332 (2017), 492–503.
  * [JO10] Aldéric Joulin and Yann Ollivier, _Curvature, concentration and error estimates for Markov chain Monte Carlo_ , The Annals of Probability 38 (2010), no. 6, 2418 – 2442.
  * [Kle14] Achim Klenke, _Probability theory: a comprehensive course_ , second ed., Springer Science & Business Media, 2014.
  * [Law14] Kody J. H. Law, _Proposals which speed up function-space MCMC_ , Journal of Computational and Applied Mathematics 262 (2014), 127–138, Selected Papers from NUMDIFF-13.
  * [ŁMN13] Krzysztof Łatuszyński, Błażej Miasojedow, and Wojciech Niemiro, _Nonasymptotic bounds on the estimation error of MCMC algorithms_ , Bernoulli 19 (2013), no. 5A, 2033–2066.
  * [ŁR14] Krzysztof Łatuszyński and Jeffrey S. Rosenthal, _The contianment condition and adapfail algorithms_ , Journal of Applied Probability 51 (2014), no. 4, 1189–1195.
  * [Mia14] Błażej Miasojedow, _Hoeffding’s inequalities for geometrically ergodic Markov chains on general state space_ , Statistics & Probability Letters 87 (2014), 115–120.
  * [MT12] Sean P. Meyn and Richard L. Tweedie, _Markov chains and stochastic stability_ , second ed., Cambridge University Press, 2012.
  * [PHŁ20] Emilia Pompe, Chris Holmes, and Krzysztof Łatuszyński, _A framework for adaptive MCMC targeting multimodal distributions_ , The Annals of Statistics 48 (2020), no. 5, 2930 – 2952.
  * [RR07] Gareth O Roberts and Jeffrey S Rosenthal, _Coupling and ergodicity of adaptive Markov chain Monte Carlo algorithms_ , Journal of Applied Probability 44 (2007), no. 2, 458–475.
  * [Rud12] Daniel Rudolf, _Explicit error bounds for Markov chain Monte Carlo_ , Dissertationes Math. 485 (2012).
  * [SV10] Eero Saksman and Matti Vihola, _On the ergodicity of the adaptive Metropolis algorithm on unbounded domains_ , Annals of Applied Probability 20 (2010), no. 6, 2178–2203.

Authors’ addresses:

Julian Hofstadler, Universität Passau, Faculty for Computer Science and
Mathematics, Innstraße 33, 94032 Passau, Germany.

Email<EMAIL_ADDRESS>
Krzysztof Łatuszyński, University of Warwick, Department of Statistics, CV47AL
Coventry, United Kingdom.

Email<EMAIL_ADDRESS>
Gareth O. Roberts, University of Warwick, Department of Statistics, CV47AL
Coventry, United Kingdom.

Email<EMAIL_ADDRESS>
Daniel Rudolf, Universität Passau, Faculty for Computer Science and
Mathematics, Innstraße 33, 94032 Passau, Germany.

Email<EMAIL_ADDRESS>