11footnotetext: Université de Lorraine, CNRS, Inria, IECL, F-54000 Nancy,
France
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Quasi-limiting estimates for periodic absorbed Markov chains

Nicolas Champagnat1, Denis Villemonais1

###### Abstract

We consider periodic Markov chains with absorption. Applying to iterates of
this periodic Markov chain criteria for the exponential convergence of
conditional distributions of aperiodic absorbed Markov chains, we obtain
exponential estimates for the periodic asymptotic behavior of the semigroup of
the Markov chain. This implies in particular the exponential convergence in
total variation of the conditional distribution of the Markov chain given non-
absorption to a periodic sequence of limit measures and we characterize the
cases where this sequence is constant, which corresponds to the cases where
the conditional distributions converge to a quasi-stationary distribution. We
also characterize the first two eignevalues of the semigroup and give a bound
for the spectral gap between these eigenvalues and the next ones. Finally, we
give ergodicity estimates in total variation for the Markov chain conditioned
to never be absorbed, often called $Q$-process, and quasi-ergodicity estimates
for the original Markov chain.

Keywords: Markov chains with absorption; periodic Markov chains; quasi-
stationary distribution; mixing property; exponential forgetting; $Q$-process;
quasi-ergodicity.

2010 Mathematics Subject Classification. 37A25, 60B10, 60F99, 60J05.

##  1 Introduction

Let $(X_{t},t\in\mathbb{Z}_{+})$ be a Markov chain in $E\cup\\{\partial\\}$
where $E$ is a measurable space, $\partial\not\in E$ and
$\mathbb{Z}_{+}:=\\{0,1,\ldots\\}$. For all $x\in E\cup\\{\partial\\}$, we
denote as usual by $\mathbb{P}_{x}$ the law of $X$ given $X_{0}=x$ and for any
probability measure $\mu$ on $E\cup\\{\partial\\}$, we define
$\mathbb{P}_{\mu}=\int_{E\cup\\{\partial\\}}\mathbb{P}_{x}\,\mu(dx)$. We also
denote by $\mathbb{E}_{x}$ and $\mathbb{E}_{\mu}$ the associated expectations.
We assume that $\partial$ is absorbing, which means that $X_{t}=\partial$ for
all $t\geq\tau_{\partial}$, $\mathbb{P}_{x}$-almost surely, where

$\tau_{\partial}=\inf\left\\{t\in I,\,X_{t}=\partial\right\\}.$

We study the sub-Markovian transition semigroup of $X$ in $E$,
$(P_{n})_{n\in\mathbb{Z}_{+}}$, defined as

$\displaystyle
P_{n}f(x)=\mathbb{E}_{x}\left(f(X_{n})\mathbbm{1}_{n<\tau_{\partial}}\right),\
\forall n\in\mathbb{Z}_{+},$

for all bounded or nonnegative measurable function $f$ on $E$ and all $x\in
E$. We also define as usual the left-action of $P_{n}$ on measures as

$\mu
P_{n}f=\mathbb{E}_{\mu}\left(f(X_{n})\mathbbm{1}_{n<\tau_{\partial}}\right)=\int_{E}P_{n}f(x)\,\mu(dx),$

for all positive measure $\mu$ on $E$ and all bounded measurable $f$.

Many references (see for example [14, 5, 6, 9, 13, 17, 1, 15, 16, 18, 2])
provide criteria allowing to characterize the asymptotic behavior of the
semigroup $(P_{n})_{n\in\mathbb{Z}_{+}}$ in the following form: there exist a
measurable function $V:E\rightarrow[1,+\infty)$, constants
$\theta_{0}\in(0,1]$, $\alpha\in(0,1)$ and $C\in\mathbb{R}_{+}$, a probability
measure $\nu_{\text{QS}}$ on $E$ such that $\nu_{\text{QS}}(V)<+\infty$ and a
measurable function $\eta:E\rightarrow\mathbb{R}_{+}$ non-identically zero
such that $\eta/V$ is bounded and $\nu_{\text{QS}}(\eta)>0$, such that

$\left|\theta_{0}^{-n}P_{n}f(x)-\eta(x)\nu_{\text{QS}}(f)\right|\leq
C\alpha^{n}V(x)$ (1.1)

for all measurable function $f:E\rightarrow\mathbb{R}$ such that $|f|\leq V$,
all $n\in\mathbb{Z}_{+}$ and all $x\in E$.

All the previously cited references assume some property of aperiodicity for
the process $X$, and (1.1) itself implies a weak form of aperiodicity that can
be formulated as follows: for all $x\in E$ such that $\eta(x)>0$ and all
measurable $A\subset E$ such that $\nu_{\text{QS}}(A)>0$,
$P_{n}\mathbbm{1}_{A}(x)>0$ for all $n$ large enough. The purpose of this note
is to explain how (1.1) should be modified for periodic Markov chains, and to
examine the implications of this modified property for quasi-limiting
estimates, spectral properties of the semigroup, ergodicity of the $Q$-process
and quasi-ergodicity of the Markov chain $X$. In the rest of this
introduction, we shall recall all these properties when (1.1) holds true.

The property (1.1) implies that, for all $x\in E$ such that $\eta(x)>0$ and
all measurable $A\subset E$,

$\left|\frac{P_{n}\mathbbm{1}_{A}(x)}{P_{n}\mathbbm{1}_{E}(x)}-\nu_{\text{QS}}(A)\right|\xrightarrow[n\rightarrow+\infty]{}0,$

which means that $\nu_{\text{QS}}$ is a _quasi-limiting distribution_ for the
process $X$ (see e.g. [19]), and thus a _quasi-stationary distribution_ (QSD)
of $X$ (see again [19]), i.e. a probability measure $\nu$ such that

$\mathbb{P}_{\nu}(X_{t}\in\cdot\mid t<\tau_{\partial})=\nu(\cdot),\quad\forall
t\in\mathbb{Z}_{+}.$

In addition, the _absorption rate_ of the QSD $\nu_{\text{QS}}$ is
$\theta_{0}$, i.e. $\mathbb{P}_{\nu_{\text{QS}}}(\tau_{\partial}\geq
n)=\theta_{0}^{n}$ (see again [19]), and $\nu_{\text{QS}}(\eta)=1$. It is also
easy to see that the measure $\nu_{\text{QS}}$ is the unique QSD satisfying
$\nu_{\text{QS}}(\eta)>0$ and $\nu_{\text{QS}}(V)<+\infty$ and that the
function $\eta$ is an eigenfunction of the semigroup, since
$P_{n}\eta=\theta_{0}^{n}\eta$ for all $n\geq 0$. After giving the setting of
our results on periodic processes and basic properties of their QSDs in
Section 2, the quasi-limiting properties recalled above are extended to the
periodic case in Section 3 below. Of course, exponential convergence of the
conditional distributions does not hold in general because of the periodicity
of the process [12, 23, 24, 20], yet we are able to obtain periodic
exponential estimates in total variation.

The property (1.1) also has consequences on the spectrum of the semigroup
$(P_{n})_{n\geq 0}$: given a nonzero function
$h:E\cup\\{\partial\\}\rightarrow\mathbb{C}$ such that $h/V$ is bounded and
$\lambda h(x)=\mathbb{E}_{x}(h(X_{1}))$ for all $x\in E\cup\\{\partial\\}$ for
some $\lambda\in\mathbb{C}$,

  * •

either $h(\partial)\neq 0$, and then $\lambda=1$ and
$P_{1}(h-h(\partial))(x)=h(x)-h(\partial)$ for all $x\in E$, so it follows
from (1.1) that the function $h$ is constant,

  * •

or $h(\partial)=0$ and

    * –

either $\nu_{\text{QS}}(h)\neq 0$ and then it follows from (1.1) that
$\lambda=\theta_{0}$ and $h=\nu_{\text{QS}}(h)\eta$,

    * –

or $\nu_{\text{QS}}(h)=0$ and then it follows from (1.1) that
$|\lambda|\leq\alpha\theta_{0}$.

This last property quantifies the spectral gap of the operator $P_{1}$. These
spectral properties are extended to the periodic case in Section 4 below.

In addition, assuming that the Markov chain $(X_{n})_{n\geq 0}$ is an adapted
stochastic process defined on a filtered probability space
$(\Omega,\\{\mathcal{F}_{m}\\}_{m\geq 0},\\{\mathbb{P}_{x}\\}_{x\in
E\cup\\{\partial\\}})$, it follows from [10] and [7] that, under (1.1), for
all $x\in E^{\prime}$ where $E^{\prime}:=\\{y\in E:\eta(x)>0\\}$, the limit

$\mathbb{Q}_{x}(A)=\lim_{n\rightarrow+\infty}\mathbb{P}_{x}(A\mid
n<\tau_{\partial})$

for all $A\in\mathcal{F}_{m}$ with $m\in\mathbb{N}$, defines a probability
measure on $\Omega$ under which the process $(\Omega,({\cal F}_{m})_{m\geq
0},(X_{n})_{n\geq 0},(\mathbb{Q}_{x})_{x\in E^{\prime}})$ is an
$E^{\prime}$-valued homogeneous Markov chain called the $Q$-process. In
addition, this process is exponentially ergodic with unique invariant measure
$\eta\,d\nu_{\text{QS}}$ in the following sense: for all measurable
$f:E^{\prime}\rightarrow\mathbb{R}$ such that $|f|\leq V/\eta$, for all $x\in
E^{\prime}$,

$\left|\mathbb{E}_{\mathbb{Q}_{x}}(f(X_{n}))-\int_{E^{\prime}}f(x)\eta(x)\nu_{\text{QS}}(dx)\right|\leq
C^{\prime}\alpha^{n}\frac{V(x)}{\eta(x)}$

for constants $C^{\prime}\in\mathbb{R}_{+}$ and $\alpha\in(0,1)$ as in (1.1).
These properties of the $Q$-process are extended to the periodic case in
Section 5 below.

To conclude, (1.1) also implies the following quasi-ergodicity property, as
shown in [21] [ou autre référence de William ? Le papier sur (E’) ?] (see also
[4, 8]): for all measurable function $f:E\rightarrow[-1,1]$ and all $x\in E$
and $n\geq 1$,

$\left|\mathbb{E}_{x}\left[\left.\frac{1}{n}\sum_{k=0}^{n}f(X_{k})\ \right|\
n<\tau_{\partial}\right]-\int_{E}f(x)\eta(x)\nu_{\text{QS}}(dx)\right|\leq\frac{C^{\prime\prime}}{n}$

for some constant $C^{\prime\prime}\in\mathbb{R}_{+}$ independent on $n$. This
quasi-ergodic property is extended to the periodic case and improved in
Section 6 below.

######  Remark 1.

All the results of this note can be easily extended to general unbounded semi-
groups (i.e. not necessarily sub-Markov) following the same approach as in
[9]. We restrict the presentation to sub-Markov semigroups to make simpler the
probabilistic interpretation of our results.

##  2 General assumptions and first properties of quasi-stationary
distributions for periodic Markov chains

In all the sequel, we shall make the standing assumption that $E$ is the
disjoint union of measurable sets $A_{0},\ldots,A_{t-1}$ such that for all
$x\in A_{i}$,

$P_{1}\mathbbm{1}_{A_{j}}(x)=\begin{cases}0&\text{if }j\neq i+1,\\\
P_{1}\mathbbm{1}_{E}(x)&\text{if }j=i+1,\end{cases}$ (2.1)

with the convention $i+1=0$ if $i=t-1$. Then $P$ is $t$ periodic and
$Q_{n}=P_{nt}$ for all $n\geq 0$ defines a sub-Markov semi-group on the set of
bounded measurable function on $A_{0}$.

######  Remark 2.

The last assumption corresponds to a particular case of periodicity, which
assumes a (weak) form of irreducibility. Other periodic situations may occur
when the state space is not irreducible and different periods may exist in
different irreducibility classes, or when the process becomes periodic after
leaving a transient set. In such situations, the results of this not may be
extended combining our arguments with e.g. those of [11].

We first observe that, under the above conditions, there is a one-to-one
correspondance between QSDs for $Q$ and $P$.

######  Proposition 2.1.

If $\nu_{\text{QS}}$ is a QSD for $P$ with absorption rate $\theta>0$, then
$\nu:=\frac{\nu_{\text{QS}}(\cdot\cap A_{0})}{\nu_{\text{QS}}(A_{0})}$ is a
QSD for $Q$ with absorption rate $\theta^{t}$.
Conversely, if a probability measure $\nu$ on $A_{0}$ is a QSD for $Q$ with
absorption rate $\theta$, then the probability measure on $E$ defined as

$\nu_{\text{QS}}=\frac{1}{\sum_{i=0}^{t-1}\theta^{-i/t}\,\nu
P_{i}\mathbbm{1}_{E}}\sum_{i=0}^{t-1}\theta^{-i/t}\,\nu P_{i}$ (2.2)

is a QSD for $P$ with absorption rate $\theta^{1/t}$, and it is the only one
such that $\nu=\frac{\nu_{\text{QS}}(\cdot\cap
A_{0})}{\nu_{\text{QS}}(A_{0})}$.

###### Proof.

Observing that, if $\nu_{\text{QS}}$ is a QSD for $P$, then
$\nu_{\text{QS}}(A_{0})>0$ (since otherwise $\nu_{\text{QS}}(A_{i})$ would be
zero for all $i$ using the QSD property), the first statement is clear. For
the second statement, we first notice that, if $\nu$ is a QSD for $Q$, then it
is clear that $\nu_{\text{QS}}$ as defined in (2.2) is a QSD for $P$. So it
only remains to check the uniqueness of this QSD. Let $\mu$ be any QSD for $P$
such that $\nu=\frac{\mu(\cdot\cap A_{0})}{\mu(A_{0})}$. It follows from the
first part of Proposition 2.1 that its absorption rate is $\theta^{1/t}$. It
then follows from (2.1) that, for all $i\in\\{0,\ldots,t-1\\}$,

$\theta^{i/t}\mu(\cdot\cap A_{i})=\mu P_{i}(\cdot\cap A_{i})=\mu(\cdot\cap
A_{0})P_{i}=\mu(A_{0})\,\nu P_{i}.$

Therefore,

$\mu=\mu(A_{0})\sum_{i=0}^{t-1}\theta^{-i/t}\,\nu P_{i},$

which entails (2.2). ∎

In the next result, we observe that the one-to-one correspondance between QSDs
for $(P_{n})_{n\in\mathbb{Z}_{+}}$ and $(Q_{k})_{k\in\mathbb{Z}_{+}}$ does not
extend to a one-to-one correspondance to the QSDs for
$(P_{n})_{n\in\mathbb{Z}_{+}}$ and $(P_{kt})_{k\in\mathbb{Z}_{+}}$, which will
explain the periodic asymptotic behaviors observed in the results of the next
section.

######  Proposition 2.2.

If $\nu$ is a QSD for $(Q_{k})_{k\in\mathbb{Z}_{+}}$, then all convex
combinations of

$\nu,\ \frac{\nu P_{1}}{\nu P_{1}\mathbbm{1}_{E}},\ldots,\ \frac{\nu
P_{t-1}}{\nu P_{t-1}\mathbbm{1}_{E}}$ (2.3)

are QSDs for $(P_{kt})_{k\in\mathbb{Z}_{+}}$. Among them, only a single one is
a QSD for $(P_{n})_{n\in\mathbb{Z}_{+}}$.

###### Proof.

It is clear that all the probability measures in (2.3) are QSDs for
$(P_{kt})_{k\in\mathbb{Z}_{+}}$ with the same absorption rate, so all convex
combinations of these measures are QSDs. The unique one among them which is a
QSD for $(P_{n})_{n\in\mathbb{Z}_{+}}$ is given by (2.2), by Proposition 2.1.
∎

In the following sections, we shall assume in addition that $Q$ satisfies the
property (1.1) on $A_{0}$, i.e. there exist a measurable function
$V:A_{0}\rightarrow[1,+\infty)$, constants $\theta_{0}\in(0,1]$,
$C\in\mathbb{R}_{+}$ and $\alpha\in(0.1)$, a nonzero measurable function
$\eta:A_{0}\rightarrow\mathbb{R}_{+}$ such that $\eta/V$ is bounded and a
probability measure $\nu$ on $A_{0}$ such that $\nu(V)<+\infty$ and
$\nu(\eta)>0$, such that

$\left|\theta_{0}^{-kt}Q_{k}f(x)-\eta(x)\nu(f)\right|\leq C_{Q}\alpha^{k}V(x)$
(2.4)

for all mesurable function $f$ on $A_{0}$ such that $|f|\leq V$, all $k\geq 0$
and all $x\in A_{0}$. Note that the absorption rate $\theta_{0}$ in (1.1) has
been modified above as $\theta_{0}^{t}$, so that the QSD $\nu_{\text{QS}}$
defined in (2.2) has absorption rate $\theta_{0}$ instead of
$\theta_{0}^{1/t}$. Note also that the last inequality implies, setting $k=1$,
that, for all $x\in A_{0}$,

$Q_{1}V(x)\leq\left(\|\eta/V\|_{\infty}\nu(V)+C_{Q}\alpha\right)V(x).$ (2.5)

##  3 Quasi-limiting behavior of periodic Markov chains under (2.4)

Our first results extends (1.1) to periodic Markov chains. Let us define

$\mathcal{B}_{V}=\left\\{f:E\rightarrow\mathbb{R}\text{ measurable, s.t.\
}\forall i\in\\{0,\ldots,t-1\\},\ |P_{i}(f\mathbbm{1}_{A_{i}})|\leq V\text{ on
}A_{0}\right\\}.$

Note that this set is non-empty since, by (2.5), it contains all functions of
the form $P_{j}g$ for any $j\in\\{0,\ldots,t-1\\}$ and any function $g$ such
that $|g|\leq\frac{V}{\|\eta/V\|_{\infty}\nu(V)+C\alpha}$ on $A_{0}$ and
$g\equiv 0$ on $E\setminus A_{0}$. Since $V\geq 1$ and $P$ is sub-Markov, it
also contains all measurable functions bounded by $1$.

######  Theorem 3.1.

Assume that $P$ satisfies (2.1) and that $Q$ satisfies (2.4). Then, there
exists $C<+\infty$ such that, for all $f\in\mathcal{B}_{V}$, $n\geq 1$,
$j\in\\{0,\ldots,t-1\\}$ and $x\in A_{k}$ for some $k\in\\{0,\ldots,t-1\\}$,

$\left|\theta_{0}^{-(nt+j)}P_{nt+j}f(x)-\theta_{0}^{-(t+j)}P_{t-k}\eta(x)\,\nu
P_{k+j}f\right|\leq C^{\prime}_{Q}\alpha^{n}P_{t-k}V(x),$ (3.1)

where we extended $\nu$ by 0 to $E\setminus A_{0}$, where the constant
$\alpha\in(0,1)$ is the same as in (2.4) and where

$C^{\prime}_{Q}=C_{Q}\theta_{0}^{-2t}\left(\|\eta/V\|_{\infty}\nu(V)+C_{Q}\alpha\right)^{2}.$

###### Proof.

Let $f\in\mathcal{B}_{V}$, $n\geq 1$ and $j\in\\{0,\ldots,t-1\\}$. We have

$\left|\theta_{0}^{-(nt+j)}P_{nt+j}f-\theta_{0}^{-(t+j)}\sum_{i=0}^{t-1}P_{t+j-i}\eta\,\nu
P_{i}f\right|\\\ \begin{aligned}
&\leq\theta_{0}^{-(t+j)}\sum_{i=0}^{t-1}\left|\theta_{0}^{-(n-1)t}P_{t+j-i}P_{(n-1)t}P_{i}(f\mathbbm{1}_{A_{i}})-P_{t+j-i}\eta\,\nu
P_{i}(f\mathbbm{1}_{A_{i}})\right|\\\
&\leq\theta_{0}^{-(t+j)}\sum_{i=0}^{t-1}P_{t+j-i}\left|\theta_{0}^{-(n-1)t}Q_{n-1}P_{i}(f\mathbbm{1}_{A_{i}})-\nu
P_{i}(f\mathbbm{1}_{A_{i}})\,\eta\right|\\\ &\leq
C_{Q}\theta_{0}^{-(t+j)}\,\alpha^{n-1}\sum_{i=0}^{t-1}P_{t+j-i}V.\end{aligned}$

Now, given $x\in A_{k}$ for some $k\in\\{0,\ldots,t-1\\}$, we observe that
only a single term in the sum in the left-hand side of the last equation is
nonzero, corresponding to the unique index $i\in\\{0,1,\ldots,t-1\\}$ such
that $j-i=-k$, $j-i=t-k$ or $j-i=2t-k$. In the first case, this term is
$P_{t-k}\eta(x)\,\nu P_{j+k}f$; in the second case, it is

$\displaystyle P_{2t-k}\eta(x)\,\nu P_{j+k-t}f$
$\displaystyle=P_{t-k}Q_{1}\eta(x)\,\nu P_{j+k-t}(f\mathbbm{1}_{A_{j+k-t}})$
$\displaystyle=P_{t-k}\eta(x)\,\theta_{0}^{t}\nu
P_{j+k-t}(f\mathbbm{1}_{A_{j+k-t}})$ $\displaystyle=P_{t-k}\eta(x)\,\nu
Q_{1}P_{j+k-t}(f\mathbbm{1}_{A_{j+k-t}})=P_{t-k}\eta(x)\,\nu P_{j+k}f;$

and the third case can be handled similarly. So (3.1) is proved using (2.5). ∎

As in the aperiodic case, the last result implies geometric estimates in
$V$-weighted total variation for the long time behavior of the conditional
distributions of the Markov chain given non-absorption.

######  Theorem 3.2.

Under the assumptions of Theorem 3.1, there exist constants $C<+\infty$ and
$\bar{\alpha}\in(0,\alpha)$ and a nonzero measurable function
$\varphi_{2}:A_{0}\to[0,1]$ such that $\varphi_{2}/\eta$ is bounded, such
that, for all probability measure $\mu$ on $E$, all $f\in\mathcal{B}_{V}$, all
$n\in\mathbb{Z}_{+}$ and all $j\in\\{0,1,\ldots,t-1\\}$,

$\displaystyle\left|\frac{\mu P_{nt+j}f}{\mu
P_{nt+j}\mathbbm{1}_{E}}-\frac{\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i+j}f}{\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i+j}\mathbbm{1}_{E}}\right|$ $\displaystyle\leq
C\,\bar{\alpha}^{n}\frac{\sum_{i=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}V}{\sum_{i=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\varphi_{2}}.$ (3.2)

If in addition $n$ is large enough so that

$C^{\prime}_{Q}\theta_{0}^{-4t}\alpha^{n}\frac{\sum_{i=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}V}{\sum_{i=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta}\leq\frac{1}{2},$
(3.3)

where the constants $C_{Q}$ and $\alpha$ are those from (2.4), there exists a
constant $C^{\prime}$ independent of $n$ and $\mu$ such that, for all
probability measure $\mu$ on $E$, all $f\in\mathcal{B}_{V}$ and all
$j\in\\{0,1,\ldots,t-1\\}$,

$\displaystyle\left|\frac{\mu P_{nt+j}f}{\mu
P_{nt+j}\mathbbm{1}_{E}}-\frac{\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i+j}f}{\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i+j}\mathbbm{1}_{E}}\right|$ $\displaystyle\leq
C^{\prime}\,\alpha^{n}\frac{\sum_{i=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}V}{\sum_{i=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta}.$ (3.4)

######  Remark 3.

As will appear in the proof below, the function $\varphi_{2}$ in the last
result can be chosen positive in a large part of the support of $\eta$. More
precisely, for any fixed $\varepsilon>0$, the function $\varphi_{2}$ can be
chosen such that

$\inf\left\\{\varphi_{2}(x),x\in E,\eta(x)\geq\varepsilon,V(x)\leq
1/\varepsilon\right\\}>0.$

###### Proof.

Let us first construct the function $\varphi_{2}$. Let $\theta_{2}\in(0,1)$ be
such that

$\left(\frac{\theta_{0}}{\theta_{2}}\right)^{t}\alpha<1$

and let $\varepsilon>0$ be small enough so that $\nu(K)\geq 1/2$, where

$K:=\left\\{\eta\geq\varepsilon,\,V\leq\frac{1}{\varepsilon}\right\\}.$

By (2.4), there exists $n_{0}\in\mathbb{N}$ such that,

$\inf_{x\in K}\theta_{2}^{-n_{0}t}\mathbb{P}_{x}(X_{n_{0}}\in K)\geq 1.$

Set for all $x\in K$

$\varphi_{2}(x):=\frac{\theta_{2}^{-t}-1}{\theta_{2}^{-n_{0}t}-1}\sum_{k=0}^{n_{0}-1}\theta_{2}^{-kt}Q_{k}\mathbbm{1}_{K}(x).$

It is then easy to check (cf. e.g. Lemma 3.?? in [6]) that, for all $x\in
A_{0}$,

$Q_{1}\varphi_{2}(x)\geq\theta_{2}^{t}\varphi_{2}(x).$ (3.5)

In addition, since $\mathbbm{1}_{K}\leq\eta/\varepsilon$ and
$Q_{1}\eta=\theta_{0}\eta$,

$\varphi_{2}(x)\leq\frac{\theta_{2}^{-t}-1}{(\theta_{2}^{-n_{0}t}-1)(1-\theta_{0}/\theta_{2})}\,\eta(x),$
(3.6)

so we have proved the properties of the function $\varphi_{2}$ stated in
Theorem 3.2 and Remark 3.

Now, assume that $n$ satisfies (3.3). It then follows from (3.1) that there
exists a constant $C$ such that

$\left|\frac{\mu P_{nt+j}f}{\mu
P_{nt+j}\mathbbm{1}_{E}}-\frac{\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i+j}f}{\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i+j}\mathbbm{1}_{E}}\right|\\\ \begin{aligned}
\leq&\frac{\theta_{0}^{-(n-1)t}\mu P_{nt+j}|f|}{\theta_{0}^{-(n-1)t}\mu
P_{nt+j}\mathbbm{1}_{E}\left[\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i+j}\mathbbm{1}_{E}\right]}\times\\\
&\qquad\qquad\times\left|\theta_{0}^{-(n-1)t}\sum_{i=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}P_{(n-1)t}P_{i+j}\mathbbm{1}_{E}-\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i+j}\mathbbm{1}_{E}\right|\\\
&+\frac{1}{\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i+j}\mathbbm{1}_{E}}\times\\\
&\qquad\qquad\times\left|\theta_{0}^{-(n-1)t}\sum_{i=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}P_{(n-1)t}P_{i+j}f-\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i+j}f\right|\\\ &\leq
C\alpha^{n}\frac{\sum_{i=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}V}{\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i+j}\mathbbm{1}_{E}}\left(1+\frac{\theta_{0}^{-(n-1)t}\mu
P_{nt+j}|f|}{\theta_{0}^{-(n-1)t}\mu
P_{nt+j}\mathbbm{1}_{E}\left[\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i+j}\mathbbm{1}_{E}\right]}\right).\end{aligned}$

Since $\nu P_{i+j}\mathbbm{1}_{E}\geq\nu P_{i+j}P_{2t-i-j}\mathbbm{1}_{E}=\nu
Q_{2}\mathbbm{1}_{E}=\theta_{0}^{2t}>0$, we deduce that

$\left|\frac{\mu P_{nt+j}f}{\mu
P_{nt+j}\mathbbm{1}_{E}}-\frac{\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i+j}f}{\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i+j}\mathbbm{1}_{E}}\right|\\\ \leq
C\alpha^{n}\frac{\sum_{i=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}V}{\sum_{i=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta}\left(1+\frac{\theta_{0}^{-(n-1)t}\mu
P_{nt+j}|f|}{\theta_{0}^{-(n-1)t}\mu
P_{nt+j}\mathbbm{1}_{E}\left[\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i+j}\mathbbm{1}_{E}\right]}\right)$

for some constant $C$. Now, we use (3.3) to deduce from (3.1) that

$\displaystyle\theta_{0}^{-(n-1)t}\mu P_{nt+j}\mathbbm{1}_{E}$
$\displaystyle\geq\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i+j}\mathbb{1}_{E}-C^{\prime}_{Q}\theta_{0}^{-(t+j)}\alpha^{n}\sum_{i=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}V$
$\displaystyle\geq\theta_{0}^{2t}\sum_{i=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta-C^{\prime}_{Q}\theta_{0}^{-2t}\alpha^{n}\sum_{i=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}V$
$\displaystyle\geq\frac{\theta_{0}^{2t}}{2}\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i+j}\mathbb{1}_{E}$

and

$\displaystyle\theta_{0}^{-(n-1)t}\mu P_{nt+j}|f|$
$\displaystyle\leq\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i+j}|f|+C^{\prime}_{Q}\theta_{0}^{-(t+j)}\alpha^{n}\sum_{i=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}V$
$\displaystyle\leq\left(\nu(V)+\frac{\theta_{0}^{3t}}{2}\right)\sum_{i=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta,$

where we used the definition of $\mathcal{B}_{V}$ in the last inequality.
Therefore,

$\left|\frac{\mu P_{nt+j}f}{\mu
P_{nt+j}\mathbbm{1}_{E}}-\frac{\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i+j}f}{\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i+j}\mathbbm{1}_{E}}\right|\leq
C\alpha^{n}\frac{\sum_{i=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}V}{\sum_{i=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta}$

for some constant $C$, so we have proved (3.4) and, thanks to (3.6), (3.2) for
$n$ satisfying (3.3).

Assume now that (3.3) is not satisfied. Then, using (3.1) in a similar way as
above,

$\left|\frac{\mu P_{nt+j}f}{\mu
P_{nt+j}\mathbbm{1}_{E}}-\frac{\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i+j}f}{\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i+j}\mathbbm{1}_{E}}\right|\leq\frac{\theta_{0}^{-(n-1)t}\mu
P_{nt+j}f}{\theta_{0}^{-(n-1)t}\mu
P_{nt+j}\mathbbm{1}_{E}}+\frac{\nu(V)}{\theta_{0}^{2t}}\\\
\leq\frac{\nu(V)\sum_{i=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta+C^{\prime}_{Q}\theta_{0}^{-(t+j)}\alpha^{n}\sum_{i=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}V}{\theta_{0}^{-(n-1)t}\sum_{i=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}Q_{n}\mathchoice{{(P_{i+j}\mathbbm{1}_{A_{i+j}})\,\smash{\vrule
height=6.00002pt,depth=3.05219pt}}_{\,A_{0}}}{{(P_{i+j}\mathbbm{1}_{A_{i+j}})\,\smash{\vrule
height=6.00002pt,depth=3.05219pt}}_{\,A_{0}}}{{(P_{i+j}\mathbbm{1}_{A_{i+j}})\,\smash{\vrule
height=4.20001pt,depth=2.58751pt}}_{\,A_{0}}}{{(P_{i+j}\mathbbm{1}_{A_{i+j}})\,\smash{\vrule
height=3.0pt,depth=2.58751pt}}_{\,A_{0}}}}+\frac{\nu(V)}{\theta_{0}^{2t}},$

where we made the abuse of notation that $A_{i+j}=A_{i+j-t}$ if $t\leq i+j\leq
2t-1$. Now, $\mathbbm{1}_{A_{i+j}}\geq P_{2t-i-j}\mathbbm{1}_{A_{0}}$, so for
all $x\in A_{0}$,

$P_{i+j}\mathbbm{1}_{A_{i+j}}\geq P_{2t}\mathbbm{1}_{A_{0}}\geq
P_{2t}\varphi_{2}\geq\theta_{2}^{2t}\varphi_{2}.$

Therefore,In additon, using (3.5) and that (3.3) is not satisfied,

$\left|\frac{\mu P_{nt+j}f}{\mu
P_{nt+j}\mathbbm{1}_{E}}-\frac{\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i+j}f}{\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i+j}\mathbbm{1}_{E}}\right|\leq
C\alpha^{n}\left(\frac{\theta_{0}}{\theta_{2}}\right)^{(n-1)t}\frac{\sum_{i=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}V}{\sum_{i=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\varphi_{2}},$

so (3.2) is proved with $\bar{\alpha}=\alpha\theta_{0}^{t}/\theta_{2}^{t}<1$.
∎

######  Remark 4.

Under the assumptions of Theorem 3.1, the conditional distributions of $X$
converge to a quasi-stationary distribution if and only if the measure

$\frac{\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i+j}}{\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i+j}\mathbbm{1}_{E}}$

does not depend on $j$. Indeed, in this case, this measure is a quasi-limiting
distribution, hence the unique quasi-stationary distribution given in
Proposition 2.1. Comparing the measures for $j=0$ and $j=1$ entails that, for
all $f\in\mathcal{B}_{V}$, the equality

$\left(\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i}f\right)\left(\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i+1}\mathbbm{1}_{E}\right)\\\
=\left(\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i+1}f\right)\left(\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i}\mathbbm{1}_{E}\right)$

should hold true. Choosing $f=\mathbbm{1}_{A_{k}}$, we obtain for all
$k\in\\{0,\ldots,t-1\\}$,

$\frac{\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-k+1}\eta}{\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-k}\eta}=\frac{\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i+1}\mathbbm{1}_{E}}{\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i}\mathbbm{1}_{E}}.$

Since the right-hand side, say $\gamma$, does not depend on $k$, we deduce
that

$\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-k}\eta=a\gamma^{-i}$

for some constant $a>0$. We deduce that

$\gamma=\frac{\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i+1}\mathbbm{1}_{E}}{\sum_{i=0}^{t-1}(\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta)\nu
P_{i}\mathbbm{1}_{E}}=\gamma\,\frac{\sum_{i=0}^{t-1}\gamma^{-(i+1)}\nu
P_{i+1}\mathbbm{1}_{E}}{\sum_{i=0}^{t-1}\gamma^{-i}\nu
P_{i}\mathbbm{1}_{E}}=\gamma\left(1+\frac{1-\gamma^{-t}\,\nu
P_{t}\mathbbm{1}_{E}}{\sum_{i=0}^{t-1}\gamma^{-i}\nu
P_{i}\mathbbm{1}_{E}}\right).$

Since $\nu P_{t}=\theta_{0}^{t}\nu$, this equality is possible only if
$\gamma=\theta_{0}$. Therefore, under the assumptions of Theorem 3.2, given
$X_{0}\sim\mu$ such that $\sum_{i=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}V<+\infty$ and
$\sum_{i=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta>0$, the conditional
distributions of $X$ converge in total variation to a quasi-stationary
distribution if and only if $\theta_{0}^{-i}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta$ does not depend of
$i\in\\{0,\ldots,t-1\\}$.

##  4 Spectral properties of periodic Markov chains under (2.4)

Let $\hat{P}_{1}f(x)=\mathbb{E}_{x}f(X_{1})$ for all $x\in
E\cup\\{\partial\\}$ and all $f\in\hat{\mathcal{B}}_{\varphi_{1}}$, where

$\hat{\mathcal{B}}_{V}=\left\\{f:E\cup\\{\partial\\}\rightarrow\mathbb{C}\text{
s.t.\ }\exists a>0,\ \text{Re}(a\mathchoice{{f\,\smash{\vrule
height=5.55557pt,depth=1.65279pt}}_{\,E}}{{f\,\smash{\vrule
height=5.55557pt,depth=1.65279pt}}_{\,E}}{{f\,\smash{\vrule
height=3.88889pt,depth=1.16167pt}}_{\,E}}{{f\,\smash{\vrule
height=2.77777pt,depth=1.16167pt}}_{\,E}})\in\mathcal{B}_{V}\text{ and
}\text{Im}(a\mathchoice{{f\,\smash{\vrule
height=5.55557pt,depth=1.65279pt}}_{\,E}}{{f\,\smash{\vrule
height=5.55557pt,depth=1.65279pt}}_{\,E}}{{f\,\smash{\vrule
height=3.88889pt,depth=1.16167pt}}_{\,E}}{{f\,\smash{\vrule
height=2.77777pt,depth=1.16167pt}}_{\,E}})\in\mathcal{B}_{V}\right\\},$

where we denoted by $\text{Re}(z)$ and $\text{Im}(z)$ the real and imaginary
parts of a complex number $z$, respectively. Theorem 3.1 can be seen as a
quasi-compactness property and it hence implies a spectral gap property, as
shown in the next result.

######  Corollary 4.1.

Under the assumptions of Theorem 3.1, each eigenfunction
$h\in\hat{\mathcal{B}}_{V}$ of $\hat{P}_{1}$ with eigenvalue
$\theta\in\mathbb{C}$ satisfies the following properties:

  1. 1.

if $h(\partial)\neq 0$ and if $\mathbb{P}_{x}(\tau_{\partial}<\infty)=1$ for
all $x\in E$, then $\theta=1$ and $h$ is constant;

  2. 2.

if $h(\partial)=0$ and there exists $i\in\\{0,\ldots,t-1\\}$ such that $\nu
P_{i}h=\nu P_{i}(\mathchoice{{h\,\smash{\vrule
height=5.55557pt,depth=2.52328pt}}_{\,A_{i}}}{{h\,\smash{\vrule
height=5.55557pt,depth=2.52328pt}}_{\,A_{i}}}{{h\,\smash{\vrule
height=3.88889pt,depth=2.05862pt}}_{\,A_{i}}}{{h\,\smash{\vrule
height=2.77777pt,depth=2.05862pt}}_{\,A_{i}}})\neq 0$, then
$\theta=\theta_{0}$ and

$\mathchoice{{h\,\smash{\vrule
height=5.55557pt,depth=1.62633pt}}_{\,E}}{{h\,\smash{\vrule
height=5.55557pt,depth=1.62633pt}}_{\,E}}{{h\,\smash{\vrule
height=3.88889pt,depth=1.16167pt}}_{\,E}}{{h\,\smash{\vrule
height=2.77777pt,depth=1.16167pt}}_{\,E}}=\nu(h)\sum_{i=0}^{t-1}\theta_{0}^{-i}P_{i}\eta,$
(4.1)

where the eigenfunction $\eta$ of $Q$ has been extended by 0 out of $A_{0}$;

  3. 3.

if $h(\partial)=0$, $\nu P_{i}h=0$ for all $i\in\\{0,\ldots,t-1\\}$, then
$|\theta|\leq\theta_{0}\alpha^{1/t}$.

###### Proof.

Observe that, for all $f\in\hat{\mathcal{B}}_{\varphi_{1}}$ all
$j\in\\{0,\ldots,t-1\\}$ and all $x\in E$,

$\hat{P}_{j}f(x)=P_{j}(\mathchoice{{f\,\smash{\vrule
height=5.55557pt,depth=1.65279pt}}_{\,E}}{{f\,\smash{\vrule
height=5.55557pt,depth=1.65279pt}}_{\,E}}{{f\,\smash{\vrule
height=3.88889pt,depth=1.16167pt}}_{\,E}}{{f\,\smash{\vrule
height=2.77777pt,depth=1.16167pt}}_{\,E}})(x)+f(\partial)(1-P_{j}\mathbbm{1}_{E}(x))$
(4.2)

and $\hat{P}_{j}f(\partial)=f(\partial)$.

Assume first that $h(\partial)\neq 0$. Then $\theta
h(\partial)=\hat{P}_{1}h(\partial)=h(\partial)$, so $\theta=1$. Moreover, for
all $x\in E$ and all $j\in\mathbb{N}$,

$h(x)=P_{j}(\mathchoice{{h\,\smash{\vrule
height=5.55557pt,depth=1.62633pt}}_{\,E}}{{h\,\smash{\vrule
height=5.55557pt,depth=1.62633pt}}_{\,E}}{{h\,\smash{\vrule
height=3.88889pt,depth=1.16167pt}}_{\,E}}{{h\,\smash{\vrule
height=2.77777pt,depth=1.16167pt}}_{\,E}})(x)+h(\partial)(1-P_{j}\mathbbm{1}_{E}(x)).$

Now, (3.1) and the fact that $\mathbb{P}_{x}(\tau_{\partial}<\infty)=1$ for
all $x\in E$ implies that $\theta_{0}<1$ and, in particular, for all $x\in E$,
$P_{j}(\mathchoice{{h\,\smash{\vrule
height=5.55557pt,depth=1.62633pt}}_{\,E}}{{h\,\smash{\vrule
height=5.55557pt,depth=1.62633pt}}_{\,E}}{{h\,\smash{\vrule
height=3.88889pt,depth=1.16167pt}}_{\,E}}{{h\,\smash{\vrule
height=2.77777pt,depth=1.16167pt}}_{\,E}})(x)$ and $P_{j}\mathbbm{1}_{E}(x)$
both converge to 0 when $j\to+\infty$ and thus $h(x)=h(\partial)$. Hence Point
1. is proved.

Assume now that $h(\partial)=0$ and that there exists $i\in\\{0,\ldots,t-1\\}$
such that $\nu P_{i}h\neq 0$. We can assume without loss of generality that
$\nu P_{i}h>0$. It then follows from (4.2) that
$P_{1}(\mathchoice{{h\,\smash{\vrule
height=5.55557pt,depth=1.62633pt}}_{\,E}}{{h\,\smash{\vrule
height=5.55557pt,depth=1.62633pt}}_{\,E}}{{h\,\smash{\vrule
height=3.88889pt,depth=1.16167pt}}_{\,E}}{{h\,\smash{\vrule
height=2.77777pt,depth=1.16167pt}}_{\,E}})=\theta\mathchoice{{h\,\smash{\vrule
height=5.55557pt,depth=1.62633pt}}_{\,E}}{{h\,\smash{\vrule
height=5.55557pt,depth=1.62633pt}}_{\,E}}{{h\,\smash{\vrule
height=3.88889pt,depth=1.16167pt}}_{\,E}}{{h\,\smash{\vrule
height=2.77777pt,depth=1.16167pt}}_{\,E}}$. Let $k\in\\{0,\ldots,t-1\\}$ be
fixed. Since $\theta_{0}^{t}\eta=P_{t}\eta=P_{k}P_{t-k}\eta$ is nonzero, there
exists $x\in A_{k}$ such that $P_{t-k}\eta(x)>0$. For such an $x\in A_{k}$, it
follow from (3.1) that

$\lim_{n\to+\infty}\theta_{0}^{-(nt+i-k)}P_{nt+j}h(x)=\lim_{n\to+\infty}\left(\frac{\theta}{\theta_{0}}\right)^{-(nt+i-k)}h(x)\geq\theta_{0}^{-(t+i-k)}P_{t-k}\eta(x)\nu
P_{i}h>0.$

Therefore, $\theta=\theta_{0}$. Since the convergence above holds for all
$x\in A_{k}$,and, we have proved that, for all $x\in A_{k}$,
$h(x)=\theta_{0}^{-(t+i-k)}P_{t-k}\eta(x)\nu P_{i}h$. In paticular,
$\nu(h)=\theta_{0}^{-i}\nu(\eta)\nu P_{i}h=\theta_{0}^{-i}\nu P_{i}h>0$, so
the previous computation could be done with $i=0$, thus entailing (4.1).

Assume finally that $h(\partial)=0$ and $\nu P_{i}h=0$ for all
$i\in\\{0,\ldots,t-1\\}$. Let $k\in\\{0,\ldots,t-1\\}$ and $x\in A_{k}$ such
that $h(x)\neq 0$. Then (3.1) entails that, for all $j\in\\{0,\ldots,t-1\\}$,

$\left|\theta_{0}^{-(nt+j)}P_{nt+j}h(x)\right|=\left(\frac{|\theta|}{\theta_{0}}\right)^{nt+j}|h(x)|\leq
C^{\prime}_{Q}\alpha^{n}P_{t-k}V(x),$

hence $|\theta|\leq\theta_{0}\alpha^{1/t}$ and Corollary 4.1 is proved. ∎

##  5 Ergodic behavior of the $Q$-process of periodic Markov chains under
(2.4)

We now study the $Q$-process of $X$, i.e. the law of $X$ conditioned to never
be absorbed. We define $E^{\prime}=\bigcup_{i=0}^{t-1}\\{x\in
A_{i},\,P_{t-i}\eta(x)>0\\}$. Let us denote by
$(\Omega,(\mathcal{F_{n}}_{n\geq 0},(X_{n})_{n\geq 0},(\mathbb{P}_{x})_{x\in
E})$ be a filtered probability a space such that, under $\mathbb{P}_{x}$, the
process $X$ has the same distribution as the Markov process with semi-group
$(P_{n})_{n\geq 0}$ and initial condition $x$.

######  Theorem 5.1.

Under the assumptions of Theorem 3.1, we have the following properties:

(i) Existence of the $Q$-process.

There exists a family $(\mathbb{Q}_{x})_{x\in E^{\prime}}$ of probability
measures on $\Omega$ defined by

$\lim_{n\rightarrow+\infty}\mathbb{P}_{x}(A\mid
n<\tau_{\partial})=\mathbb{Q}_{x}(A)$

for all $x\in E^{\prime}$, for all ${\cal F}_{m}$-measurable set $A$ and for
all $m\geq 0$. The process $(\Omega,({\cal F}_{n})_{n\geq 0},(X_{n})_{n\geq
0},(\mathbb{Q}_{x})_{x\in E^{\prime}})$ is an $E^{\prime}$-valued homogeneous
Markov chain.

(ii) Semigroup.

The semigroup of the Markov process $X$ under $(\mathbb{Q}_{x})_{x\in
E^{\prime}}$ is given for all bounded measurable function $\varphi$ on
$E^{\prime}$, all $j\in\\{0,\ldots,t-1\\}$ and all $n\geq 0$ by

$\displaystyle\widetilde{P}_{nt+j}\varphi(x)=\sum_{i=0}^{t-1}\mathbbm{1}_{A_{i}\cap
E^{\prime}}(x)\theta_{0}^{-(nt+j)}\frac{P_{nt+j}(\varphi\eta_{2t-i-j})(x)}{\eta_{t-i}(x)},$
(5.1)

where, for all $k\in\mathbb{Z}_{+}$,

$\eta_{k}:=\theta_{0}^{-k}P_{k}\eta,$

where the function $\eta$ has been extended by 0 to $E\setminus A_{0}$.

(iii) Exponential contraction in total variation.

The Markov process $X$ under $(\mathbb{Q}_{x})_{x\in E^{\prime}}$ admits as
unique invariant probability measure

$\displaystyle\frac{1-\theta_{0}}{1-\theta_{0}^{t}}\sum_{i=0}^{t-1}\nu
P_{i}(\,\cdot\,\eta_{t-i}).$ (5.2)

In addition, there exist constants $C<+\infty$ and $\alpha\in(0,1)$ such that,
for all $j\in\\{0,\ldots,t-1\\}$, all $n\geq 0$, all probability measure
$\mu^{\prime}$ on $E^{\prime}$ such that
$\mathchoice{{\mu^{\prime}\,\smash{\vrule
height=4.68889pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu^{\prime}\,\smash{\vrule
height=4.68889pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu^{\prime}\,\smash{\vrule
height=3.3pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu^{\prime}\,\smash{\vrule
height=2.61111pt,depth=2.05862pt}}_{\,A_{i}}}(\frac{P_{t-i}V}{\eta_{t-i}})<+\infty$
for all $i\in\\{0,\ldots,t-1\\}$ and all measurable
$h:E^{\prime}\rightarrow\mathbb{R}$ such that, for all
$k\in\\{0,\ldots,t-1\\}$ and all $x\in A_{0}$, $P_{k}(h\eta_{t-k})(x)\leq
V(x)$ (or, equivalently, $\widetilde{P}_{k}h(x)\leq V(x)/\eta(x)$),

$\displaystyle\left|\mu^{\prime}\widetilde{P}_{nt+j}h-\sum_{i=0}^{t-1}\theta_{0}^{t-i}\mu^{\prime}(A_{i})\nu
P_{i+j}(h\eta_{2t-i-j})\right|\leq
C\alpha^{n}\sum_{i=0}^{t-1}\mathchoice{{\mu^{\prime}\,\smash{\vrule
height=4.68889pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu^{\prime}\,\smash{\vrule
height=4.68889pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu^{\prime}\,\smash{\vrule
height=3.3pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu^{\prime}\,\smash{\vrule
height=2.61111pt,depth=2.05862pt}}_{\,A_{i}}}\left(\frac{P_{t-i}V}{\eta_{t-i}}\right).$
(5.3)

Furthermore, for all probability measure $\mu^{\prime}$ on $E^{\prime}$ and
all $j\in\\{0,\ldots,t-1\\}$,

$\displaystyle\left\|\mu^{\prime}\widetilde{P}_{nt+j}-\sum_{i=0}^{t-1}\theta_{0}^{t-i}\mu^{\prime}(A_{i})\nu
P_{i+j}(\,\cdot\,\eta_{2t-i-j})\right\|_{TV}\xrightarrow[n\rightarrow\infty]{}0,$
(5.4)

where $\|\cdot\|_{TV}$ denotes the total variation norm.

###### Proof.

We introduce $\Gamma_{m}=\mathbbm{1}_{m<\tau_{\partial}}$ and define for all
$x\in E^{\prime}$ and $m,\geq 0$ the probability measure

$\displaystyle
Q^{\Gamma,x}_{m}=\frac{\Gamma_{m}}{\mathbb{E}_{x}\left(\Gamma_{m}\right)}\mathbb{P}_{x},$

so that the $Q$-process exists if and only if $Q_{m}^{\Gamma,x}$ admits a
proper limit when $m\rightarrow\infty$. Fix $j\in\\{0,\ldots,t-1\\}$ and
$k\geq 0$. Let $k_{0}\in\\{j-t+1,\ldots,j\\}$ be such that $k=n_{0}t+k_{0}$
for some integer $n_{0}$. For all $n\geq 0$ such that $nt+j\geq k$, we have by
the Markov property

$\displaystyle\frac{\mathbb{E}_{x}\left(\Gamma_{nt+j}\mid{\cal
F}_{k}\right)}{\mathbb{E}_{x}\left(\Gamma_{nt+j}\right)}$
$\displaystyle=\mathbbm{1}_{k<\tau_{\partial}}\frac{\mathbb{P}_{X_{k}}\left(nt+j-k<\tau_{\partial}\right)}{\mathbb{P}_{x}\left(nt+j<\tau_{\partial}\right)}$
$\displaystyle=\theta_{0}^{-k}\mathbbm{1}_{k<\tau_{\partial}}\frac{\theta_{0}^{-((n-n_{0})t+j-k_{0})}P_{(n-n_{0})t+j-k_{0}}\mathbbm{1}_{E}(X_{k})}{\theta_{0}^{-(nt+j)}P_{nt+j}\mathbbm{1}_{E}(x)}.$

Assume that $x\in A_{i}\cap E^{\prime}$ for some $i\in\\{0,\ldots,t-1\\}$. By
Theorem 3.1, almost surely,

$\displaystyle\lim_{n\rightarrow+\infty}\frac{\mathbb{E}_{x}\left(\Gamma_{nt+j}\mid{\cal
F}_{k}\right)}{\mathbb{E}_{x}\left(\Gamma_{nt+j}\right)}$
$\displaystyle=\theta_{0}^{-k}\mathbbm{1}_{k<\tau_{\partial}}\frac{\theta_{0}^{-(2t+j-k_{0})}P_{2t-(i+k_{0})}\eta(X_{k})\,\nu
P_{i+j}\mathbbm{1}_{E}}{\theta_{0}^{-(t+j)}P_{t-i}\eta(x)\,\nu
P_{i+j}\mathbbm{1}_{E}}$
$\displaystyle=\theta_{0}^{-(n_{0}+1)t}\mathbbm{1}_{k<\tau_{\partial}}\frac{P_{2t-(i+k_{0})}\eta(X_{k})}{P_{t-i}\eta(x)}=:M_{k}.$

Since the limit is independent of $j\in\\{0,\ldots,t-1\\}$, we deduce that
$\frac{\mathbb{E}_{x}\left(\Gamma_{m}\mid{\cal
F}_{k}\right)}{\mathbb{E}_{x}\left(\Gamma_{m}\right)}\rightarrow M_{k}$ almost
surely when $m\rightarrow+\infty$. Since in addition

$\mathbb{E}_{x}M_{k}=\theta_{0}^{-(t+k-k_{0})}\frac{P_{2t-i+k-k_{0}}\eta(x)}{P_{t-i}\eta(x)}=1,$

we can apply the penalization’s theorem of Roynette, Vallois and Yor [22,
Theorem 2.1], which implies that $M$ is a martingale under $\mathbb{P}_{x}$
and that $Q_{m}^{\Gamma,x}(A)$ converges to
$\mathbb{E}_{x}\left(M_{k}\mathbbm{1}_{A}\right)$ for all $A\in{\cal F}_{k}$
when $m\rightarrow\infty$. This means that $\mathbb{Q}_{x}$ is well defined
and

$\displaystyle\mathchoice{{\frac{d\mathbb{Q}_{x}}{d\mathbb{P}_{x}}\,\smash{\vrule
height=10.37114pt,depth=3.46376pt}}_{\,{\cal
F}_{k}}}{{\frac{d\mathbb{Q}_{x}}{d\mathbb{P}_{x}}\,\smash{\vrule
height=7.26666pt,depth=2.57079pt}}_{\,{\cal
F}_{k}}}{{\frac{d\mathbb{Q}_{x}}{d\mathbb{P}_{x}}\,\smash{\vrule
height=5.28888pt,depth=2.10611pt}}_{\,{\cal
F}_{k}}}{{\frac{d\mathbb{Q}_{x}}{d\mathbb{P}_{x}}\,\smash{\vrule
height=5.28888pt,depth=2.10611pt}}_{\,{\cal F}_{k}}}$ $\displaystyle=M_{k}.$
(5.5)

Note that, in view of the expression of $M_{k}$ and by definition of
$E^{\prime}$, $(X_{n},n\geq 0)$ is $E^{\prime}$-valued $\mathbb{Q}_{x}$-almost
surely for all $x\in E^{\prime}$. The fact that $X$ is Markov under
$(\mathbb{Q}_{x})_{x\in E^{\prime\prime}}$ can be easily deduced from the last
formula (see e.g. [5, Section 6.1]).

Point (ii) is a direct consequence of (5.5) and of the definition of
$M_{nt+j}$.

We can now prove (5.3): this a direct consequence of (3.1) with

$\mu(dx)=\sum_{i=0}^{t-1}\frac{1}{\eta_{t-i}(x)}\mathchoice{{\mu^{\prime}\,\smash{\vrule
height=4.68889pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu^{\prime}\,\smash{\vrule
height=4.68889pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu^{\prime}\,\smash{\vrule
height=3.3pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu^{\prime}\,\smash{\vrule
height=2.61111pt,depth=2.05862pt}}_{\,A_{i}}}(dx).$

In particular, for all $x\in E^{\prime}$,

$\left\|\delta_{x}\widetilde{P}_{nt+j}-\sum_{i=0}^{t-1}\theta_{0}^{t-i}\mathbbm{1}_{A_{i}}(x)\nu
P_{i+j}(h\eta_{2t-i-j})\right\|_{TV}\xrightarrow[n\to+\infty]{}0,$

and so (5.4) follows from the dominated convergence theorem.

It only remains to check that (5.2) is the only invariant distribution for
$\widetilde{P}$. Note that, because of (5.4), all invariant measure must be of
the form

$\sum_{i=0}^{t-1}a_{i}\nu P_{i+j}(\,\cdot\,\eta_{2t-i-j})$

for some $a_{i}\geq 0$ such that the last measure does not depend on
$j\in\\{0,\ldots,t-1\\}$. Since $\nu P_{k}$ has support in $A_{k}$,
identifying the last measure for $j=0$ and $j=1$, we deduce that all the
constants $a_{i}$ must be equal, so that there is a unique invariant measure
given by

$\frac{\sum_{i=0}^{j-1}\nu P_{i}(\,\cdot\,\eta_{t-i})}{\sum_{i=0}^{j-1}\nu
P_{i}\eta_{t-i}}=\frac{\sum_{i=0}^{j-1}\nu
P_{i}(\,\cdot\,\eta_{t-i})}{\sum_{i=0}^{j-1}\theta_{0}^{i}}.$

This ends the proof of Theorem 5.1. ∎

##  6 Quasi-ergodic behavior of periodic Markov chains under (2.4)

Related to the asymptotic behavior of the $Q$-process is the so-called quasi-
ergodicity [8], given in the next result.

######  Theorem 6.1.

Under the assumptions of Theorem 3.1, there exists a constant $C<+\infty$ such
that, for all bounded measurable $f:E\rightarrow[-1,1]$, all probability
measure $\mu$ on $E$ such that $\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}(\eta_{t-i})>0$ for some
$i\in\\{0,\ldots,t-1\\}$ and $\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}V<+\infty$ for all
$i\in\\{0,\ldots,t-1\\}$, for all $N\geq 0$

$\left|\mathbb{E}_{\mu}\left[\left.\frac{1}{N+1}\sum_{m=0}^{N}f(X_{m})\quad\right|\
N<\tau_{\partial}\right]-\frac{\theta_{0}^{-t}}{t}\sum_{\ell=0}^{t-1}\nu
P_{\ell}(fP_{t-\ell}\eta)\right|\\\
\leq\frac{C\sum_{i=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}V}{(N+1)\,\sum_{i=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}(\eta_{t-i})}.$ (6.1)

######  Remark 5.

Notice that, for $f\equiv 1$, $\sum_{k=0}^{t-1}\nu
P_{k}(fP_{t-k}\eta)=\theta_{0}^{t}\sum_{k=0}^{t-1}\nu(\eta)=t\theta_{0}^{t}$,
so the measure $\nu_{\text{QE}}:=\frac{\theta_{0}^{-t}}{t}\sum_{k=0}^{t-1}\nu
P_{k}(\cdot P_{t-k}\eta)$ is a probability measure, called the _quasi-ergodic
distribution_.

In the last result, we do not recover the full the quasi-ergodic theorem of
[4] since they obtain convergence for all $f\in L^{1}(\nu_{\text{QE}})$.
However, it does not seem that condition (2.4) is sufficient to imply the
conditions of [4] (see [3]). However, (2.4) allows to improve the convergence
in theorem 6.1 into what could be called a convergence in conditional
probability.

######  Corollary 6.2.

Under the assumptions of Theorem 3.1, for all bounded measurable
$f:E\rightarrow\mathbb{R}$, all probability measure $\mu$ on $E$ such that
$\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}(\eta_{t-i})>0$ for some
$i\in\\{0,\ldots,t-1\\}$ and $\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}V<+\infty$ for all
$i\in\\{0,\ldots,t-1\\}$,

$\lim_{N\rightarrow
0}\mathbb{P}_{\mu}\left(\left.\left|\frac{1}{N+1}\sum_{m=0}^{N}f(X_{m})-\frac{\theta_{0}^{-t}}{t}\sum_{\ell=0}^{t-1}\nu
P_{\ell}(fP_{t-\ell}\eta)\right|>\varepsilon\ \right|\
N<\tau_{\partial}\right)=0,\quad\forall\varepsilon>0.$ (6.2)

###### Proof of Theorem 6.1.

In all the proof, the constant $C$ denotes a constant that may change from
line to line. First notice that

$\displaystyle\sum_{j=0}^{t-1}\theta_{0}^{-(i+j)}\nu P_{i+j}(f\eta_{2t-i-j})$
$\displaystyle=\theta_{0}^{-2t}\sum_{j=0}^{t-1}\nu P_{i+j}(fP_{2t-i-j}\eta)$
$\displaystyle=\theta_{0}^{-t}\left(\sum_{j=0}^{t-1-i}\nu
P_{i+j}(fP_{t-i-j}\eta)+\sum_{j=t-i}^{t-1}\nu
P_{i+j-t}(fP_{2t-i-j}\eta)\right)$
$\displaystyle=\theta_{0}^{-t}\sum_{k=0}^{t-1}\nu P_{k}(fP_{t-k}\eta).$

Hence, introducing $n\geq 0$ and $k\in\\{0,\ldots,t-1\\}$ such that $N=nt+k$
and denoting
$\bar{\mu}(\bar{\eta})=\sum_{i=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}(\eta_{t-i})\,\theta_{0}^{-(k+i)}\nu
P_{k+i}\mathbbm{1}_{E}$, it follows from Theorem 3.1 that

$\displaystyle\left|\mathbb{E}_{\mu}\left[\left.\frac{1}{N+1}\sum_{m=0}^{N}f(X_{m})\
\right|\
nt+k<\tau_{\partial}\right]-\frac{\theta_{0}^{-t}}{t}\sum_{\ell=0}^{t-1}\nu
P_{\ell}(fP_{t-\ell}\eta)\right|$
$\displaystyle=\left|\mathbb{E}_{\mu}\left[\left.\frac{1}{nt+k+1}\sum_{m=0}^{nt+k}f(X_{m})\
\right|\ nt+k<\tau_{\partial}\right]\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.-\frac{\sum_{i=0}^{t-1}\sum_{j=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}(\eta_{t-i})\,\theta_{0}^{-(k+i)}\nu
P_{k+i}\mathbbm{1}_{E}\,\theta_{0}^{-(i+j)}\nu
P_{i+j}(f\eta_{2t-i-j})}{t\,\sum_{i=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}(\eta_{t-i})\,\theta_{0}^{-(k+i)}\nu
P_{k+i}\mathbbm{1}_{E}}\right|$ $\displaystyle\leq\frac{|\theta_{0}^{-N}\mu
P_{N}\mathbbm{1}_{E}-\bar{\mu}(\bar{\eta})|}{\theta_{0}^{-N}\mu
P_{N}\mathbbm{1}_{E}\,\bar{\mu}(\bar{\eta})}\,\frac{1}{N+1}\sum_{m=0}^{N}\theta_{0}^{-N}\mu
P_{N}\mathbbm{1}_{E}$
$\displaystyle+\frac{1}{(N+1)\bar{\mu}(\bar{\eta})}\sum_{i=0}^{t-1}\sum_{j=0}^{t-1}\sum_{m=0}^{\lfloor(N-j)/t\rfloor}\left|\theta_{0}^{-(mt+j)}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{mt+j}\left(f\theta_{0}^{-[(n-m)t+k-j]}P_{(n-m)t+k-j}\mathbbm{1}_{E}\right)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\theta_{0}^{-(mt+j)}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{mt+j}(f\eta_{2t-i-j})\,\theta_{0}^{-(k+i)}\nu
P_{k+i}\mathbbm{1}_{E}\Big{|}$
$\displaystyle+\frac{1}{(N+1)\bar{\mu}(\bar{\eta})}\sum_{i=0}^{t-1}\sum_{j=0}^{t-1}\sum_{m=0}^{\lfloor(N-j)/t\rfloor}\theta_{0}^{-(k+i)}\nu
P_{k+i}\mathbbm{1}_{E}\left|\theta_{0}^{-(mt+j)}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{mt+j}(f\eta_{2t-i-j})\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}(\eta_{t-i})\,\theta_{0}^{-(i+j)}\nu
P_{i+j}(f\eta_{2t-i-j})\Big{|}$
$\displaystyle+\frac{1}{\bar{\mu}(\bar{\eta})}\sum_{i=0}^{t-1}\sum_{j=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}(\eta_{t-i})\,\theta_{0}^{-(k+i)}\nu
P_{k+i}\mathbbm{1}_{E}\,\theta_{0}^{-(i+j)}\nu
P_{i+j}(f\eta_{2t-i-j})\left|\frac{\lfloor(N-j)/t\rfloor+1}{N+1}-\frac{1}{t}\right|.$
(6.3)

It can be checked using Theorem 3.1 that the first term of the right-hand-side
is bounded by

$C\alpha^{n}\frac{\sum_{i=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}V}{(N+1)\bar{\mu}(\bar{\eta})}$

and that each of the three other terms are bounded by

$\frac{C}{N+1}\,\frac{\sum_{i=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}V}{(N+1)\bar{\mu}(\bar{\eta})}.$

The only non-immediate bound is for the second term of the right-hand-side of
(6.3), which is bounded by

$\frac{C}{(N+1)\bar{\mu}(\bar{\eta})}\sum_{i=0}^{t-1}\sum_{j=0}^{t-1}\sum_{m=0}^{\lfloor(N-j)/t\rfloor}C\alpha^{n}\theta_{0}^{-(mt+j)}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{mt+j}(fP_{2t-i-j}V)\\\
\leq\frac{C^{\prime}\alpha^{n}}{(N+1)\bar{\mu}(\bar{\eta})}\sum_{i=0}^{t-1}\sum_{j=0}^{t-1}\sum_{m=0}^{\lfloor(N-j)/t\rfloor}\theta_{0}^{-(mt+j)}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{(m+1)t+t-i}V.$

by a first application of Theorem 3.1. A second application of Theorem 3.1
then proves that

$\theta_{0}^{-(mt+j)}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{(m+1)t+t-i}V\leq\theta_{0}^{-(t-i)}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}\eta\nu(V)+C\alpha^{n}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}V\leq
C^{\prime}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}V.$

Noting that

$\bar{\mu}(\bar{\eta})\geq\sum_{i=0}^{t-1}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}(\eta_{t-i})\,\inf_{0\leq\ell\leq
2t}\theta_{0}^{-\ell}\nu P_{\ell}\mathbbm{1}_{E},$

we have proved Theorem 6.1. ∎

###### Proof of Corollary 6.2.

In all the proof, the constant $C$ denotes a constant that may change from
line to line. The result follows from Chebychev’s bound and

$\lim_{N\rightarrow+\infty}\mathbb{E}_{\mu}\left[\left.\left(\frac{1}{N+1}\sum_{m=0}^{N}f(X_{m})-\frac{\theta_{0}^{-t}}{t}\sum_{\ell=0}^{t-1}\nu
P_{\ell}(fP_{t-\ell}\eta)\right)^{2}\ \right|\ N<\tau_{\partial}\right]=0,$
(6.4)

which follows from a similar computation as before. First set

$\bar{f}=f-\frac{\theta_{0}^{-t}}{t}\sum_{\ell=0}^{t-1}\nu
P_{\ell}(fP_{t-\ell}\eta)=f-\nu_{\text{QE}}(f).$

We have

$\mathbb{E}_{\mu}\left[\left.\left(\frac{1}{N+1}\sum_{m=0}^{N}\bar{f}(X_{m})\right)^{2}\
\right|\
N<\tau_{\partial}\right]=\frac{1}{(N+1)^{2}}\sum_{m=0}^{N}\mathbb{E}_{\mu}\left[\bar{f}^{2}(X_{m})\mid
N<\tau_{\partial}\right]\\\ +\frac{2}{(N+1)^{2}}\sum_{0\leq n<m\leq
N}\mathbb{E}_{\mu}\left[\bar{f}(X_{n})\bar{f}(X_{m})\mid
N<\tau_{\partial}\right]$ (6.5)

We shall use the bound, for all $0\leq n<m\leq N$ and all
$i\in\\{0,\ldots,t-1\\}$,

$\mathbb{E}_{\mathchoice{{\mu\,\smash{\vrule
height=2.41112pt,depth=1.76631pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=1.76631pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.68777pt,depth=1.44104pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.20555pt,depth=1.44104pt}}_{\,A_{i}}}}\left[\bar{f}(X_{n})\bar{f}(X_{m})\mathbbm{1}_{N<\tau_{\partial}}\right]\\\
\begin{aligned} &=\mathbb{E}_{\mathchoice{{\mu\,\smash{\vrule
height=2.41112pt,depth=1.76631pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=1.76631pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.68777pt,depth=1.44104pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.20555pt,depth=1.44104pt}}_{\,A_{i}}}}\left[\bar{f}(X_{n})\bar{f}(X_{m})\mathbbm{1}_{m<\tau_{\partial}}\left(\mathbb{P}_{X_{m}}(N-m<\tau_{\partial})-\theta_{0}^{N-m-
k-j}\eta_{t-k}(X_{m})\nu P_{k+j}\mathbbm{1}_{E}\right)\right]\\\
&+\theta_{0}^{N-m-k-j}\nu P_{k+j}\mathbbm{1}_{E}\
\mathbb{E}_{\mathchoice{{\mu\,\smash{\vrule
height=2.41112pt,depth=1.76631pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=1.76631pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.68777pt,depth=1.44104pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.20555pt,depth=1.44104pt}}_{\,A_{i}}}}\left[\bar{f}(X_{n})\mathbbm{1}_{n<\tau_{\partial}}\left(\mathbb{E}_{X_{n}}\left(\bar{f}(X_{m-n})\eta_{t-k}(X_{m-n})\right)\phantom{\theta_{0}^{k}}\right.\right.\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.\left.-\theta_{0}^{m-n-k^{\prime}-j^{\prime}}\eta_{t-k^{\prime}}(X_{n})\nu
P_{k^{\prime}+j^{\prime}}(\bar{f}\eta_{t-k})\right)\right]\\\
&+\theta_{0}^{N-n-k-k^{\prime}-j-j^{\prime}}\nu P_{k+j}\mathbbm{1}_{E}\,\nu
P_{k^{\prime}+j^{\prime}}(\bar{f}\eta_{t-k})\
\mathbb{E}_{\mathchoice{{\mu\,\smash{\vrule
height=2.41112pt,depth=1.76631pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=1.76631pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.68777pt,depth=1.44104pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.20555pt,depth=1.44104pt}}_{\,A_{i}}}}\left(\bar{f}(X_{n})\eta_{t-k^{\prime}}(X_{n})\right),\end{aligned}$

where $j\in\\{0,\ldots,t-1\\}$ is such that $N-m-j\in t\mathbb{Z}$,
$k\in\\{0,\ldots,t-1\\}$ is such that $i+m-k\in t\mathbb{Z}$,
$j^{\prime}=k-k^{\prime}$ (such that $m-n-j^{\prime}\in t\mathbb{Z}$) and
$k^{\prime}\in\\{0,\ldots,t-1\\}$ is such that $i+n-k^{\prime}\in
t\mathbb{Z}$.

Let us first deal with the last term of the last equation: for all fixed
$n\in\\{0,\ldots,N-t\\}$ and all $p\in\mathbb{Z}$ such that $pt-i>n$ et
$(p+1)t-i-1\leq N$, using that $j^{\prime}=k-k^{\prime}$, that $k^{\prime}$
only depends on $n$ and that $\theta_{0}^{-k-j}\nu
P_{k+j}\mathbbm{1}_{E}=\theta_{0}^{-N-i}\nu P_{N+i}\mathbbm{1}_{E}$ since
$N+i-k-j\in t\mathbb{Z}$,

$\sum_{m=pt-i}^{(p+1)t-i-1}\theta_{0}^{N-n-k-k^{\prime}-j-j^{\prime}}\nu
P_{k+j}\mathbbm{1}_{E}\,\nu P_{k^{\prime}+j^{\prime}}(\bar{f}\eta_{t-k})\
\mathbb{E}_{\mathchoice{{\mu\,\smash{\vrule
height=2.41112pt,depth=1.76631pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=1.76631pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.68777pt,depth=1.44104pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.20555pt,depth=1.44104pt}}_{\,A_{i}}}}\left(\bar{f}(X_{n})\eta_{t-k^{\prime}}(X_{n})\right)\\\
=\theta_{0}^{-n-i}\,\nu
P_{N+i}\mathbbm{1}_{E}\,\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{n}(\bar{f}\eta_{t-k^{\prime}})\sum_{m=pt-i}^{(p+1)t-i-1}\theta_{0}^{-k}\nu
P_{k}(\bar{f}\eta_{t-k})=0,$

since $\nu_{\text{QE}}(\bar{f})=0$.

Combining the last two inequalities and using Theorem 3.1, we deduce that, for
all $i\in\\{0,\ldots,t-1\\}$,

$\frac{2}{(N+1)^{2}}\sum_{0\leq n<m\leq
N}\mathbb{E}_{\mathchoice{{\mu\,\smash{\vrule
height=2.41112pt,depth=1.76631pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=1.76631pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.68777pt,depth=1.44104pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.20555pt,depth=1.44104pt}}_{\,A_{i}}}}\left[\bar{f}(X_{n})\bar{f}(X_{m})\mathbbm{1}_{N<\tau_{\partial}}\right]\\\
\begin{aligned} &\leq\frac{C}{(N+1)^{2}}\sum_{0\leq n<m\leq N}\theta_{0}^{N-n-
k-k^{\prime}-j-j^{\prime}}\nu P_{k+j}\mathbbm{1}_{E}\,\nu
P_{k^{\prime}+j^{\prime}}(\bar{f}\eta_{t-k})\ \mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{n}(\bar{f}\eta_{t-k^{\prime}})\\\
&+\frac{C\|f\|_{\infty}^{2}}{(N+1)^{2}}\sum_{0\leq n<m\leq
N}\alpha^{N-m}\theta_{0}^{N-m}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{m+t-k}V+\alpha^{m-n}\theta_{0}^{N-n-
k-j}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{n+t-k^{\prime}}V\\\
&\leq\frac{CtN\,\|f\|_{\infty}^{2}\,\theta_{0}^{N}}{(N+1)^{2}}\,\sup_{0\leq\ell<2t}(\nu
P_{\ell}\mathbbm{1}_{E})\,\nu(V)\,\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}(\eta_{t-i})\\\
&+\frac{C\|f\|_{\infty}^{2}\,\theta_{0}^{N}}{(N+1)^{2}}\,\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}V\,\sum_{0\leq n<m\leq
N}\left(\alpha^{N-m}+\alpha^{m-n}\right),\end{aligned}$

where we used in the last inequality that
$\theta_{0}^{-m}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{m+t-k}V\leq
C\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{t-i}V$ because $m+t-k=\ell
t-i$ for some integer $\ell$ and $\theta_{0}^{\ell t}Q_{\ell}V(x)\leq CV(x)$
for all $x\in A_{0}$ by (2.4), and similarly for
$\theta_{0}^{-n}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}P_{n+t-k^{\prime}}V$. Using that

$\sum_{0\leq n<m\leq
N}\left(\alpha^{N-m}+\alpha^{m-n}\right)\leq\frac{2N}{1-\alpha},$

we have proved that

$\frac{2}{(N+1)^{2}}\sum_{0\leq n<m\leq
N}\mathbb{E}_{\mu}\left[\bar{f}(X_{n})\bar{f}(X_{m})\mathbbm{1}_{N<\tau_{\partial}}\right]\leq\frac{C(\mu)\,\|f\|^{2}_{\infty}\,\theta_{0}^{N}}{N+1},$

where the constant $C(\mu)$ depends on $\mu$.

Now, it follows from Theorem 3.1 that, for $N$ large enough, given
$i\in\\{0,\ldots,t-1\\}$ such that $\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}(\eta_{t-i})>0$,

$\mathbb{P}_{\mu}(N<\tau_{\partial})\geq\mathbb{P}_{\mathchoice{{\mu\,\smash{\vrule
height=2.41112pt,depth=1.76631pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=1.76631pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.68777pt,depth=1.44104pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.20555pt,depth=1.44104pt}}_{\,A_{i}}}}(N<\tau_{\partial})\geq\frac{1}{2}\mathchoice{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=3.44444pt,depth=2.52328pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=2.41112pt,depth=2.05862pt}}_{\,A_{i}}}{{\mu\,\smash{\vrule
height=1.72221pt,depth=2.05862pt}}_{\,A_{i}}}(\eta_{t-i})\theta_{0}^{N-i-\ell}\nu
P_{i+\ell}\mathbbm{1}_{E}$

where $\ell\in\\{0,\ldots,t-1\\}$ is such that $N-\ell\in t\mathbb{Z}$. Hence,
there exists a constant $c(\mu)>0$ depending only on $\mu$ such that, for all
$N$, $\mathbb{P}_{\mu}(N<\tau_{\partial})\geq c(\mu)\theta_{0}^{N}$.
Therefore, (6.4) follows from (6.5) and Corollary 6.2 is proved. ∎

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