# Harris recurrent Markov chains and nonlinear monotone cointegrated models

Patrice<EMAIL_ADDRESS>[
Cécile<EMAIL_ADDRESS>[ Carlos
<EMAIL_ADDRESS>[ MODAL’X, UMR
CNRS 9023, Université Paris Nanterrepresep=, ]e1,e2  LTCI, Telecom Paris,
Institut Polytechnique de Parispresep=, ]e3

###### Abstract

In this paper, we study a nonlinear cointegration-type model of the form
$Z_{t}=f_{0}(X_{t})+W_{t}$ where $f_{0}$ is a monotone function and $X_{t}$ is
a Harris recurrent Markov chain. We use a nonparametric Least Square Estimator
to locally estimate $f_{0}$, and under mild conditions, we show its strong
consistency and obtain its rate of convergence. New results (of the Glivenko-
Cantelli type) for localized null recurrent Markov chains are also proved.

62G05,

62M05,

62G30,

monotone regression,

isotonic regression,

nonlinear cointegration,

nonparametric estimation,

null recurrent Markov chain,

###### keywords:

[class=MSC]

###### keywords:

and

##  1 Introduction and motivations

###  1.1 Linear and nonlinear cointegation models

Linear cointegration introduced by [16] and developed by [14] and [21, 22], is
a concept used in statistics and econometrics to describe a long-term
relationship between two or more time series. In general, these time series
are non-stationary, integrated of order 1, that is, they behave roughly as
random walks. In traditional linear cointegration analysis, variables are
assumed to have a linear relationship, which means their long-term
equilibrium, as time grows, is characterized by a constant linear combination.
This concept has since been extensively studied, particularly in the field of
econometrics [14, 30, 31, 21, 22]. Notice that, when there is indeed a
significant linear relationship, the link is monotone in each of the
variables.

However, in some cases, the relationship between variables may exhibit
nonlinear behavior, which cannot be adequately captured by linear
cointegration models. The incorporation of nonlinearities allows for a more
comprehensive understanding of long-term relationships between variables. [20]
have developed an approach for analyzing nonlinear cointegration through
threshold cointegration models. These models assume that the linear
relationship between variables differs after some changepoints, leading to
different long-run equilibrium states (for instance according to some latent
regimes). Threshold cointegration models provide a framework for capturing
nonlinearities in the data and estimating the changepoints. Refer to [32] for
examples and discussions on the importance to introduce nonlinearities in
cointegration applications and for further references.

Another method for analyzing nonlinear cointegration is through the use of
smooth transition cointegration models introduced by [17]. These models assume
a ECM (Error Correction Model) form and allow for smooth transitions between
different regimes in the data. Most estimators of non-linear cointegration may
be seen as Nadaraya-Watson estimators of the link function. For instance, Wang
and Phillips [36, 37, 35] show that it is possible to estimate and perform
asymptotic inference in specific nonparametric cointegration regression models
using kernel regression techniques. Furthermore, they established that the
self-normalized kernel regression estimators converge to a standard normal
distribution limit, even when the explanatory variable is integrated. These
findings indicate that the estimators can consistently capture the underlying
relationship between variables, even in cases where the explanatory variable
exhibits non-stationary behavior. The problem of estimating $f_{0}$ under the
Markovian assumptions has also been tackled using local linear M-type
estimators in [8, 26] using smoothing techniques.

These results have been partially extended in the framework of general
$\beta$-recurrent Markov chain and not just integrated $I(1)$ time series by
[23, 7]. Consider the simple framework where we observe two Markov chains,
$Z_{t}$ and $X_{t}$. [23] are essentially interested in the study of nonlinear
cointegrated models such as,

$Z_{t}=f_{0}(X_{t})+W_{t},$ (1)

where $f_{0}$ is a nonlinear function. $X_{t}$ and $W_{t}$ are independent
processes, and $X_{t}$ is a positive or $\beta$-null recurrent Markov chain.
Despite the fact that there is no stationary probability measure for $X_{t}$,
they apply the Nadaraya-Watson method to estimate $f_{0}$ and established the
asymptotic theory of the proposed estimator. The rate of convergence of the
estimators at some point $x$ is essentially linked to the local properties of
the $\beta$ -null recurrent chain $X_{t}$ and typically of the order of the
square root of the number of visits of the chain in a neighborhood of the
point $x$.

###  1.2 Monotone cointegration models: motivations

Monotonicity in cointegration is a rather natural assumption in many economic
applications, for instance for modeling demand as a function of income or
prices (see for instance [11]) or other variables. Suppose, for example, we
are interested in analyzing the long-term relationship between ice cream sales
and the average monthly temperature. These two non-stationary variables may be
modeled by some $\beta$-recurrent Markov chains. We hypothesize that as the
average monthly temperature increases, the demand for ice cream also
increases: however the rate of increase may vary according to the season. In
that case, the nonlinear relationship between the two variables will be
monotone. In microeconomics, the same phenomenon is expected for Engel curves,
describing how real expenditure varies with household income (see [11]).
Expenditures and income (or their log in most models) may be considered as
non-stationary variables. However considering a linear cointegration between
them may be misleading, since the relationship may change along the life
cycle. By Engle’s law, the relationship between the two variables should be
monotone. Other types of examples of monotone non-linear cointegration
phenomenon may be found in [32].

The purpose of this paper is to propose a simple estimator that is
automatically monotone, does not require strong smoothness assumptions (we
only require continuity of the link function), and operates under general
Markovian assumptions. We establish a nonparametric estimation theory of the
nonparametric least squares estimator (LSE) for the function $f_{0}$ in the
model (1) under the constraints that $f_{0}$ is monotone non-increasing. Here,
$\\{W_{t}\\}$ is an unobserved process such that $E(W_{t}|X_{t})=0$ to ensure
identifiability of $f_{0}$. Since a minimal condition for undertaking
asymptotic analysis on $f_{0}(x_{0})$ at a given point $x_{0}$ is that, as the
number of observations on $\\{X_{t}\\}$ increases, there must be infinitely
many observations in the neighborhood of $x_{0}$, the process $\\{X_{t}\\}$
will be assumed to be a Harris recurrent Markov chain (cf section 2). We
consider at the same time the stationary and $\beta$ null recurrent non-
stationary framework. To our knowledge, it is the first time that such an
estimator is proposed in the literature in such a large framework.

###  1.3 The estimator

Let $C$ be a set whose interior contains our point of interest $x_{0}$. Having
observed $\left(X_{t},Z_{t}\right)\\}_{t=0}^{n}$, we denote by $T_{n}(C)$ the
number of times that X visited $C$ up to time $n$ and by
$\sigma_{C}\left(i\right)$ the time of the $i$-th visit. Then, we consider the
nonparametric LSE defined as the minimizer of

$f\mapsto\sum\limits_{i=1}^{T_{n}(C)}{{{\left({{Z_{{\sigma_{C}\left(i\right)}}}-f\left({{X_{{\sigma_{C}\left(i\right)}}}}\right)}\right)}^{2}}}$
(2)

over the set of non-increasing functions $f$ on $\mathbb{R}$. The
nonparametric LSE $\widehat{f}_{n}$ has a well known characterization, as
follows. Let $m$ be the number of unique values of
$X_{\sigma_{C}\left(1\right)},\dots,X_{\sigma_{C}\left(T_{n}(C)\right)}$, and
$Y_{1}<\dots<Y_{m}$ be the corresponding order statistics. Then,
$\widehat{f}_{n}(Y_{k})$ is the left-hand slope at
$\sum_{i=1}^{T_{n}(C)}\mathbb{I}{\left\\{X_{\sigma_{C}\left(i\right)}\leqslant
Y_{k}\right\\}}$ of the least concave majorant of the set of points

$\left\\{(0,0),\
\left(\sum_{i=1}^{T_{n}(C)}\mathbb{I}{\left\\{X_{\sigma_{C}\left(i\right)}\leqslant
Y_{k}\right\\}},\sum_{i=1}^{T_{n}(C)}Z_{\sigma_{C}\left(i\right)}\mathbb{I}{\left\\{X_{\sigma_{C}\left(i\right)}\leqslant
Y_{k}\right\\}}\right),\ k=1,\dots,m\right\\},$ (3)

and it can be computed using simple algorithms as discussed in [3]. Thus, the
constrained LSE is uniquely defined at the observation points, however, it is
not uniquely defined between these points: any monotone interpolation of these
values is a constrained LSE. As is customary, we consider in the sequel the
piecewise-constant and left-continuous LSE that is constant on every interval
$(Y_{k-1},Y_{k}]$, $k=2,\dots,m$ and also on $(-\infty,Y_{1}]$ and on
$[Y_{m},\infty)$.

The use of a localized estimator is due to the fact that we need to control
the behavior of the chain around $x_{0}$, and, to do this, we need to estimate
the asymptotic ”distribution” of X in a vicinity of $x_{0}$. For Harris
recurrent Markov chains, the long-term behavior of the chain is given by its
invariant measure (see Section 2). In the positive recurrent case, the
invariant measure is finite and it can be estimated by simply considering the
empirical cumulative distribution function of the $X_{t}$. However, in the
null recurrent case, the invariant measure is only $\sigma$-finite, hence, we
need to localize our analysis in a set big enough such that the chain visits
it infinitely often, but small enough that the restriction of the invariant
measure to it is finite. Moreover, contrary to the bandwidth in kernel type
estimators, $C$ does not depend on $n$, and the rate of convergence of the
estimator does not depend on $C$.

###  1.4 Outline

Since our paper draws quite heavily on the theory of Harris recurrent Markov
chains, we have added a small introduction to the subject as well as the main
results that we use throughout the paper in Section 2. In Section 3, we show
that under very general assumptions, our estimator $\widehat{f}_{n}$ is
strongly consistent, while its rate of convergence is presented in Section 4.
In Section 5, we present new results concerning the localized empirical
process of Harris recurrent Markov chains that have emerged during our
investigation and we believe are interesting in their own right. Section 6
contains the proofs of our main results.

##  2 Markov chain theory and notation

In this section, we present the notation and main results related to Markov
chains that are needed to present our main results. For further details, we
refer the reader to the first section of the Supplementary Material [4] and
the books [29, 27, 12].

Consider a time-homogeneous irreducible Markov Chain, denoted as
$\textbf{X}=X_{0},X_{1},X_{2},\ldots$, defined on a probability space
$\left(E,\mathcal{E},\mathbb{P}\right)$, where $\mathcal{E}$ is countably
generated. The irreducibility measure of the chain is represented by $\psi$.
The transition kernel of the chain is denoted as $P\left(x,A\right)$ and its
initial distribution is represented by $\lambda$. If the initial measure of
the chain is specified, we use $\mathbb{P}_{\lambda}$ (and
$\mathbb{E}_{\lambda}$) to denote the probability (and the expectation)
conditioned on the law of the initial state
$\mathcal{L}\left(X_{0}\right)=\lambda$.

For any set $C\in\mathcal{E}$, we will denote by $\sigma_{C}$ and $\tau_{C}$,
respectively, the times of first visit and first return of the chain to the
set $C$, i.e. $\tau_{C}=\operatorname{inf}\left\\{n\geqslant 1:X_{n}\in
C\right\\}$ and $\sigma_{C}=\operatorname{inf}\left\\{n\geqslant 0:X_{n}\in
C\right\\}$. The subsequent visit and return times
$\sigma_{C},\tau_{C}\left(k\right)$, $k\geqslant 1$ are defined inductively.

Given that our methods will only deal with the values of X in a fixed set $C$,
if $A$ is a measurable set, we will write $\mathbb{I}_{C}\\{X_{t}\in A\\}$
instead of $\mathbb{I}\\{X_{t}\in A\cap C\\}$ and if $A=E$, then we will
simply write $\mathbb{I}_{C}\left(X_{t}\right)$. We will use
$T_{n}\left(C\right)$ to denote the random variable that counts the number of
times the chain has visited the set $C$ up to time $n$, that is
$T_{n}\left(C\right)=\sum_{t=0}^{n}{\mathbb{I}_{C}\left(X_{t}\right)}$.
Similarly, we will write $T\left(C\right)$ for the total of numbers of visits
the chain X to $C$. The set $C$ is called recurrent if
$\mathbb{E}_{x}T\left(C\right)=+\infty$ for all $x\in C$ and the chain X is
recurrent if every set $A\in\mathcal{E}$ such that $\psi\left(A\right)>0$ is
recurrent. A recurrent chain is called Harris recurrent if for all $x\in E$
and all $A\in\mathcal{E}$ with $\psi(A)>0$ we have
$\mathbb{P}\left({{X_{n}}\in A\;\text{infinitely
often}\left|{{X_{0}}=x}\right.}\right)=1$.

Denote by $\mathcal{E}^{+}$ the class of nonnegative measurable functions with
positive $\psi$ support. A function $s\in\mathcal{E}^{+}$ is called small if
there exists an integer $m_{0}\geqslant 1$ and a measure
$\nu\in\mathscr{M}{\left({\mathcal{E}}\right)_{+}}$ such that

${P^{m_{0}}}\left({x,A}\right)\geqslant
s\left(x\right)\nu\left(A\right)\quad\forall x\in E,A\in\mathcal{E}.$ (4)

When a chain possesses a small function $s$, we say that it satisfies the
minorization inequality $M\left(m_{0},s,\nu\right)$. A set $A\in\mathcal{E}$
is said to be small if the function $\mathbb{I}_{A}$ is small. Similarly, a
measure $\nu$ is small if there exist $m_{0}$, and $s$ that satisfy (4). By
Theorem 2.1 in [29], every irreducible Markov chain possesses a small function
and Proposition 2.6 of the same book shows that every measurable set $A$ with
$\psi\left(A\right)>0$ contains a small set. In practice, finding such a set
consists in most cases in exhibiting an accessible set, for which the
probability that the chain returns to it in $m$ steps is uniformly bounded
below. Moreover, under quite wide conditions a compact set will be small, see
[15].

An irreducible chain possesses an accessible atom, if there is a set
$\boldsymbol{\alpha}\in\mathcal{E}$ such that for all $x,y$ in
$\boldsymbol{\alpha}$: $P(x,.)=P(y,.)$ and $\psi(\boldsymbol{\alpha})>0$. When
an accessible atom exists, the stochastic stability properties of X amount to
properties concerning the speed of return time to the atom only. Moreover, it
follows from the strong Markov property that the sample paths may be divided
into independent blocks of random length corresponding to consecutive visits
to $\boldsymbol{\alpha}$. The sequence
$\left\\{\tau_{\boldsymbol{\alpha}}(j)\right\\}_{j\geqslant 1}$ defines
successive times at which the chain forgets its past, called regeneration
times. Similarly, the sequence of i.i.d. blocks
$\\{\mathcal{B}_{j}\\}_{j\geqslant 1}$ are named regeneration blocks. The
random variable $T\left({n}\right)=T_{n}\left(\boldsymbol{\alpha}\right)-1$
counts the number of i.i.d. blocks up to time $n$. This term is called number
of regenerations up to time $n$.

If X does not possess an atom but is Harris recurrent (and therefore satisfies
a minorization inequality $M\left(m_{0},s,\nu\right)$), a splitting technique,
introduced in [28, 29], allows us to extend in some sense the probabilistic
structure of X in order to artificially construct an atomic chain (named the
split chain and denoted by ${\check{\textbf{X}}}$) that inherits the
communication and stochastic stability properties from X. One of the main
results derived from this construction is the fact that every Harris recurrent
Markov chain admits a unique (up to multiplicative constant) invariant measure
(see Proposition 10.4.2 in [27]), that is, a measure $\pi$ such that

$\pi\left(B\right)=\int{P\left({x,B}\right)d\pi\left(x\right)}.\quad\forall
B\in\mathcal{E}.$

The invariant measure is finite if and only if
$\mathbb{E}_{\check{\boldsymbol{\alpha}}}{\tau_{\check{\boldsymbol{\alpha}}}}<+\infty$,
in this case we say the chain is positive recurrent, otherwise, we say the
chain is null recurrent. A null recurrent chain is called $\beta$-null
recurrent (c.f. Definition 3.2 in [24]) if there exists a small nonnegative
function $h$, a probability measure $\lambda$, a constant
$\beta\in\left({0,1}\right)$ and a slowly varying function $L_{h}$ such that

${\mathbb{E}_{\lambda}}\left({\sum\limits_{t=0}^{n}{h\left({{X_{t}}}\right)}}\right)\sim\frac{1}{{\Gamma\left({1+\beta}\right)}}{n^{\beta}}{L_{h}}\left(n\right)\quad\textnormal{as}\;n\to\infty.$

As argued in [24], is not a too severe restriction to assume $m_{0}=1$.
Therefore, throughout this paper we assume that X satisfies the minorization
inequality $M(1,s,\nu)$, i.e, there exist a measurable function $s$ and a
probability measure $\nu$ such that $0\leqslant s\left(x\right)\leqslant 1$,
$\int_{E}{s(x)d\nu(x)}>0$ and

${P}\left({x,A}\right)\geqslant s\left(x\right)\nu\left(A\right).$ (5)

######  Remark 2.1.

The extensions to the case where $m_{0}>1$ of the results that will be
presented in this paper can be carried out (although they involve some
complicated notations/proofs) using the $m$-skelethon or the resolvent chains,
as described in [9, 10] and Chapter 17 of [27]. However, they are not treated
in this paper.

The following theorem is a compendium of the main properties of Harris’s
recurrent Markov chains that will be used throughout the paper. Among other
things, it shows that the asymptotic behavior of $T\left({n}\right)$ is
similar to the function $u\left(n\right)$ defined as

$u\left(n\right)=\begin{cases}n,&\text{if }\textbf{X}\text{ is positive
recurrent}\\\ n^{\beta}L\left(n\right),&\text{if }\textbf{X}\text{ is
}\beta\text{-null recurrent}\end{cases}.$ (6)

######  Theorem 2.1.

Suppose X is a Harris recurrent, irreducible Markov chain, with initial
measure $\lambda$, that satisfies the minorization condition (5). Let
$T\left({n}\right)$ be the number of complete regenerations until time $n$ of
the split chain ${\check{\textbf{X}}}$ , let $C\in\mathcal{E}$ be a small set
and $\pi$ be an invariant measure for X. Then,

  1. 1.

$0<\pi\left(C\right)<+\infty$.

  2. 2.

$\frac{T\left({n}\right)}{T_{n}(C)}$ converges almost surely to a positive
constant.

  3. 3.

$\frac{T\left({n}\right)}{u\left(n\right)}$ converges almost surely to a
positive constant if X is positive recurrent and converges in distribution to
a Mittag-Leffler111The Mittag-Leffler distribution with index $\beta$ is a
non-negative continuous distribution, whose moments are given by
$\mathbb{E}\left(M^{m}_{\beta}\left(1\right)\right)=\frac{m!}{\Gamma\left(1+m\beta\right)}\;\;m\geqslant
0.$ See (3.39) in [24] for more details. random variable with index $\beta$ if
X is $\beta$-null recurrent.

##  3 Consistency

The aim of the section is to show that for an arbitrary $x_{0}$ in the support
of $f_{0}$, the LSE $\widehat{f}_{n}(x_{0})$ is consistent. We make the
following assumptions on the processes $\textbf{X}=\\{X_{t}\\}$ and
$\textbf{W}=\\{W_{t}\\}$.

  * (A1)

X is a Harris recurrent Markov chain whose kernel $P(x,A)$ satisfies the
minorization condition (5).

Let $\mathcal{F}_{n}=\sigma\left(\left\\{X_{0},\ldots,X_{n}\right\\}\right)$
be sigma algebra generated by the chain X up to time $n$.

  * (A2)

For each $n$, the random variables $W_{1},\ldots,W_{n}$ are conditionally
independent given ${{\mathcal{F}_{n}}}$,
$\mathbb{E}{\left(W_{t}|\mathcal{F}_{n}\right)}=0$ and
$\operatorname{Var}\left(W_{t}|\mathcal{F}_{n}\right)\leqslant\sigma^{2}$ for
some $\sigma>0$.

It follows from Assumption (A1) that the Markov Chain X admits a unique (up to
a multiplicative constant) $\sigma$-finite invariant measure $\pi$. Let $C$ be
a set such that $0<\pi\left(C\right)<\infty$ and $x_{0}\in C$. We denote by
$F_{n}$ the process defined by

$F_{n}(y)=\frac{1}{T_{n}(C)}\sum_{i=1}^{T_{n}(C)}{\mathbb{I}{\\{X_{\sigma_{C}\left(i\right)}\leqslant
y\\}}}=\frac{1}{T_{n}(C)}\sum_{t=0}^{n}\mathbb{I}_{C}{\\{X_{t}\leqslant y\\}}$
(7)

for all $y\in\mathbb{R}$, which is a localized version of the empirical
distribution function of the $X_{t}$’s. It is proved in Lemma 5.1 that $F_{n}$
converges almost surely to the distribution function $F$ supported on $C$ and
defined by

${F}\left(y\right)=\frac{{\pi\left({C\cap\left({-\infty,y}\right]}\right)}}{{\pi\left(C\right)}}.$
(8)

Our next two assumptions guarantee that there is a compact $C$, that is a
small set and contains $x_{0}$ as an interior point. Sets like this can be
found under very wide conditions (cf [15]).

  * (A3)

There is $\delta=\delta(x_{0})$ such that the set
$C=\left[{{x_{0}}-\delta,{x_{0}}+\delta}\right]$ is small.

  * (A4)

$x_{0}$ belongs to the interior of the support of $X_{t}$.

Notice that by part 1 of Theorem 2.1, (A3) guarantees that $\pi(C)$ is finite
and positive, and hence, $F$ is properly defined.

In addition to the assumptions on the processes $\\{X_{t}\\}$ and
$\\{W_{t}\\}$, we need smoothness assumptions on $F$ and on $f_{0}$. In
particular, we will assume that $F$ and $f_{0}$ are continuous and strictly
monotone in $C$. This implies that $f_{0}$ and $F$ are invertible in $C$, so
we can find neighborhoods of $f_{0}(x_{0})$ and $F(x_{0})$ respectively, over
which the inverse functions are uniquely defined. We denote by $f_{0}^{-1}$
and $F^{-1}$ respectively the inverses of $f_{0}$ and $F$ over such a
neighborhood of $f_{0}(x_{0})$ and $F(x_{0})$ respectively. The function
$f_{0}$ is assumed to be monotone on its whole support.

  * (A5)

$F$ is locally continuous and strictly increasing in the sense that for all
$x^{\prime}$ in $C$, for all $\varepsilon>0$, there exists $\gamma>0$ such
that $|F^{-1}(u)-x^{\prime}|>\gamma$ for all $u$ such that
$|u-F(x^{\prime})|\geqslant\varepsilon$.

  * (A6)

$f_{0}$ is non-increasing, and $f_{0}$ is locally strictly decreasing in the
sense that for all $x^{\prime}$ in $C$, for all $\varepsilon>0$, there exists
$\gamma>0$ such that $|f_{0}(x^{\prime})-f_{0}(y)|>\gamma$ for all $y$ such
that $|y-x^{\prime}|\geqslant\varepsilon$.

  * (A7)

$f_{0}$ continuous in $C$.

Assumptions (A1), (A3) and (A5) ensure that $X_{t}$ visits infinitely many
times any small enough neighborhood of $x_{0}$ with probability 1, and
guarantee that $x_{0}$ is not at the boundary of the recurrent states.
Assumptions (A1) and (A3) and Lemma 3.2 in [24] imply that $T_{n}(C)\to\infty$
almost surely.

######  Theorem 3.1.

Suppose that assumptions (A1)-(A7) are satisfied. Then, as $n\to\infty$, one
has

$\widehat{f}_{n}(x_{0})=f_{0}(x_{0})+o_{P}(1),$ (9)

and

$\widehat{f}_{n}^{-1}(f_{0}(x_{0}))=x_{0}+o_{P}(1).$ (10)

##  4 Rates of convergence

To compute rates of convergence, we need stronger assumptions than for
consistency. We replace assumption (A1) for the following stronger version

  * (B1)

$\\{X_{t}\\}$ is a positive or $\beta$-null recurrent, aperiodic and
irreducible Markov Chain whose kernel $P(x,A)$ satisfies the minorization
condition (5).

We replace, (A5), (A6) and (A7), for the following slightly more restrictive
assumption

  * (B2)

The function $f_{0}$ is non-increasing, the functions $f_{0}$ and $F_{C}$ are
differentiable in $C$, and the derivatives $F_{C}^{\prime}$ and
$f_{0}^{\prime}$ are bounded, in absolute value, above and away from zero in
$C$.

Let $\lambda$ be the initial measure of X. Our next hypothesis imposes some
control on the behavior of the chain in the first regenerative block.

  * (B3)

There exists a constant $K$ and a neighborhood $V$ of 0, such that

$\mathbb{E}_{\lambda}\left(\sum\limits_{t=0}^{{\tau_{\check{\boldsymbol{\alpha}}}}}{\left({{\mathbb{I}_{C}}\left\\{{{X_{t}}\leqslant{x_{0}}+\gamma}\right\\}-{\mathbb{I}_{C}}\left\\{{{X_{t}}\leqslant{x_{0}}-\gamma}\right\\}}\right)}\right)\leqslant
K\gamma\quad\forall\gamma\in V.$

Assumption (B3)is satisfied if we assume that the initial measure of the chain
is the small measure used for the construction of the split chain (see
equation 4.16c in [29]). In the positive recurrent case, taking $\lambda$
equal to the unique invariant probability measure of the chain also satisfies
(B3).

And finally, we need to control the number of times the chain visits $C$ in a
regeneration block.

  * (B4)

$\ell_{C}(\mathcal{B}_{1})=\sum_{t\in\mathcal{B}_{1}}\mathbb{I}_{C}{\left\\{X_{t}\right\\}}$
has finite second moment.

######  Theorem 4.1.

Assume that (A2), (A3), (A4), (B1), (B2), (B3) and (B4) hold. Then, as
$n\to\infty$, one has

$\widehat{f}_{n}(x_{0})=f_{0}(x_{0})+O_{P}(u\left(n\right)^{-1/3}),$ (11)

with $u\left(n\right)$ as defined in (6).

The rate $u\left(n\right)$ comes from Lemmas 5.3 and 6.7, and as it can be
seen from Theorem 2.1, it is a deterministic approximation of
$T\left({n}\right)$. Note that in the positive recurrent case,
$u\left(n\right)=n$, hence we obtain the same rate $n^{-1/3}$ as in the i.i.d.
case [18, Chapter 2]. In the $\beta$-null recurrent case, however, the rate of
convergence is $n^{\beta/3}L^{1/3}\left(n\right)$ which is slower than the
usual rate. This is due to the null-recurrence of the chain because it takes
longer for the process to return to a neighborhood of the point $x_{0}$ and it
is these points in the neighborhood of $x_{0}$ which are used in nonparametric
estimation.

##  5 Localized Markov chains

Given the localized nature of our approach, in this section, we present some
results that are particularly useful in this scenario. These results are well
known for positive recurrent chains but are new in the null recurrent case.
The detailed proofs of these results can be found in Section 2 of the
Supplementary material [4].

The first result can be viewed as an extension of the Glivenko-Cantelli
theorem to the localized scenario.

######  Lemma 5.1.

Assume that (A1) and (A3) hold. Then, there exists a stationary
$\sigma$-finite measure $\pi$, and $F$ defined by (8), such that,

$\mathop{\sup}\limits_{y\in\mathbb{R}}\left|{{F_{n}}\left(y\right)-F\left(y\right)}\right|\to
0\quad a.s.$ (12)

as $n\to\infty$. If (A5) is also satisfied, then, for all sufficiently small
$\varepsilon>0$, as $n\to\infty$ we have

$\sup\limits_{\left|p-F\left(x_{0}\right)\right|\leqslant\varepsilon}\left|{F_{n}^{-1}\left(p\right)-F^{-1}\left(p\right)}\right|\to
0\quad a.s.$ (13)

Our next result (Lemma 5.2), which is an extension of Lemma 2 in [5] to the
localized $\beta$-null recurrent case, deals with the properties of classes of
functions defined over the regeneration blocks. Before presenting the result,
we need some machinery.

Recall that $E\subseteq\mathbb{R}$ denotes the state space of $X$. Define
$\widehat{E}=\cup_{k=1}^{\infty}E^{k}$ (i.e. the set of finite subsets of $E$)
and let the localized occupation measure $M_{C}$ be given by

$\displaystyle M_{C}(B,dy)=\sum_{x\in B\cap C}\delta_{x}(y),\qquad\text{for
every }B\in\widehat{E}.$

The function that gives the size of the localized blocks is
$\ell_{C}:\widehat{E}\to\mathbb{N}$

$\displaystyle\ell_{C}(B)=\int\,M_{C}(B,\mathrm{d}y),\qquad\text{ for every
}B\in\widehat{E}.$

Let $\widehat{\mathcal{E}}$ denote the smallest $\sigma$-algebra formed by the
elements of the $\sigma$-algebras $\mathcal{E}^{k}$, $k\geqslant 1$, where
$\mathcal{E}^{k}$ stands for the classical product $\sigma$-algebra. Let
$\widehat{Q}$ denote a probability measure on
$(\widehat{E},\widehat{\mathcal{E}})$. If $B(\omega)$ is a random variable
with distribution $Q^{\prime}$, then $M_{C}(B(\omega),\mathrm{d}y)$ is a
random measure, i.e., $M_{C}(B(\omega),\mathrm{d}y)$ is a (counting) measure
on $(E,\mathcal{E})$, almost surely, and for every $A\in\mathcal{E}$,
$M_{C}(B(\omega),A)=\int_{A}\,M_{C}(B(\omega),\mathrm{d}y)$ is a measurable
random variable (valued in $\mathbb{N}$). Henceforth
$\ell(B(\omega))\times\int f(y)\,M_{C}(B(\omega),\mathrm{d}y)$ is a random
variable and, provided that $\widehat{Q}(\ell^{2})<\infty$, the map $Q_{C}$,
defined by

$\displaystyle
Q_{C}(A)={E}_{\widehat{Q}}\left(\ell_{C}(B)\times\int_{A}\,M_{C}(B,\mathrm{d}y)\right)/E_{\widehat{Q}}(\ell_{C}^{2}),\qquad\text{for
every }A\in\mathcal{E},$ (14)

is a probability measure on $(E,\mathcal{E})$. The notation $E_{Q_{C}}$ stands
for the expectation with respect to the underlying measure $Q_{C}$. Introduce
the following notations: for any function $g:E\to\mathbb{R}$, let
$\widehat{g}_{C}:\widehat{E}\to\mathbb{R}$ be given by

$\widehat{g}_{C}\left(B\right)=\int{g\left(y\right){M_{C}}\left({B,dy}\right)}=\sum\limits_{x\in
B\cap C}{g\left(x\right)}=\sum\limits_{x\in B}{{g_{C}}\left(x\right)},$ (15)

and for any class $\mathcal{G}$ of real-valued functions defined on $E$,
denote the localized version of the sums on the blocks by
$\widehat{G}_{C}=\\{\widehat{g}_{C}\,:\,g\in\mathcal{G}\\}$.

Notice that, for any function $g$,

${\mathbb{E}_{{Q_{C}}}}\left(g\right)=\frac{{{\mathbb{E}_{{\widehat{Q}}}}\left({{\ell_{C}}\left(B\right)\times\int{g\left(y\right){M_{C}}\left({B,dy}\right)}}\right)}}{{{\mathbb{E}_{{\widehat{Q}}}}\left({\ell_{C}^{2}}\right)}}=\frac{{{\mathbb{E}_{{\widehat{Q}}}}\left({{\ell_{C}}\left(B\right)\widehat{g}_{C}\left(B\right)}\right)}}{{{\mathbb{E}_{{\widehat{Q}}}}\left({\ell_{C}^{2}}\right)}}.$
(16)

######  Lemma 5.2.

Let $\widehat{Q}$ be a probability measure on
$(\widehat{E},\widehat{\mathcal{E}})$ such that
$0<\|\ell_{C}\|_{L^{2}(\widehat{Q})}<\infty$ and $\mathcal{G}$ be a class of
measurable real-valued functions defined on $(E,\mathcal{E})$. Then we have,
for every $0<\varepsilon<\infty$,

$\mathcal{N}\left({\varepsilon{{\left\|{{\ell_{C}}}\right\|}_{{L_{2}}\left({{\widehat{Q}}}\right)}},\widehat{\mathcal{G}}_{C},{L^{2}}\left({{\widehat{Q}}}\right)}\right)\leqslant\mathcal{N}\left({\varepsilon,\mathcal{G},{L^{2}}\left(Q\right)}\right),$

where $Q$ is given in (14). Moreover, if $\mathcal{G}$ belongs to the
Vapnik–Chervonenkis (VC) class of functions with constant envelope $U$ and
characteristic $(\textbf{C},v)$, then $\widehat{\mathcal{G}}$ is VC with
envelope $U\ell_{C}$ and characteristic $(\textbf{C},v)$.

######  Remark 5.1.

For a probability measure $\mu$, and a class of functions $\mathcal{H}$, the
covering number
$\mathcal{N}\left({\varepsilon,\mathcal{H},{L^{r}}\left(\mu\right)}\right)$ is
the minimum number of $L^{r}\left(\mu\right)$ $\varepsilon$-balls needed to
cover $\mathcal{H}$. For more details about this concept and the VC class of
functions, see [25].

To put into perspective Lemma 5.2, consider a class of bounded functions
$\mathcal{G}$ that is VC with finite envelope. Lemma 5.2 tells us that the
class of unbounded functions $\widehat{\mathcal{G}}_{C}$ is also VC. If we
also have that (B4) holds, then Theorem 2.5 in [25] tells us that
$\widehat{\mathcal{G}}_{C}$ is a Donsker class. A reasoning like this is used
in the proof of the following result, which is a stronger version of Lemma 5.1
under assumptions (B1) and (B2) and has some interest on its own.

######  Lemma 5.3.

Assume that (B1), (B2), (A3), (A4) and (B4) hold. Then, for all sufficiently
small $\varepsilon>0$ we have,

${T_{n}}(C)\mathop{\sup}\limits_{|y-x_{0}|\leqslant\varepsilon}{\left|{F_{n}(y)-F(y)}\right|^{2}}={O_{p}}\left(1\right)$
(17)

when $n$ goes to $+\infty$. If (B2) is also satisfied, as $n\to\infty$ we have

${T_{n}}(C)\mathop{\sup}\limits_{|p-F(x_{0})|\leqslant\varepsilon}{\left|{F_{n}^{-1}(p)-{F^{-1}}(p)}\right|^{2}}={O_{p}}\left(1\right).$
(18)

##  6 Proofs

In this section, we provide the proof of Theorems 3.1 and 4.1. These proofs
make use of several intermediate lemmas, whose proofs can be found in Sections
7 and 8.

###  6.1 Proof of Theorem 3.1

Recall that we consider the piecewise constant and left-continuous LSE
$\widehat{f}_{n}$, that is constant on every interval $(Y_{k-1},Y_{k}]$,
$k=2,\dots,m$ and also on $(-\infty,Y_{1}]$ and on $[Y_{m},\infty)$. With
$\delta>0$ fixed, we denote by $T_{n}(C)$ the number of times the Markov Chain
X visits the set $C:=[x_{0}-\delta,x_{0}+\delta]$ until time $n$:

$T_{n}(C)=\sum_{t=0}^{n}\mathbb{I}{\\{X_{t}\in C\\}}.$ (19)

Let $l_{k}=\sum_{t=1}^{n}\mathbb{I}_{C}{\left\\{X_{t}\leqslant
Y_{k}\right\\}}$ for all $k\in\\{1,\dots,m\\}$ and $l_{0}=0$.

Our aim is to provide a characterization of $\widehat{f}_{n}(x_{0})$. Recall
from (7) that the localized empirical distribution function $F_{n}$ is defined
as

$F_{n}(y)=\frac{1}{T_{n}(C)}\sum_{i=0}^{T_{n}(C)}{\mathbb{I}{\\{X_{\sigma_{C}\left(i\right)}\leqslant
y\\}}}=\frac{1}{T_{n}(C)}\sum_{t=0}^{n}\mathbb{I}_{C}{\\{X_{t}\leqslant y\\}}$

for $y\in\mathbb{R}$. $F_{n}$ is 0 on $\left({-\infty,{Y_{1}}}\right)$, so,
with an arbitrary random variable $Y_{0}<Y_{1}$ we have
$F_{n}(y)=F_{n}(Y_{0})=0$ for all $y<Y_{1}$. Let $\cal K$ be the set

$\mathcal{K}:=\left\\{F_{n}(Y_{k}),\ k=0,\dots,m\right\\}$ (20)

and let $\Lambda_{n}$ be the continuous piecewise-linear process on
$[F_{n}(Y_{0}),F_{n}(Y_{m})]$ with knots at the points in $\cal K$ and values

$\Lambda_{n}\left(F_{n}(Y_{k})\right)=\frac{1}{T_{n}(C)}\sum_{t=0}^{n}Z_{t}\mathbb{I}_{C}{\left\\{X_{t}\leqslant
Y_{k}\right\\}}$ (21)

at the knots. The characterization of $\widehat{f}_{n}$ in Lemma 6.2 involves
the least concave majorant of $\Lambda_{n}$. Note that we use $T_{n}(C)$ as a
normalization in the definitions of the processes $F_{n}$ and $\Lambda_{n}$
since this choice ensures that $F_{n}$ and $\Lambda_{n}$ converge to fixed
functions, see Lemma 5.1.

######  Lemma 6.1.

For all
$y\in\left[{F_{n}\left(Y_{0}\right),{F_{n}}\left({{Y_{m}}}\right)}\right]$,

${\Lambda_{n}}\left(y\right)={L_{n}}\left(y\right)+{M_{n}}\left(y\right),$

where,

${L_{n}}\left(y\right)=\int\limits_{0}^{y}{f\circ
F_{{}_{n}}^{-1}\left(u\right)du},$ (22)

and $M_{n}$ is a piece-wise linear processes with knots at $F_{n}(Y_{k})$ for
$k\in\\{0,\dots,m\\}$ such that

$M_{n}(F_{n}(Y_{k}))=\frac{1}{T_{n}(C)}\sum_{t=0}^{n}W_{t}\mathbb{I}_{C}{\\{X_{t}\leqslant
Y_{k}\\}}.$

Moreover, $M_{n}$ can be written as

$M_{n}(y)=\left\\{\begin{array}[]{crl}0&,&\textnormal{if}\;y=0\\\
R_{n}^{j}(y)+M_{n}^{j}&,&\textnormal{otherwise}\end{array}\right.$ (23)

where,

$\displaystyle M_{n}^{j}$
$\displaystyle=M_{n}(F_{n}(Y_{j}))=\frac{1}{T_{n}(C)}\sum_{t=0}^{n}W_{t}\mathbb{I}_{C}{\\{X_{t}\leqslant
Y_{j}\\}},$ (24) $\displaystyle R_{n}^{j}\left(y\right)$
$\displaystyle=\frac{\sum\limits_{t=0}^{n}{{W_{t}}\mathbb{I}_{C}{\\{X_{t}=Y_{j+1}\\}}}}{{{l_{j+1}}-{l_{j}}}}\left({y-F_{n}\left(Y_{j}\right)}\right),$
(25)

and $j$ is such that $Y_{j+1}=F_{n}^{-1}\left(y\right)$.

In the next lemma, we give an alternative characterization of the monotone
nonparametric LSE $\widehat{f}_{n}$ at the observation points
$Y_{1},\dots,Y_{m}$.

######  Lemma 6.2.

Let $C=[x-\delta,x+\delta]$ for some fixed $\delta>0$. Let
$\widehat{\lambda}_{n}$ be the left-hand slope of the least concave majorant
of $\Lambda_{n}$. Then,

$\widehat{f}_{n}(Y_{k})=\widehat{\lambda}_{n}\circ F_{n}(Y_{k}),\quad\forall
k\in\\{1,\dots,m\\}.$ (26)

with probability 1 for $n$ big enough.

We consider below the generalized inverse function of $\widehat{f}_{n}$ since
it has a more tractable characterization than $\widehat{f}_{n}$ itself. To
this end, let us define precisely the generalized inverses of all processes of
interest. Since $\widehat{\lambda}_{n}$ is a non-increasing left-continuous
step function on $(F_{n}(Y_{0}),F_{n}(Y_{m})]$ that can have jumps only at the
points $F_{n}(Y_{k})$, $k\in\\{1,\dots,m\\}$, we define its generalized
inverse $\widehat{U}_{n}(a)$, for $a\in\mathbb{R}$, as the greatest
$y\in(F_{n}(Y_{0}),F_{n}(Y_{m})]$ that satisfies
$\widehat{\lambda}_{n}(y)\geqslant a$, with the convention that the supremum
of an empty set is $F_{n}(Y_{0})$. Then for every $a\in\mathbb{R}$ and
$y\in(F_{n}(Y_{0}),F_{n}(Y_{m})]$, one has

$\widehat{\lambda}_{n}(y)\geqslant a\mbox{ if and only if
}\widehat{U}_{n}(a)\geqslant y.$ (27)

Likewise, since $\widehat{f}_{n}$ is a left-continuous non-increasing step
function on $\mathbb{R}$ that can have jumps only at the observation times
$Y_{1}<\dots<Y_{m}$, we define the generalized inverse
$\widehat{f}_{n}^{-1}(a)$, for $a\in\mathbb{R}$, as the greatest
$y\in[Y_{0},Y_{m}]$ that satisfies $\widehat{f}_{n}(y)\geqslant a$, with the
convention that the supremum of an empty set is $Y_{0}$. We then have

$\widehat{f}_{n}(y)\geqslant a\mbox{ if and only if
}\widehat{f}_{n}^{-1}(a)\geqslant y$ (28)

for all $a\in\mathbb{R}$ and $y\in(Y_{0},Y_{m}]$. On the other hand, since
$F_{n}$ is a right-continuous non-decreasing step function on $\mathbb{R}$
with range $[F_{n}(Y_{0}),F_{n}(Y_{m})]$, we define the generalized inverse
$F_{n}^{-1}(a)$, for $a\leqslant F_{n}(Y_{m})$, as the smallest
$y\in[Y_{0},Y_{m}]$ which satisfies $F_{n}(y)\geqslant a$. Note that the
infimum is achieved for all $a\leqslant F_{n}(Y_{m})$. We then have

$F_{n}(y)\geqslant a\mbox{ if and only if }F_{n}^{-1}(a)\leqslant y$ (29)

for all $a\leqslant F_{n}(Y_{m})$ and $y\in[Y_{0},Y_{m}]$, and thanks to Lemma
6.2 we have

$\widehat{f}_{n}^{-1}=F_{n}^{-1}\circ\widehat{U}_{n}$ (30)

on $\mathbb{R}$. Moreover, one can check that

$\widehat{U}_{n}(a)=\mathop{\mbox{\sl\em
argmax}}_{p\in[F_{n}(Y_{0}),F_{n}(Y_{m})]}\\{\Lambda_{n}(p)-ap\\},\mbox{ for
all }a\in\mathbb{R},$ (31)

where argmax denotes the greatest location of maximum (which is achieved on
the set $\cal K$ in (20)). Thus, the inverse process $\widehat{U}_{n}$ is a
location process that is more tractable than $\widehat{f}_{n}$ and
$\widehat{\lambda}_{n}$ themselves. A key idea in the following proofs is to
derive properties of $\widehat{U}_{n}$ from its argmax characterization (31),
then, to translate these properties to $\widehat{f}_{n}^{-1}$ thanks to (30),
and finally to translate them to $\widehat{f}_{n}$ thanks to (28).

To go from $\widehat{U}_{n}$ to $\widehat{f}_{n}^{-1}$ using (30) requires to
approximate $F_{n}^{-1}$ by a fixed function. Hence, in the sequel, we are
concerned by the convergence of the process $F_{n}$ given in (7), where
$\delta>0$ is chosen sufficiently small, and by the convergence of the
corresponding inverse function $F_{n}^{-1}$.

It is stated in Lemma 5.1 that under (A1) and (A3), $F_{n}$ converges to a
fixed distribution function $F$ that depends on $C$, hence on $\delta$. If,
moreover, $F$ is strictly increasing in $C$, then we can find a neighborhood
of $F(x_{0})$ over which the (usual) inverse function $F^{-1}$ is uniquely
defined, and $F_{n}^{-1}$ converges to $F^{-1}$.

In the following lemma, we show that $F(x_{0})$ belongs to the domain of
$\Lambda_{n}$ with probability tending to one as $n\to\infty$.

######  Lemma 6.3.

Assume that (A1), (A3), (A4) and (A5) hold. Then, we can find $\varepsilon>0$
such that the probability that $Y_{1}+\varepsilon\leqslant x_{0}\leqslant
Y_{m}-\varepsilon$ tends to one as $n\to\infty$. Moreover, the probability
that $F_{n}(Y_{1})\leqslant F(x_{0})\leqslant F_{n}(Y_{m})$ tends to one as
$n\to\infty$.

We will also need to control the noise $\\{W_{t}\\}$. The following lemma
shows that the noise is negligible under our assumptions.

######  Lemma 6.4.

Assume that (A1) and (A2) hold. Let
$\mathcal{F}_{n}=\sigma\left(\left\\{X_{1},\ldots,X_{n}\right\\}\right)$.
Then,

$\sum\limits_{t=0}^{n}{{W_{t}}{\mathbb{I}_{C}}{\left\\{{{X_{t}}={A_{n}}}\right\\}}}={o_{P}}\left({T_{n}(C)}\right),$

and

$\mathop{\sup}\limits_{u>{A_{n}}}\left|{\sum\limits_{t=0}^{n}{{W_{t}}{\mathbb{I}_{C}}\left\\{{{X_{t}}\in\left({{A_{n}},u}\right]}\right\\}}}\right|={o_{P}}\left({T_{n}(C)}\right).$

for any sequence of random variables $A_{n}$, independent of the process
$\\{W_{t}\\}$, that is adapted to the filtration $\\{\mathcal{F}_{n}\\}$.

With the above lemmas, we can prove convergence of $\widehat{U}_{n}$ to
$F\left(x_{0}\right)$ given by (31).

######  Lemma 6.5.

Suppose that assumptions (A1)-(A7) are satisfied. Then, as $n\to\infty$, one
has

$\widehat{U}_{n}(f_{0}(x_{0}))=F(x_{0})+o_{P}(1).$ (32)

Now we proceed to the proof of (10). Fix $\varepsilon>0$ arbitrarily small. It
follows from (30) and (29) that

$\displaystyle\mathbb{P}\left(\widehat{f}_{n}^{-1}(a)>x_{0}+\varepsilon\right)$
$\displaystyle\leqslant$
$\displaystyle\mathbb{P}\left(F_{n}^{-1}\circ\widehat{U}_{n}(a)>x_{0}+\varepsilon\right)$
$\displaystyle\leqslant$
$\displaystyle\mathbb{P}\left(\widehat{U}_{n}(a)\geqslant
F_{n}(x_{0}+\varepsilon)\right)$ $\displaystyle\leqslant$
$\displaystyle\mathbb{P}\left(\widehat{U}_{n}(a)\geqslant
F(x_{0}+\varepsilon)-K_{n}\right),$

where

$K_{n}=\sup_{|y-x_{0}|\leqslant\varepsilon}|F_{n}(y)-F(y)|.$

With $\nu:=F(x_{0}+\varepsilon)-F(x_{0})$, we obtain

$\displaystyle\mathbb{P}\left(\widehat{f}_{n}^{-1}(a)>x_{0}+\varepsilon\right)$
$\displaystyle\leqslant$
$\displaystyle\mathbb{P}\left(\widehat{U}_{n}(a)\geqslant F(x_{0})+\nu-
K_{n}\right),$

and $\nu$ is strictly positive since $F$ is strictly increasing in the
neighborhood of $x_{0}$. Hence, it follows from (12) that for sufficiently
small $\varepsilon>0$ one has

$\displaystyle\mathbb{P}\left(\widehat{f}_{n}^{-1}(a)>x_{0}+\varepsilon\right)$
$\displaystyle\leqslant$
$\displaystyle\mathbb{P}\left(\widehat{U}_{n}(a)\geqslant
F(x_{0})+\nu/2\right)+o(1),$

so it follows from (32) that the probability that
$\widehat{f}_{n}^{-1}(a)>x_{0}+\varepsilon$ tends to zero as $n\to\infty$.
Similarly, the probability that $\widehat{f}_{n}^{-1}(a)<x_{0}-\varepsilon$
tends to zero as $n\to\infty$ so we conclude that the probability that
$|\widehat{f}_{n}^{-1}(a)-x_{0}|>\varepsilon$ tends to zero as $n\to\infty$.
This completes the proof of (10).

To prove (9), fix $\varepsilon>0$ sufficiently small so that $F$ and $f_{0}$
are continuous and strictly increasing in the neighborhood of
$x^{\prime}:=f_{0}^{-1}(f_{0}(x_{0})+\varepsilon)$. Equation (10) shows that

$\widehat{f}_{n}^{-1}(f_{0}(x_{0})+\varepsilon)=f_{0}^{-1}(f_{0}(x_{0})+\varepsilon)+o_{P}(1),$
(33)

as $n\to\infty$. Now, it follows from the switch relation (27) that

$\displaystyle\mathbb{P}\left(\widehat{f}_{n}(x_{0})>f_{0}(x_{0})+\varepsilon\right)$
$\displaystyle\leqslant\mathbb{P}\left(\widehat{f}_{n}^{-1}(f_{0}(x_{0})+\varepsilon)\geqslant
x\right)$
$\displaystyle\leqslant\mathbb{P}\left(\widehat{f}_{n}^{-1}(f_{0}(x_{0})+\varepsilon)\geqslant
f_{0}^{-1}(f_{0}(x_{0})+\varepsilon)+\nu\right),$ (34)

where $\nu:=x-f_{0}^{-1}(f_{0}(x_{0})+\varepsilon)>0$. It follows from (33)
that the probability on the right-hand side tends to zero as $n\to\infty$.
Hence, the probability on the left-hand side tends to zero as well as
$n\to\infty$.

Similarly, the probability that
$\widehat{f}_{n}(x_{0})<f_{0}(x_{0})-\varepsilon$ tends to zero as
$n\to\infty$ so we conclude that the probability that
$|\widehat{f}_{n}(x_{0})-f_{0}(x_{0})|>\varepsilon$ tends to zero as
$n\to\infty$. This completes the proof of Theorem 3.1.

###  6.2 Proof of Theorem 4.1

The proof of Theorem 4.1, uses similar ideas as the ones used in the proof of
Theorem 3.1 but under stronger assumptions (and therefore using stronger
lemmas).

The first intermediate result is the following stronger version of Lemma 6.4.

######  Lemma 6.6.

Assume that (A2), (A3), (A4), (B1), (B2) and (B3) hold. Then, there exists
$K>0$, $\gamma_{0}>0$ that do not depend on $n$ and
$N_{\gamma_{0}}\in\mathbb{N}$, such that for all $\gamma\in[0,\gamma_{0}]$ and
$n\geqslant N_{\gamma_{0}}$ one has

$\displaystyle\mathbb{E}_{\lambda}\left(\sup_{|y-x_{0}|\leqslant\gamma}\left|\sum_{t=0}^{n}W_{t}\left(\mathbb{I}_{C}{\\{X_{t}\leqslant
y\\}}-\mathbb{I}_{C}{\\{X_{t}\leqslant x_{0}\\}}\right)\right|^{2}\right)$
$\displaystyle\leqslant Ku\left(n\right)\gamma$ (35)
$\displaystyle\mathbb{E}_{\lambda}\left(\sup_{|y-x_{0}|\leqslant\gamma}\left|\sum_{t=0}^{n}W_{t}\mathbb{I}_{C}{\\{X_{t}=y\\}}\right|^{2}\right)$
$\displaystyle\leqslant Ku\left(n\right)\gamma$ (36)

Then, we need to quantify how well we can approximate $T_{n}(C)$ by
$u\left(n\right)$.

######  Lemma 6.7.

Assume that (B1) and (A3) hold. Then we have

  * a)

As $n\to\infty$ we have

$\frac{u\left(n\right)}{T_{n}(C)}=O_{P}(1).$

  * b)

Let $\alpha$ and $\eta$ be positive constants, then there positive exists
constants $N_{\eta}$, $\underline{c}_{\eta}$ and $\overline{c}_{\eta}$, such
that

$\mathbb{P}\left({{{\left({\frac{{T_{n}\left(C\right)}}{{a\left(n\right)}}}\right)}^{\alpha}}\in\left[{\underline{c}_{\eta},\overline{c}_{\eta}}\right]}\right)\geqslant
1-\eta,\quad\forall n\geqslant{N_{\eta}}.$

With the above lemmas (including Lemma 5.3 and the ones used in Section 6.1),
we can obtain the rate of convergence of $\widehat{U}_{n}$ given by (31).

######  Lemma 6.8.

Assume that (A2), (A3), (A4), (B1), (B2), (B3) and (B4) hold. Then, as
$n\to\infty$, one has

$\widehat{U}_{n}(f_{0}(x_{0}))=F(x_{0})+O_{P}\left(u\left(n\right)^{-1/3}\right),$
(37)

and

$\widehat{f}_{n}^{-1}(f_{0}(x_{0}))=x+O_{P}\left(u\left(n\right)^{-1/3}\right).$
(38)

Inspecting the proof of Lemma 6.8, one can see that the convergences in (37)
and (38) hold in a uniform sense in the neighborhood of $x_{0}$. More
precisely, there exists $\gamma>0$, independent on $n$, such that for all
$\eta>0$ we can find $K_{1}>0$ such that

$\sup_{|a-f_{0}(x_{0})|\leqslant\gamma}\mathbb{P}\left(\left|\widehat{U}_{n}(a)-F\circ
f_{0}^{-1}(a)\right|>K_{1}{u\left(n\right)^{-1/3}}\right)\leqslant\eta$

and

$\sup_{|a-f_{0}(x_{0})|\leqslant\gamma}\mathbb{P}\left(\left|\widehat{f}_{n}^{-1}(a)-f_{0}^{-1}(a)\right|>K_{1}{u\left(n\right)^{-1/3}}\right)\leqslant\eta.$

Let $\varepsilon=K_{1}{u\left(n\right)^{-1/3}}$ where $K_{1}>0$ does not
depend on $n$, and recall (6.1) where
$\nu=x-f_{0}^{-1}(f_{0}(x_{0})+\varepsilon)>0$. It follows from the assumption
(B2) that $f_{0}^{-1}$ has a derivative that is bounded in sup-norm away from
zero in a neighborhood of $f_{0}(x_{0})$. Hence, it follows from the Taylor
expansion that there exists $K_{2}>0$ that depends only on $f_{0}$ such that
$\nu\geqslant K_{2}\varepsilon$, provided that $n$ is sufficiently large to
ensure that $f_{0}(x_{0})+\varepsilon$ belongs to this neighborhood of
$f_{0}(x_{0})$. Hence,

$\displaystyle\mathbb{P}\left(\widehat{f}_{n}(x_{0})>f_{0}(x_{0})+\varepsilon\right)$
$\displaystyle\leqslant$
$\displaystyle\mathbb{P}\left(\widehat{f}_{n}^{-1}(f_{0}(x_{0})+\varepsilon)\geqslant
f_{0}^{-1}(f_{0}(x_{0})+\varepsilon)+K_{2}\varepsilon\right).$
$\displaystyle\leqslant$
$\displaystyle\sup_{|a-f_{0}(x_{0})|\leqslant\gamma}\mathbb{P}\left(\left|\widehat{f}_{n}^{-1}(a)-f_{0}^{-1}(a)\right|>K_{2}K_{1}{u\left(n\right)^{-1/3}}\right),$

provided that $n$ is sufficiently large to ensure that
$f_{0}(x_{0})+\varepsilon$ belongs to the above neighborhood of
$f_{0}(x_{0})$, and that $\gamma\geqslant C{u\left(n\right)^{-1/3}}$. For
fixed $\eta>0$ one can choose $K_{2}>0$ such that the probability on the
right-hand side of the previous display is smaller than or equal to $\eta$ and
therefore,

$\lim_{n\to\infty}\mathbb{P}\left(\widehat{f}_{n}(x_{0})>f_{0}(x_{0})+K_{2}{u\left(n\right)^{-1/3}}\right)\leqslant\eta.$

Similarly, for all fixed $\eta>0$, one can find $K_{3}$ that does not depend
on $n$ such that

$\lim_{n\to\infty}\mathbb{P}\left(\widehat{f}_{n}(x_{0})<f_{0}(x_{0})-K_{3}{u\left(n\right)^{-1/3}}\right)\leqslant\eta.$

Hence, for all fixed $\eta>0$, there exists $K>0$ that independent of $n$ such
that

$\lim_{n\to\infty}\mathbb{P}\left(|\widehat{f}_{n}(x_{0})-f_{0}(x_{0})|>K{u\left(n\right)^{-1/3}}\right)\leqslant\eta.$

This completes the proof of Theorem 4.1.

##  7 Technical proofs for Section 6.1

Proof of Lemma 6.1. Combining (21) and (1) yields

$\Lambda_{n}(F_{n}(Y_{k}))=\frac{1}{T_{n}(C)}\sum_{t=0}^{n}f_{0}(X_{t})\mathbb{I}_{C}{\\{X_{t}\leqslant
Y_{k}\\}}+\frac{1}{T_{n}(C)}\sum_{t=0}^{n}W_{t}\mathbb{I}_{C}{\\{X_{t}\leqslant
Y_{k}\\}}.$

The first term on the right-hand side of the previous display can be rewritten
as follows:

$\displaystyle\frac{1}{T_{n}(C)}\sum_{t=0}^{n}f_{0}(X_{t})\mathbb{I}_{C}{\\{X_{t}\leqslant
Y_{k}\\}}$
$\displaystyle=\frac{1}{T_{n}(C)}\sum_{j=1}^{m}f_{0}(Y_{l})(l_{j}-l_{j-1})\mathbb{I}_{C}{\\{Y_{j}\leqslant
Y_{k}\\}}$
$\displaystyle=\sum_{j=1}^{k}\int_{l_{j-1}/T_{n}(C)}^{l_{j}/T_{n}(C)}f_{0}\circ
F_{n}^{-1}(u)du,$

using that $F_{n}^{-1}(u)=Y_{j}$ for all
$u\in(l_{j-1}/T_{n}(C),l_{j}/T_{n}(C)]$. Hence, for all $k$ in
$\\{0,\dots,m\\}$

$\Lambda_{n}(F_{n}(Y_{k}))=\int_{0}^{l_{k}/T_{n}(C)}f_{0}\circ
F_{n}^{-1}(u)du+\frac{1}{T_{n}(C)}\sum_{t=0}^{n}W_{t}\mathbb{I}_{C}{\\{X_{t}\leqslant
Y_{k}\\}}.$ (39)

Combining (39) with the piece-wise linearity of $\Lambda_{n}$ yields

${\Lambda_{n}}\left({{F_{n}}\left({{Y_{k}}}\right)}\right)={L_{n}}\left({{F_{n}}\left({{Y_{k}}}\right)}\right)+{M_{n}}\left({{F_{n}}\left({{Y_{k}}}\right)}\right),$

where $L_{n}$ and $M_{n}$ are piece-wise linear processes with knots at
$F_{n}(Y_{k})$ for $k$ in $\\{0,\dots,m\\}$ and such that

$L_{n}(F_{n}(Y_{k}))=\int_{0}^{l_{k}/T_{n}(C)}f_{0}\circ F_{n}^{-1}(u)du$

and

$M_{n}(F_{n}(Y_{k}))=\frac{1}{T_{n}(C)}\sum_{t=0}^{n}W_{t}\mathbb{I}_{C}{\\{X_{t}\leqslant
Y_{k}\\}}.$

In order to ease the notation, we will write $F_{n}^{i}=F_{n}(Y_{i})$,
$L_{n}^{i}={L_{n}}\left({{F_{n}}\left({{Y_{i}}}\right)}\right)$ and
$M_{n}^{i}={M_{n}}\left({{F_{n}}\left({{Y_{i}}}\right)}\right)$. Let
$y\in\left({{F_{n}}\left({{Y_{0}}}\right),{F_{n}}\left({{Y_{m}}}\right)}\right]$,
take $j$ such that $Y_{j+1}=F_{n}^{-1}\left(y\right)$, then
${F_{n}}\left({{Y_{j}}}\right)<y\leqslant{F_{n}}\left({{Y_{j+1}}}\right)$.
With this notation,

$\displaystyle{L_{n}}\left(y\right)$
$\displaystyle=\frac{{L_{n}^{j+1}-L_{n}^{j}}}{{F_{n}^{j+1}-F_{n}^{j}}}\left({y-F_{n}^{j}}\right)+L_{n}^{j},$
$\displaystyle{M_{n}}\left(y\right)$
$\displaystyle=\frac{{M_{n}^{j+1}-M_{n}^{j}}}{{F_{n}^{j+1}-F_{n}^{j}}}\left({y-F_{n}^{j}}\right)+M_{n}^{j}.$

Notice that

$\displaystyle L_{n}^{j+1}-L_{n}^{j}$
$\displaystyle=\int\limits_{\frac{{{l_{j}}}}{{T_{n}(C)}}}^{\frac{{{l_{j+1}}}}{{T_{n}(C)}}}{f_{0}\circ{F_{n}^{-1}}\left(u\right)du}=\frac{{{l_{j+1}}-{l_{j}}}}{{T_{n}(C)}}f\left({{Y_{j+1}}}\right),$
$\displaystyle F_{n}^{j+1}-F_{n}^{j}$
$\displaystyle=\frac{{{l_{j+1}}-{l_{j}}}}{{T_{n}(C)}},$

therefore,

${L_{n}}\left(y\right)=f_{0}\left({{Y_{j+1}}}\right)\left({y-F_{n}^{j}}\right)+L_{n}^{j}=\int\limits_{\frac{{{l_{j}}}}{{T_{n}(C)}}}^{y}{f_{0}\circ
F_{{}_{n}}^{-1}\left(u\right)du}+L_{n}^{j}=\int\limits_{0}^{y}{f_{0}\circ
F_{{}_{n}}^{-1}\left(u\right)du},$

which proves (22).

For $M_{n}$ we have,

$M_{n}^{j+1}-M_{n}^{j}=\frac{1}{{T_{n}(C)}}\sum\limits_{t=0}^{n}{{W_{t}}\mathbb{I}_{C}{\\{X_{t}=Y_{j+1}\\}}},$

then,

${M_{n}}\left(y\right)=\frac{\sum\limits_{t=1}^{n}{{W_{t}}\mathbb{I}_{C}{\\{X_{t}=Y_{j+1}\\}}}}{{{l_{j+1}}-{l_{j}}}}\left({y-F_{n}^{j}}\right)+M_{n}^{j}=R_{n}^{j}(y)+M_{n}^{j}.$

and this completes the proof.

Proof of Lemma 6.2. By definition, with $l_{0}=0$, and
$l_{k}=\sum_{t=0}^{n}\mathbb{I}_{C}{\left\\{X_{t}\leqslant Y_{k}\right\\}}$
for all $k\in\\{1,\dots,m\\}$, we have $F_{n}(Y_{k})=al_{k}$ for all
$k\in\\{0,\dots,m\\}$, where $a=1/T_{n}(C)$ and does not depend on $k$.
Moreover,

$\Lambda_{n}\left(F_{n}(Y_{k})\right)=a\sum_{t=0}^{n}Z_{t}\mathbb{I}_{C}{\left\\{X_{t}\leqslant
Y_{k}\right\\}}$

Since $\widehat{f}_{n}(Y_{k})$ is the left-hand slope at $l_{k}$ of the least
concave majorant of the set of points in (3), the equality in (26) follows
from Lemma 2.1 in [13].

Proof of Lemma 6.3. The first assertion follows from Assumption (A4) and the
second immediately follows from the first one by (12) combined with the strict
monotonicity of $F$ in $C$.

Proof of Lemma 6.4. Let
$\mathcal{F}_{n}=\sigma\left(\left\\{X_{0},\ldots,X_{n}\right\\}\right)$ be
sigma algebra generated by the chain $\left\\{X_{t}\right\\}$ up to time $n$.
Denote by $\mathbb{P}_{\mathcal{F}_{n}}$ the probability conditioned to
$\mathcal{F}_{n}$. Take $\varepsilon>0$.

By Chebyshev’s inequality, we have

${\mathbb{P}_{{\mathcal{F}_{n}}}}\left({\left|{\frac{{\sum\limits_{t=0}^{n}{{W_{t}}{\mathbb{I}_{C}}\left\\{{{X_{t}}={A_{n}}}\right\\}}}}{{{T_{n}}\left(C\right)}}}\right|>\varepsilon}\right)\leqslant\frac{{{\sigma^{2}}\sum\limits_{t=0}^{n}{{\mathbb{I}_{C}}\left\\{{{X_{t}}={A_{n}}}\right\\}}}}{{{\varepsilon^{2}}{T_{n}}^{2}\left(C\right)}}\leqslant\frac{{{\sigma^{2}}}}{{{\varepsilon^{2}}{T_{n}}\left(C\right)}},$

which implies the first part of the Lemma because $T_{n}(C)\to\infty$ with
probability 1.

For the second part, let $\gamma_{n}(u)$ be the number of times the chain
visits ${\left({{A_{n}},u}\right]}\cap C$ up to time $n$ and
${A_{n}}\left(u\right)=\left\\{{t\leqslant
n:{X_{t}}\in\left({{A_{n}},u}\right]}\cap
C\right\\}=\left\\{a_{1},\ldots,a_{\gamma_{n}(u)}\right\\}$ the times of those
visits. Using that
$\gamma_{n}=\mathop{\sup}\limits_{u>{A_{n}}}{\gamma_{n}}\left(u\right)\leqslant{T_{n}}\left(C\right)$
and Kolmogorov’s inequality (Th 3.1.6, pp 122 in [19]) we obtain,

$\displaystyle{\mathbb{P}_{{\mathcal{F}_{n}}}}\left({\mathop{\sup}\limits_{u>{A_{n}}}\left|{\frac{{\sum\limits_{t=0}^{n}{{W_{t}}{\mathbb{I}_{C}}\left\\{{{X_{t}}\in\left({{A_{n}},u}\right]}\right\\}}}}{{{T_{n}}\left(C\right)}}}\right|>\varepsilon}\right)$
$\displaystyle={\mathbb{P}_{{\mathcal{F}_{n}}}}\left({\mathop{\sup}\limits_{u>{A_{n}}}\left|{\sum\limits_{i=1}^{{\gamma_{n}}\left(u\right)}{\frac{{{W_{{t_{{a_{i}}}}}}}}{{{T_{n}}\left(C\right)}}}}\right|>\varepsilon}\right)$
$\displaystyle\leqslant{\mathbb{P}_{{\mathcal{F}_{n}}}}\left({\mathop{\sup}\limits_{1\leqslant
k\leqslant\gamma_{n}}\left|{\sum\limits_{i=1}^{k}{\frac{{{{W}_{t_{a_{i}}}}}}{{{T_{n}}\left(C\right)}}}}\right|>\varepsilon}\right)$
$\displaystyle\leqslant\frac{{{\sigma^{2}}}}{{{\varepsilon^{2}}{T_{n}}\left(C\right)}}.$

which by the same argument as before, implies the second part of the Lemma.

Proof of Lemma 6.5. In the sequel, we set $a=f_{0}(x_{0})$. We begin with the
proof of (32).

Fix $\varepsilon>0$ arbitrarily, and let $\nu>0$ and $\gamma>0$ be such that
$|F^{-1}(u)-x_{0}|>\nu$ for all $u$ such that
$|u-F(x_{0})|\geqslant\varepsilon/2$, and $|f_{0}(x_{0})-f_{0}(y)|>\gamma$ for
all $y$ such that $|y-x_{0}|\geqslant\nu/2$. Note that existence of $\nu$ and
$\gamma$ is ensured by assumptions (A5) and (A6).

By Lemma 6.3, we can assume without loss of generality that $F(x_{0})$ belongs
to the domain $[F_{n}(Y_{1}),F_{n}(Y_{m})]$ of $\Lambda_{n}$, since this
occurs with probability tending to one. Therefore, we can find $j(x_{0})$ such
that ${Y_{j(x_{0})}}={F_{n}}^{-1}\left({F\left({{x_{0}}}\right)}\right)$. It
follows from the characterization in (31) that the event
$E_{n}^{1}:=\\{\widehat{U}_{n}(a)>F(x_{0})+\varepsilon\\}$ is contained in the
event that there exists $p\in\cal K$ such that $p>F(x_{0})+\varepsilon$ and

${\Lambda_{n}}\left(p\right)-ap\geqslant{\Lambda_{n}}\left({F\left({{x_{0}}}\right)}\right)-aF\left({{x_{0}}}\right),$

where we recall that $a=f_{0}\left(x_{0}\right)$.

By Lemma 6.1, $E_{n}^{1}$ is contained in the event that there exists
$p\in\cal K$ such that $p>F(x_{0})+\varepsilon$ and

$L_{n}(p)+M_{n}(p)-ap\geqslant L_{n}(F(x_{0}))+M_{n}(F(x_{0}))-aF(x_{0})$ (40)

Using (22) in (40) we obtain that $E_{n}^{1}$ is contained in the event that
there exists $p\in\cal K$ such that $p>F(x_{0})+\varepsilon$ and

$\int_{t_{0}/T_{n}(C)}^{p}f_{0}\circ
F_{n}^{-1}(u)du+S_{n}-ap\geqslant\int_{t_{0}/T_{n}(C)}^{F(x_{0})}f_{0}\circ
F_{n}^{-1}(u)du-aF(x_{0}),$

where

$\displaystyle{S_{n}}=\mathop{\sup}\limits_{p>F(x_{0})+\varepsilon,\;p\in\mathcal{K}}\left\\{{{M_{n}}(p)-{M_{n}}\left({F\left({{x_{0}}}\right)}\right)}\right\\}.$

Let $j$ and $k$ such that $Y_{j+1}=F_{n}^{-1}\left(F\left(x_{0}\right)\right)$
and $p=F_{n}(Y_{k})$. By equation (23) we have
${M_{n}}\left(p\right)-{M_{n}}\left({F\left({{x_{0}}}\right)}\right)=M_{n}^{k}-M_{n}^{j}-R_{n}^{j}(F(x_{0}))$,
therefore,

$\displaystyle{S_{n}}$
$\displaystyle=\mathop{\sup}_{\begin{subarray}{l}p>F\left({{x_{0}}}\right)+\varepsilon\\\
p\in\mathcal{K}\end{subarray}}{\left\\{{M_{n}^{k}-M_{n}^{j}}\right\\}-R_{n}^{j}\left({F\left({{x_{0}}}\right)}\right)}$
$\displaystyle\begin{split}&\leqslant\mathop{\sup}_{\begin{subarray}{l}p>F\left({{x_{0}}}\right)+\varepsilon\\\
p\in\mathcal{K}\end{subarray}}{\left|\frac{1}{T_{n}(C)}\sum_{t=0}^{n}W_{t}\left(\mathbb{I}_{C}{\\{X_{t}\leqslant
F_{n}^{-1}(p)\\}}-\mathbb{I}_{C}{\\{X_{t}\leqslant
F_{n}^{-1}(F(x_{0}))\\}}\right)\right|}\\\
&\qquad+\left|{R_{n}^{j}\left({F_{n}\left(Y_{j+1}\right)}\right)}\right|\\\
\end{split}$
$\displaystyle\begin{split}&\leqslant\mathop{\sup}_{\begin{subarray}{l}p>F\left({{x_{0}}}\right)+\varepsilon\\\
p\in\mathcal{K}\end{subarray}}{\left|\frac{1}{T_{n}(C)}\sum_{t=0}^{n}W_{t}\mathbb{I}_{C}{\left\\{X_{t}\in\left({F_{n}^{-1}\left({F\left({{x_{0}}}\right)}\right);F_{n}^{-1}(p)}\right]\right\\}}\right|}\\\
&\qquad+\frac{\left|\sum\limits_{t=0}^{n}{{W_{t}}\mathbb{I}_{C}{\left\\{X_{t}=F_{n}^{-1}\left({F\left({{x_{0}}}\right)}\right)\right\\}}}\right|}{T_{n}(C)}.\end{split}$

Hence,

$\begin{split}T_{n}(C)S_{n}&\leqslant\mathop{\sup}_{\begin{subarray}{l}p>F\left({{x_{0}}}\right)+\varepsilon\\\
p\in\mathcal{K}\end{subarray}}{\left|\sum_{t=0}^{n}W_{t}\mathbb{I}_{C}{\left\\{X_{t}\in\left({F_{n}^{-1}\left({F\left({{x_{0}}}\right)}\right);F_{n}^{-1}(p)}\right]\right\\}}\right|}\\\
&\qquad+\left|\sum\limits_{t=0}^{n}{{W_{t}}\mathbb{I}_{C}{\left\\{X_{t}=F_{n}^{-1}\left({F\left({{x_{0}}}\right)}\right)\right\\}}}\right|.\end{split}$

Therefore, the event $E_{n}^{1}$ is contained in the event that there exists
$p>F(x_{0})+\varepsilon$ such that

$\int_{F(x_{0})}^{p}f_{0}\circ F_{n}^{-1}(u)du+S_{n}\geqslant a(p-F(x_{0})).$

Now, let $E_{n}^{2}$ be the event that

$\sup_{|u-F(x_{0})|\leqslant\varepsilon}|F_{n}^{-1}(u)-F^{-1}(u)|\leqslant\eta$

where $\eta\in(0,\nu/4)$ is such that
$|f_{0}(y)-f_{0}(x_{0})|\leqslant\gamma/2$ for all $y$ such that
$|x_{0}-y|\leqslant\eta$. Note that the existence of $\eta$ is ensured by
assumption (A7). Then, it follows from the monotonicity of $f_{0}$ and $F_{n}$
that on $E_{n}^{2}$,

$\displaystyle\int_{F(x_{0})}^{p}f_{0}\circ F_{n}^{-1}(u)du$
$\displaystyle\leqslant$
$\displaystyle\int_{F(x_{0})}^{F(x_{0})+\varepsilon/2}f_{0}(F^{-1}(u)-\eta)du+\int_{F(x_{0})+\varepsilon/2}^{p}f_{0}(F_{n}^{-1}(F(x_{0})+\varepsilon/2))du.$

Hence, it follows from the definitions of $\eta$, $\nu$ and $\gamma$ that on
$E_{n}^{2}$,

$\displaystyle\int_{F(x_{0})}^{p}f_{0}\circ F_{n}^{-1}(u)du$
$\displaystyle\leqslant$
$\displaystyle\frac{\varepsilon}{2}f_{0}(x_{0})+\frac{\gamma\varepsilon}{4}+(p-F(x_{0})-\varepsilon/2)f_{0}(F^{-1}(F(x_{0})+\varepsilon/2)-\eta)$
$\displaystyle\leqslant$
$\displaystyle\frac{\varepsilon}{2}f_{0}(x_{0})+\frac{\gamma\varepsilon}{4}+(p-F(x_{0})-\varepsilon/2)f_{0}(x_{0}+\nu/2)$
$\displaystyle\leqslant$
$\displaystyle\frac{\varepsilon}{2}f_{0}(x_{0})+\frac{\gamma\varepsilon}{4}+(p-F(x_{0})-\varepsilon/2)(f_{0}(x_{0})-\gamma).$

This implies that on $E_{n}^{2}$,

$\displaystyle\int_{F(x_{0})}^{p}f_{0}\circ F_{n}^{-1}(u)du$
$\displaystyle\leqslant$ $\displaystyle
a(p-F(x_{0}))-(p-F(x_{0})-3\varepsilon/4)\gamma$ $\displaystyle\leqslant$
$\displaystyle a(p-F(x_{0}))-\varepsilon\gamma/4$

for all $p>F(x_{0})+\varepsilon$. Hence, the event $E_{n}^{1}\cap E_{n}^{2}$
is contained in the event $\\{S_{n}\geqslant\varepsilon\gamma/4\\}$. Now, on
$E_{n}^{2}$, for all $p>F(x_{0})+\varepsilon$ we have

$\displaystyle F_{n}^{-1}(p)$ $\displaystyle\geqslant$ $\displaystyle
F_{n}^{-1}(F(x_{0})+\varepsilon)$ $\displaystyle\geqslant$ $\displaystyle
F^{-1}(F(x_{0})+\varepsilon)-\eta$ $\displaystyle\geqslant$ $\displaystyle
x+\nu-\eta$ $\displaystyle\geqslant$ $\displaystyle
F_{n}^{-1}(F(x_{0}))+\nu-2\eta$ $\displaystyle\geqslant$ $\displaystyle
F_{n}^{-1}(F(x_{0}))+\nu/2,$

since $\nu>4\eta$. Therefore,

$\displaystyle T_{n}(C)S_{n}\leqslant$
$\displaystyle\sup_{u>F_{n}^{-1}(F(x_{0}))+\nu/2}\left|\sum_{t=0}^{n}W_{t}\mathbb{I}_{C}{\\{X_{t}\in(F_{n}^{-1}(F(x_{0})),u]\\}}\right|+$
$\displaystyle+\left|\sum\limits_{t=0}^{n}{{W_{t}}\mathbb{I}_{C}{\\{X_{t}=F_{n}^{-1}\left({F\left({{x_{0}}}\right)}\right)\\}}}\right|.$

Hence, it follows from Lemma 6.4 that $S_{n}$ converges in probability to zero
as $n\to\infty$, so that the probability of the event
$\\{S_{n}\geqslant\varepsilon\gamma/4\\}$ tends to zero as $n\to\infty$. It
follows from Lemma 5.1 that for $\varepsilon$ sufficiently small, the
probability of the event $E_{n}^{2}$ tends to one as $n\to\infty$, so we
conclude that the probability of $E_{n}^{1}$ tends to zero as $n\to\infty$.
Similarly, the probability of the event
$\\{\widehat{U}_{n}(a)<F(x_{0})-\varepsilon\\}$ tends to zero as $n\to\infty$,
so that

$\lim_{n\to\infty}\mathbb{P}(|\widehat{U}_{n}(a)-F(x_{0})|>\varepsilon)=0$

for all $\varepsilon>0$. This completes the proof of (32).

##  8 Technical proofs for Section 6.2

Proof of Lemma 6.6. Let
$\mathcal{F}_{n}=\sigma\left(\left\\{X_{0},\ldots,X_{n}\right\\}\right)$ be
sigma algebra generated by the chain X up to time $n$. Denote by
$\mathbb{E}_{\mathcal{F}_{n}}$ the expected value conditioned to
$\mathcal{F}_{n}$. Take $0<\gamma\leqslant\delta$ and define
${I_{0}}=\left[{{x_{0}}-\gamma,{x_{0}}}\right]$,
${I_{1}}=\left[{{x_{0}},{x_{0}}+\gamma}\right]$ and

$\begin{gathered}{S_{0}\left(\gamma\right)}=\mathop{\sup}\limits_{y\in{I_{0}}}\left|{\sum\limits_{t=0}^{n}{{W_{t}}\left({{\mathbb{I}}\left\\{{{X_{t}}\leqslant
y}\right\\}-{\mathbb{I}}\left\\{{{X_{t}}\leqslant{x_{0}}}\right\\}}\right)}}\right|^{2}\hfill\\\
{S_{1}\left(\gamma\right)}=\mathop{\sup}\limits_{y\in{I_{1}}}\left|{\sum\limits_{t=0}^{n}{{W_{t}}\left({{\mathbb{I}}\left\\{{{X_{t}}\leqslant
y}\right\\}-{\mathbb{I}}\left\\{{{X_{t}}\leqslant{x_{0}}}\right\\}}\right)}}\right|^{2}\hfill\\\
\end{gathered}$

then,

$\displaystyle S\left(\gamma\right)$
$\displaystyle=\mathop{\sup}\limits_{\left|{y-{x_{0}}}\right|\leqslant\gamma}\left|{\sum\limits_{t=0}^{n}{{W_{t}}\left({{\mathbb{I}}\left\\{{{X_{t}}\leqslant
y}\right\\}-{\mathbb{I}}\left\\{{{X_{t}}\leqslant{x_{0}}}\right\\}}\right)}}\right|^{2}=\max{\Bigl{(}S_{0}\left(\gamma\right),S_{1}\left(\gamma\right)\Bigr{)}},$
$\displaystyle\leqslant S_{0}\left(\gamma\right)+S_{1}\left(\gamma\right)$
(41)

Following the notation of section 2, let

$\alpha_{n}^{(0)}\left(\gamma\right)=\mathop{\sup}\limits_{y\in{I_{0}}}{T_{n}}\left({\left[{y,{x_{0}}}\right]}\right)\quad,\quad\alpha_{n}^{(1)}\left(\gamma\right)=\mathop{\sup}\limits_{y\in{I_{1}}}{T_{n}}\left({\left[{{x_{0}},y}\right]}\right),$

with this notation,
${S_{0}}=\mathop{\sup}\limits_{y\in{I_{0}}}{\left|{\sum\limits_{i=1}^{{T_{n}}\left({\left[{y,{x_{0}}}\right]}\right)}{{W_{{\sigma_{[y,x_{0})}(i)}}}}}\right|^{2}}$
and
$S_{1}=\mathop{\sup}\limits_{y\in{I_{1}}}\left|{\sum\limits_{{i=1}}^{{T_{n}}\left({\left[{{x_{0}},y}\right]}\right)}{{W_{{{\sigma_{[x_{0},y]}(i)}}}}}}\right|^{2}$.

By Doob’s maximal inequality (Th 10.9.4 in [19]), we have, for $j=0,1$,

$\displaystyle{\mathbb{E}_{{\mathcal{F}_{n}}}}S_{j}\left(\gamma\right)$
$\displaystyle\leqslant
4{\mathbb{E}_{{\mathcal{F}_{n}}}}{\left({\sum\limits_{i=1}^{{\alpha^{(j)}_{n}}}{{W_{{t_{i}}}}}}\right)^{2}}=4\sigma^{2}\alpha^{(j)}_{n}\left(\gamma\right)$
$\displaystyle\leqslant
4{\sigma^{2}}{T_{n}}\left({\left[{{x_{0}}-\gamma,{x_{0}}+\gamma}\right]}\right)$
$\displaystyle\leqslant
4{\sigma^{2}}\sum\limits_{t=0}^{n}{\left({{\mathbb{I}}\left\\{{{X_{t}}\leqslant{x_{0}}+\gamma}\right\\}-{\mathbb{I}}\left\\{{{X_{t}}<{x_{0}}-\gamma}\right\\}}\right)}.$
(42)

Therefore, by (41) and (8)

${\mathbb{E}_{{\mathcal{F}_{n}}}}S\left(\gamma\right)\leqslant
8{\sigma^{2}}\sum\limits_{t=0}^{n}{\Bigl{(}{{\mathbb{I}}\left\\{{{X_{t}}\leqslant{x_{0}}+\gamma}\right\\}-{\mathbb{I}}\left\\{{{X_{t}}<{x_{0}}-\gamma}\right\\}}\Bigr{)}}.$
(43)

Define,

  * •

$h\left({y,\gamma}\right)=\mathbb{I}{\left\\{{y\in\left[{{x_{0}}-\gamma,{x_{0}}+\gamma}\right]}\right\\}}$,

  * •

$h\left({{\mathcal{B}_{j}},\gamma}\right)=\begin{cases}\sum\limits_{t=0}^{{\tau_{\alpha}}}{h\left({{X_{t}},\gamma}\right)}&,\quad
j=0\hfill\\\
\sum\limits_{t={\tau_{A}}(j)+1}^{{\tau_{A}}\left({j+1}\right)}{h\left({{X_{t}},\gamma}\right)}&,\quad
j\geqslant 1\hfill\\\ \end{cases}$

  * •

${Z_{n}}\left(\gamma\right)=\sum\limits_{t=0}^{n}{h\left({{X_{t}},\gamma}\right)}$

  * •

$\ell\left({{\mathcal{B}_{j}}}\right)=\begin{cases}{\tau_{\boldsymbol{\alpha}}}&,\quad
j=0\hfill\\\
{\tau_{\boldsymbol{\alpha}}}(j+1)-{\tau_{\boldsymbol{\alpha}}}(j)&,\quad
j\geqslant 1\hfill\\\ \end{cases}$

  * •

$\widetilde{T}\left(n\right)=\min\left\\{{k:\sum\limits_{i=0}^{k}{\ell\left({{\mathcal{B}_{j}}}\right)}\geqslant
n}\right\\}$.

  * •

${\mathcal{G}_{k}}=\sigma\left({\left\\{{\left({h\left({{\mathcal{B}_{j}},\gamma}\right),\ell\left({{\mathcal{B}_{j}}}\right)}\right)}\right\\}_{j=0}^{k}}\right)$
for $k\geqslant 0$.

By the Strong Markov property,
$\left\\{{\left({h\left({{\mathcal{B}_{j}},\gamma}\right),\ell\left({{\mathcal{B}_{j}}}\right)}\right)}\right\\}_{j=1}^{+\infty}$
is an i.i.d. sequence which is independent of
${\left({h\left({{\mathcal{B}_{0}},\gamma}\right),\ell\left({{\mathcal{B}_{0}}}\right)}\right)}$
(and, therefore, of the initial measure $\lambda$). For $n$ fixed, the random
variable $\widetilde{T}\left(n\right)$ is a stopping time for the sequence
$\left\\{{\left({h\left({{\mathcal{B}_{j}},\gamma}\right),\ell\left({{\mathcal{B}_{j}}}\right)}\right)}\right\\}_{j=0}^{+\infty}$,
in effect

$\displaystyle\left\\{{\widetilde{T}\left(n\right)=0}\right\\}$
$\displaystyle=\left\\{{\ell\left({{\mathcal{B}_{0}}}\right)\geqslant
n}\right\\}\in{\mathcal{G}_{0}},$
$\displaystyle\left\\{{\widetilde{T}\left(n\right)=k}\right\\}$
$\displaystyle=\bigcap\limits_{j=0}^{k-1}{\left\\{{\sum\limits_{i=0}^{j}{\ell\left({{\mathcal{B}_{i}}}\right)}<n}\right\\}}\bigcap\left\\{{\sum\limits_{i=0}^{k}{\ell\left({{\mathcal{B}_{j}}}\right)}\geqslant
n}\right\\}\in{\mathcal{G}_{k}}\quad\forall k\geqslant 1.$

For each $n$ and $\gamma$ we have that

$\displaystyle{Z_{n}}\left(\gamma\right)$
$\displaystyle=\sum\limits_{t=0}^{{\tau_{\alpha}}}{h\left({{X_{t}},\gamma}\right)}+\sum\limits_{j=1}^{T\left(n\right)}{h\left({{\mathcal{B}_{j}},\gamma}\right)}+\sum\limits_{t={t_{\alpha}}\left({T\left(n\right)}\right)+1}^{n}{h\left({{X_{t}},\gamma}\right)}$
$\displaystyle\leqslant{h\left({{\mathcal{B}_{0}},\gamma}\right)}+\sum\limits_{j=1}^{\widetilde{T}\left(n\right)}{h\left({{\mathcal{B}_{j}},\gamma}\right)}.$
(44)

where the last inequality is justified by the fact that,
$T\left(n\right)\leqslant\widetilde{T}\left(n\right)$ and
$h\left(y,\gamma\right)$ is a nonnegative function. Because
$\ell\left({{\mathcal{B}_{j}}}\right)\geqslant 1$ for all $j$, we have that,

$\sum\limits_{j=1}^{\widetilde{T}\left(n\right)}{h\left({{\mathcal{B}_{j}},\gamma}\right)}=\sum\limits_{j=1}^{n}{h\left({{\mathcal{B}_{j}},\gamma}\right)\mathbb{I}\left\\{{\widetilde{T}\left(n\right)\geqslant
j}\right\\}},$

then,

$\mathbb{E}\left({\sum\limits_{j=1}^{\widetilde{T}\left(n\right)}{h\left({{\mathcal{B}_{j}},\gamma}\right)}}\right)=\sum\limits_{j=1}^{n}{\mathbb{E}\left({h\left({{\mathcal{B}_{j}},\gamma}\right)\mathbb{I}\left\\{{\widetilde{T}\left(n\right)\geqslant
j}\right\\}}\right)}.$ (45)

For each $j$ we have,

$\mathbb{E}_{\lambda}\left({h\left({{\mathcal{B}_{j}},\gamma}\right)\mathbb{I}\left\\{{\widetilde{T}\left(n\right)\geqslant
j}\right\\}}\right)=\mathbb{E}_{\lambda}\left({\mathbb{E}\left({h\left({{\mathcal{B}_{j}},\gamma}\right)\mathbb{I}\left\\{{\widetilde{T}\left(n\right)\geqslant
j}\right\\}\left|{{\mathcal{G}_{j-1}}}\right.}\right)}\right)$

Notice that $\mathbb{I}\left\\{{\widetilde{T}\left(n\right)\geqslant
j}\right\\}=1-\mathbb{I}\left\\{{\widetilde{T}\left(n\right)\leqslant
j-1}\right\\}\in{\mathcal{G}_{j-1}}$ and
${h\left({{\mathcal{B}_{j}},\gamma}\right)}$ is independent of
${\mathcal{G}_{j-1}}$, therefore,

$\displaystyle\mathbb{E}_{\lambda}\left({h\left({{\mathcal{B}_{j}},\gamma}\right)\mathbb{I}\left\\{{\widetilde{T}\left(n\right)\geqslant
j}\right\\}}\right)=\mathbb{E}_{\lambda}\left({\mathbb{I}\left\\{{\widetilde{T}\left(n\right)\geqslant
j}\right\\}}\right)\mathbb{E}\left({h\left({{\mathcal{B}_{j}},\gamma}\right)}\right).$

Plugging this into equation (45) we get,

$\mathbb{E}_{\lambda}\left({\sum\limits_{j=1}^{\widetilde{T}\left(n\right)}{h\left({{\mathcal{B}_{j}},\gamma}\right)}}\right)=\sum\limits_{j=1}^{n}{\mathbb{E}\left({h\left({{\mathcal{B}_{j}},\gamma}\right)}\right)\mathbb{P}_{\lambda}\left({\widetilde{T}\left(n\right)\geqslant
j}\right)}\leqslant\mathbb{E}\left({h\left({{\mathcal{B}_{1}},\gamma}\right)}\right)\mathbb{E}_{\lambda}\widetilde{T}\left(n\right).$

Then, by taking expectation in (8) we obtain

$\displaystyle\mathbb{E}_{\lambda}{Z_{n}}\left(\gamma\right)$
$\displaystyle\leqslant\mathbb{E}_{\lambda}h\left({{\mathcal{B}_{0}},\gamma}\right)+\mathbb{E}\left({h\left({{\mathcal{B}_{1}},\gamma}\right)}\right)\mathbb{E}_{\lambda}\widetilde{T}\left(n\right)$
$\displaystyle\leqslant\mathbb{E}_{\lambda}h\left({{\mathcal{B}_{0}},\gamma}\right)+\mathbb{E}\left({h\left({{\mathcal{B}_{1}},\gamma}\right)}\right)\mathbb{E}_{\lambda}\left(T\left(n\right)+1\right).$
(46)

By Theorem 1.1 and the fact that $F$ is Lipschitz we can find $K_{1}$
independent of $\gamma$ such that,

$\displaystyle\mathbb{E}\left({h\left({{\mathcal{B}_{1}},\gamma}\right)}\right)$
$\displaystyle=\int{h\left({t,\gamma}\right)d\pi\left(t\right)=K_{\pi}\pi\left(C\right)\left({{F}\left({{x_{0}}+\gamma}\right)-{F}\left({{x_{0}}-\gamma}\right)}\right)}$
$\displaystyle\leqslant K_{1}\gamma.$ (47)

If X is positive recurrent, by Theorem 1.1,
$\frac{T\left({n}\right)}{u\left(n\right)}$ converges almost surely to a
positive constant $K_{2}>0$. Moreover,
$\frac{T\left(n\right)}{u\left(n\right)}\leqslant 1$ therefore, by the
Dominated Convergence Theorem we obtain that
$\mathbb{E}_{\lambda}T\left(n\right)\sim\frac{u\left(n\right)}{K_{2}}$. If X
is $\beta$-null recurrent, by Lemma 3.3 in [24],
$\mathbb{E}_{\lambda}T\left(n\right)\sim\frac{{u\left(n\right)}}{{\Gamma\left({1+\beta}\right)}}$,
hence, for both positive and $\beta$-null recurrent chains, we can find
$K_{2}$ and $N$, both independent of $\gamma$, such that
$\mathbb{E}_{\lambda}T\left(n\right)\leqslant{K_{2}}u\left(n\right)$ for all
$n\geqslant N$. Using this with (8) and (47) we get,

$\frac{{\mathbb{E}_{\lambda}{Z_{n}}\left(\gamma\right)}}{{u\left(n\right)\gamma}}\leqslant\frac{{\mathbb{E}_{\lambda}h\left({{\mathcal{B}_{0}},\gamma}\right)}}{{u\left(n\right)\gamma}}+{K_{1}}{K_{2}}\quad\forall
n\geqslant N,\;\forall\gamma\in\left({0,\delta}\right].$ (48)

Combining (48) with assumption (B3) and the fact that
$Z_{n}\left(0\right)\equiv 0$ we obtain that there exist positive constants
$K_{3}$ and $\gamma_{0}$ such that

$\mathbb{E}_{\lambda}{Z_{n}}\left(\gamma\right)\leqslant
u\left(n\right)\gamma\quad\forall n\geqslant
N,\;\forall\gamma\in\left({0,\gamma_{0}}\right].$

Equation (35) now follows after taking expectation in (43). The proof of (36)
follows the same reasoning, but using

${S_{j}}\left(\gamma\right)=\mathop{\sup}\limits_{y\in{I_{j}}}{\left|{\sum\limits_{t=0}^{n}{{W_{t}}\left({\mathbb{I}_{C}\left\\{{{X_{t}}=y}\right\\}}\right)}}\right|^{2}}{\text{
}}.$

Proof of Lemma 6.7. a) If X is positive recurrent, Theorem 1.1 implies that
there exists a positive constant $K$ such that
$\frac{T_{n}\left(C\right)}{u\left(n\right)}$ converges almost surely to
$K\pi(C)$, which is not zero by (A3).

On the other hand, if X is $\beta$-null recurrent, Theorem 1.1 and Slutsky’s
Theorem implies that there exists a constant $K>0$ such that
$\frac{{{T_{n}}\left(C\right)}}{{u\left(n\right)}}$ converges in distribution
to $KM_{\beta}(1)$ where $M_{\beta}(1)$ denotes a Mittag-Leffler distribution
with parameter $\beta$. This distribution is continuous and strictly positive
with probability 1, then, by the Continuous Mapping Theorem,
$\frac{{u\left(n\right)}}{{{T_{n}}\left(C\right)}}$ converges in distribution
to a multiple of $\frac{1}{{{M_{\beta}}}}$, therefore,
$\frac{{u\left(n\right)}}{{{T_{n}}\left(C\right)}}$ is bounded in probability
by Theorem 2.4 in [33].

b) Let X be positive recurrent, then, we can find $N_{\eta}$ such that

$\mathbb{P}\left({\left|{{{\left({\frac{{{T_{n}}\left(C\right)}}{{u\left(n\right)}}}\right)}^{\alpha}}-K^{\alpha}\pi{{\left(C\right)}^{\alpha}}}\right|\leqslant{{\left({\frac{{K\pi\left(C\right)}}{2}}\right)}^{\alpha}}}\right)\geqslant
1-\eta,\quad\forall n\geqslant{N_{\eta}}.$

hence,

$\mathbb{P}\left({{{\left({\frac{{{T_{n}}\left(C\right)}}{{u\left(n\right)}}}\right)}^{\alpha}}\in\left[{\frac{{{K^{\alpha}\pi{{\left(C\right)}^{\alpha}}}}}{2},\frac{{3K^{\alpha}\pi{{\left(C\right)}^{\alpha}}}}{2}}\right]}\right)\geqslant
1-\eta,\quad\forall n\geqslant{N_{\eta}}.$

Now let X be $\beta$-null recurrent. Let
$Z={\left({K{M_{\beta}}\left(1\right)}\right)^{\alpha}}$, This random variable
is continuous and positive, therefore, we can find positive constants
$\underline{c}_{\eta}$ and $\overline{c}_{\eta}$ such that

$\mathbb{P}\left({Z\in\left[{\underline{c}_{\eta},\overline{c}_{\eta}}\right]}\right)\geqslant
1-\frac{\eta}{2}.$ (49)

By the Continuous Mapping Theorem,
${{{\left({\frac{{{T_{n}}\left(C\right)}}{{u\left(n\right)}}}\right)}^{\alpha}}}$
converges in distribution to $Z$, therefore, we can find
$N_{\eta}\in\mathbb{N}$ such that

$\left|{\mathbb{P}\left({{{\left({\frac{{T_{n}(C)}}{{u\left(n\right)}}}\right)}^{\alpha}}\in\left[{{\underline{c}_{\eta}},{{\overline{c}}_{\eta}}}\right]}\right)-\mathbb{P}\left({Z\in\left[{{\underline{c}_{\eta}},{{\overline{c}}_{\eta}}}\right]}\right)}\right|\leqslant\frac{\eta}{2},\quad\forall
n\geqslant{N_{\eta}},$ (50)

Combining (49) and (50) we obtain that

$\mathbb{P}\left({{{\left({\frac{{T_{n}(C)}}{{u\left(n\right)}}}\right)}^{\alpha}}\in\left[{{\underline{c}_{\eta}},{{\overline{c}}_{\eta}}}\right]}\right)\geqslant
1-\eta,\quad\forall n\geqslant{N_{\eta}}.$ (51)

Proof of Lemma 6.8. Fix $\varepsilon\in(0,1)$ small enough so that
$F^{\prime}$ and $|f_{0}^{\prime}|$ are bounded from above and away from zero
on $[F^{-1}(F(x_{0})-2\varepsilon),F^{-1}(F(x_{0})+2\varepsilon)]$, see the
assumption (B2). Then, the proper inverse functions of $F$ and $f_{0}$ are
well defined on $[F(x_{0})-2\varepsilon,F(x_{0})+2\varepsilon]$ and

$[f_{0}\circ F^{-1}(F(x_{0})-2\varepsilon),f_{0}\circ
F^{-1}(F(x_{0})+2\varepsilon)]$

respectively. We denote the inverses on that intervals by $F^{-1}$ and
$f_{0}^{-1}$ respectively. Let

$U_{n}(a)=\mathop{\mbox{\sl\em
argmax}}_{|p-F(x_{0})|\leqslant\varepsilon}\\{\Lambda_{n}(p)-ap\\}$ (52)

where $a=f_{0}(x_{0})$ and where the supremum is restricted to
$p\in[F_{n}(Y_{0}),F_{n}(Y_{m})]$. We will show below that

$U_{n}(a)=F(x_{0})+O_{P}({u\left(n\right)^{-1/3}}),$ (53)

as $n\to\infty$. Combining (31) to Lemma 6.5 ensures that $\widehat{U}_{n}(a)$
coincides with $U_{n}(a)$ with a probability that tends to one as
$n\to\infty$, so (37) follows from (53).

We turn to the proof of (53). Fix $\eta>0$ arbitrarily and let

$\gamma_{n}=K_{0}{u\left(n\right)^{-1/3}}$ (54)

for some $K_{0}\geqslant 1$ sufficiently large so that

$\gamma_{n}\geqslant\frac{1}{\sqrt{u\left(n\right)}}.$ (55)

Then, by part ii) of Lemma 6.7, we can find positive constants
$\underline{c}_{\eta}$, $\bar{c}_{\eta}$ and $N_{\eta}$ such that

$\mathbb{P}\left(T_{n}(C)^{2/3}\gamma_{n}u\left(n\right)^{-1/3}\in[K_{0}\underline{c}_{\eta},K_{0}\bar{c}_{\eta}]\right)\geqslant
1-\eta/2\quad\forall n\geqslant N_{\eta},$ (56)

Let $\underline{c}=K_{0}\underline{c}_{\eta}$ and
$\bar{c}=K_{0}\overline{c}_{\eta}$. It follows from (18) that for sufficiently
small $\varepsilon>0$, we can find $K_{1}>0$ such that

$\mathbb{P}\left(T_{n}(C)\sup_{|p-F(x_{0})|\leqslant
2\varepsilon}|F_{n}^{-1}(p)-F^{-1}(p)|^{2}\leqslant K_{1}\right)\geqslant
1-\eta/2$

for all $n$. Hence for $n\geqslant\mathbb{N}_{\eta}$,

$\mathbb{P}(\mathcal{E}_{n})\geqslant 1-\eta,$

where $\mathcal{E}_{n}$ denotes the intersection of the events

$T_{n}(C)^{2/3}\gamma_{n}u\left(n\right)^{-1/3}\in[\underline{c},\bar{c}]$
(57)

and

$T_{n}(C)\sup_{|p-F(x_{0})|\leqslant
2\varepsilon}|F_{n}^{-1}(p)-F^{-1}(p)|^{2}\leqslant K_{1}.$ (58)

Combining equations (57) and (58), we obtain that, in $\mathcal{E}_{n}$,

$\mathop{\sup}\limits_{\left|{p-{F}\left({{x_{0}}}\right)}\right|\leqslant
2\varepsilon}{\left|{F_{n}^{-1}\left(p\right)-{F^{-1}}\left(p\right)}\right|^{2}}\leqslant{K_{2}}a{\left(n\right)^{-1}}$
(59)

where $K_{2}={K_{1}}{\left({\frac{{{K_{0}}}}{{\underline{c}}}}\right)^{3/2}}$
is independent of $n$ and $K_{0}$.

By Lemma 6.3, we can assume without loss of generality that $F(x_{0})$ belongs
to $[F_{n}(Y_{0}),F_{n}(Y_{m})]$, since this occurs with probability that
tends to one. Hence, by (52), the event
$\\{|U_{n}(a)-F(x_{0})|\geqslant\gamma_{n}\\}$ is contained in the event that
there exists $p\in[F_{n}(Y_{0}),F_{n}(Y_{m})]$ with
$|p-F(x_{0})|\leqslant\varepsilon$, $|p-F(x_{0})|\geqslant\gamma_{n}$ and

$\Lambda_{n}(p)-ap\geqslant\Lambda_{n}(F(x_{0}))-aF(x_{0}).$ (60)

Obviously, the probability is equal to zero if $\gamma_{n}>\varepsilon$ so we
assume in the sequel that $\gamma_{n}\leqslant\varepsilon$. For all
$p\in[F(x_{0})-\varepsilon,F(x_{0})+\varepsilon]$ define

$\Lambda\left(p\right)=\int_{F\left({x_{0}}\right)}^{p}f_{0}\circ
F^{-1}(u)du.$

Let $c>0$ such that $|f_{0}^{\prime}|/F^{\prime}>2c$ on the interval
$[F^{-1}(F(x_{0})-2\varepsilon),F^{-1}(F(x_{0})+2\varepsilon)]$. Since
$\Lambda^{\prime}(F(x_{0}))=a$ and $\Lambda^{\prime\prime}=f_{0}^{\prime}\circ
F^{-1}/F^{\prime}\circ F^{-1}$, it then follows from Taylor’s expansion that

$\Lambda(p)-\Lambda(F(x_{0}))\leqslant(p-F(x_{0}))a-c(p-F(x_{0}))^{2}$

for all $p\in[F(x_{0})-\varepsilon,F(x_{0})+\varepsilon]$ and therefore, (60)
implies that

$\Delta_{n}(p)-\Delta_{n}(F(x_{0}))-c(p-F(x_{0}))^{2}\geqslant 0$

for all such $p$’s, where we set $\Delta_{n}:=\Lambda_{n}-\Lambda$. Hence, for
all $n\geqslant N_{\eta}$,

$\displaystyle\mathbb{P}\left(\left|U_{n}\left(a\right)-F\left(x_{0}\right)\right|\geqslant\gamma_{n}\right)$
$\displaystyle\qquad\leqslant\eta+\mathbb{P}\left(\sup_{|p-F(x_{0})|\in[\gamma_{n},\varepsilon]}\\{\Delta_{n}(p)-\Delta_{n}(F(x_{0}))-c(p-F(x_{0}))^{2}\\}\geqslant
0\mbox{ and }{\mathcal{E}_{n}}\right)$
$\displaystyle\qquad\leqslant\eta+\sum_{j}\mathbb{P}\left(\sup_{|u|\in[\gamma_{n}2^{j},\gamma_{n}2^{j+1}]}\\{\Delta_{n}(F(x_{0})+u)-\Delta_{n}(F(x_{0}))\\}\geqslant
c(\gamma_{n}2^{j})^{2}\mbox{ and }{\mathcal{E}_{n}}\right)$
$\displaystyle\qquad{\leqslant\eta+\sum_{j}\mathbb{P}\left(\sup_{|u|\leqslant\gamma_{n}2^{j+1}}|\Delta_{n}(F(x_{0})+u)-\Delta_{n}(F(x_{0}))|\geqslant
c(\gamma_{n}2^{j})^{2}\mbox{ and }\mathcal{E}_{n}\right)}$ (61)

where the sums are taken over all integers $j\geqslant 0$ such that
$\gamma_{n}2^{j}\leqslant\varepsilon$. Recall that we have (39) for all
$k\in\\{0,\dots,m\\}$. Since $\Lambda_{n}$ is piecewise-linear with knots at
$F_{n}(Y_{0}),\dots,F_{n}(Y_{m})$, by (22) and (23) we get that for every $j$
in the above sum,

$\displaystyle\sup_{|u|\leqslant\gamma_{n}2^{j+1}}|\Delta_{n}(F(x_{0})+u)-\Delta_{n}(F(x_{0}))|$
$\displaystyle\qquad\leqslant\mathop{\sup}\limits_{|u|\leqslant{\gamma_{n}}{2^{j+1}}}\left|{\int\limits_{F\left(x_{0}\right)}^{{F}\left({{x_{0}}}\right)+u}{\left({{f_{0}}\circ
F_{n}^{-1}\left(y\right)-{f_{0}}\circ{F^{-1}}\left(y\right)}\right)dy}}\right|$
$\displaystyle\qquad\qquad+\mathop{\sup}\limits_{|u|\leqslant{\gamma_{n}}{2^{j+1}}}\left|{{M_{n}}\left({{F}\left({{x_{0}}}\right)+u}\right)-{M_{n}}\left({{F}\left({{x_{0}}}\right)}\right)}\right|.$
(62)

Moreover, $|f_{0}^{\prime}|$ is bounded above on
$[F^{-1}(F(x_{0})-2\varepsilon),F^{-1}(F(x_{0})+2\varepsilon)]$, so we obtain
that for every $j$ with $\gamma_{n}2^{j}\leqslant\varepsilon$, the first term
on the right-hand side of (8) satisfies

$\displaystyle\sup_{|u|\leqslant\gamma_{n}2^{j+1}}\left|\int_{F(x_{0})}^{F(x_{0})+u}\left(f_{0}\circ
F_{n}^{-1}(p)-f_{0}\circ F^{-1}(p)\right)dp\right|$
$\displaystyle\qquad\leqslant\int_{F(x_{0})-\gamma_{n}2^{j+1}}^{F(x_{0})+\gamma_{n}2^{j+1}}\left|f_{0}\circ
F_{n}^{-1}(p)-f_{0}\circ F^{-1}(p)\right|dp$ $\displaystyle\qquad\leqslant
K_{3}\gamma_{n}2^{j}\sup_{|p-F(x_{0})|\leqslant
2\varepsilon}\left|F_{n}^{-1}(p)-F^{-1}(p)\right|,$

for some $K_{3}>0$ that does not depend on $n$. Hence, it follows from the
previous display and (59) that

$\displaystyle\mathbb{E}\left(\sup_{|u|\leqslant\gamma_{n}2^{j+1}}\left|\int_{F(x_{0})}^{F(x_{0})+u}(f_{0}\circ
F_{n}^{-1}(p)-f_{0}\circ
F^{-1}(p)dp\right|^{2}\mathbb{I}(\mathcal{E}_{n})\right)$
$\displaystyle\qquad\leqslant
K_{3}^{2}\gamma_{n}^{2}2^{2j}\mathbb{E}\left(\sup_{|p-F(x_{0})|\leqslant
2\varepsilon}|F_{n}^{-1}(p)-F^{-1}(p)|^{2}\mathbb{I}({\mathcal{E}_{n}})\right)$
$\displaystyle\qquad\leqslant
K_{3}^{2}\gamma_{n}^{2}{2^{2j}}{K_{2}}{u\left(n\right)}^{-1}.$

Hence, taking $K_{4}=K_{3}^{2}K_{2}$ we get that for all $j$ with
$\gamma_{n}2^{j}\leqslant\varepsilon\leqslant 1$.

$\mathbb{E}\left(\sup_{|u|\leqslant\gamma_{n}2^{j+1}}\left|\int_{F(x_{0})}^{F(x_{0})+u}(f_{0}\circ
F_{n}^{-1}(p)-f_{0}\circ
F^{-1}(p)dp\right|^{2}\mathbb{I}(\mathcal{E}_{n})\right)\leqslant{K_{4}}{\gamma_{n}}{2^{j}}u\left(n\right)^{-1}.$
(63)

By equations (23) and (24) in Lemma 6.1, the second term on the right-hand
side of (8) satisfies,

$\mathop{\sup}\limits_{|u|\leqslant{\gamma_{n}}{2^{j+1}}}\left|{{M_{n}}\left({{F}\left({{x_{0}}}\right)+u}\right)-{M_{n}}\left({{F}\left({{x_{0}}}\right)}\right)}\right|\leqslant{I^{n,j}_{1}}+{I^{n,j}_{2}},$
(64)

where $I_{1}^{n,j}$ and $I_{2}^{n,j}$ are given by

$\displaystyle I^{n,j}_{1}$
$\displaystyle=\frac{1}{T_{n}(C)}\sup_{|u|\leqslant\gamma_{n}2^{j+1}}\left|\sum_{t=0}^{n}W_{t}\Bigl{(}\mathbb{I}_{C}{\\{X_{t}\leqslant
F_{n}^{-1}(F(x_{0})+u)\\}}-\mathbb{I}_{C}{\\{X_{t}\leqslant
F_{n}^{-1}(F(x_{0}))\\}}\Bigl{)}\right|,$ $\displaystyle{I^{n,j}_{2}}$
$\displaystyle=\frac{2}{{{T_{n}}\left(C\right)}}\sup_{|u|\leqslant\gamma_{n}2^{j+1}}\left|\sum\limits_{t=0}^{n}{{W_{t}}\Bigl{(}{\mathbb{I}_{C}{\left\\{{{X_{t}}=F_{n}^{-1}\left({{F}\left({{x_{0}}}\right)+u}\right)}\right\\}}}\Bigl{)}}\right|.$

For $I^{n,j}_{1}$, it follows from the triangle inequality that

$I^{n,j}_{1}\leqslant\frac{2}{T_{n}(C)}\sup_{|u|\leqslant\gamma_{n}2^{j+1}}\left|\sum_{t=0}^{n}W_{t}\left(\mathbb{I}_{C}{\\{X_{t}\leqslant
F_{n}^{-1}(F(x_{0})+u)\\}}-\mathbb{I}_{C}{\\{X_{t}\leqslant
x_{0}\\}}\right)\right|.$

Combining (59) and the fact that $F^{-1}$ is Lipschitz in
$[F(x_{0})-2\varepsilon,F(x_{0})+2\varepsilon]$ we can find
$K_{5}=\max{\left(\sqrt{K_{2}},\sup\left(F^{-1}\right)\right)}$ independent of
$n$ such that, on $\mathcal{E}_{n}$,

$\sup_{|p-F(x_{0})|\leqslant
2\varepsilon}|F_{n}^{-1}(p)-F^{-1}(p)|\leqslant\frac{K_{5}}{\sqrt{u\left(n\right)}}$

and $|F^{-1}(F(x_{0})+u)-x|=|F^{-1}(F(x_{0})+u)-F^{-1}(F(x_{0}))|\leqslant
K_{5}|u|/2$ for all $u$ with $|u|\leqslant 2\varepsilon$. Hence, on
$\mathcal{E}_{n}$

$\displaystyle I^{n,j}_{1}$
$\displaystyle\leqslant\frac{2}{T_{n}(C)}\sup_{|y-x_{0}|\leqslant
K_{5}\gamma_{n}2^{j}+K_{5}/\sqrt{u\left(n\right)}}\left|\sum_{t=0}^{n}W_{t}\left(\mathbb{I}_{C}{\\{X_{t}\leqslant
y\\}}-\mathbb{I}_{C}{\\{X_{t}\leqslant x_{0}\\}}\right)\right|,$
$\displaystyle I^{n,j}_{2}$
$\displaystyle\leqslant\frac{2}{{{T_{n}}\left(C\right)}}\sup_{|y-x_{0}|\leqslant
K_{5}\gamma_{n}2^{j}+K_{5}/\sqrt{u\left(n\right)}}\left|\sum\limits_{t=0}^{n}{{W_{t}}\Bigl{(}{\mathbb{I}_{C}{\left\\{{{X_{t}}=y}\right\\}}}\Bigl{)}}\right|.$

It follows from (55) that $\gamma_{n}2^{j}\geqslant\gamma_{n}\geqslant
1/\sqrt{u\left(n\right)}$ for all $j\geqslant 0$, then, on $\mathcal{E}_{n}$

$\displaystyle I^{n,j}_{1}$
$\displaystyle\leqslant\frac{2}{T_{n}(C)}\sup_{|y-x_{0}|\leqslant
2K_{5}\gamma_{n}2^{j}}\left|\sum_{t=0}^{n}W_{t}\left(\mathbb{I}_{C}{\\{X_{t}\leqslant
y\\}}-\mathbb{I}_{C}{\\{X_{t}\leqslant x_{0}\\}}\right)\right|,$
$\displaystyle I^{n,j}_{2}$
$\displaystyle\leqslant\frac{2}{{{T_{n}}\left(C\right)}}\sup_{|y-x_{0}|\leqslant
2K_{5}\gamma_{n}2^{j}}\left|\sum\limits_{t=0}^{n}{{W_{t}}\Bigl{(}{\mathbb{I}_{C}{\left\\{{{X_{t}}=y}\right\\}}}\Bigl{)}}\right|.$

By Lemma 6.6, we conclude that there exists $K_{6}>0$ and $N_{\eta}^{\prime}$
such that, for $n\geqslant N_{\eta}^{\prime}$

$\mathbb{E}\Bigl{(}\left(I^{n,j}_{1}+I^{n,j}_{2}\right)^{2}\mathbb{I}(\mathcal{E}_{n})\Bigr{)}\leqslant
K_{6}\gamma_{n}2^{j}u\left(n\right)^{-1}$ (65)

Combining (8), (63), (64) and (65), we conclude that there exists $K_{7}>0$,
independent of $n$ and $K_{0}$, such that for all $n\geqslant
N_{\eta}^{\prime}$ and $j\geqslant 0$ where
$\gamma_{n}2^{j}\leqslant\varepsilon$, one has

$\displaystyle\mathbb{E}\left(\sup_{|u|\leqslant\gamma_{n}2^{j+1}}|\Delta_{n}(F(x_{0})+u)-\Delta_{n}(F(x_{0}))|^{2}{\mathbb{I}(\mathcal{E}_{n})}\right)\leqslant{K_{7}\gamma_{n}2^{j}u\left(n\right)^{-1}}.$

Combining this with (8) and the Markov inequality, we conclude that there
exist $K_{8}>0$ and $N_{\eta}^{\prime\prime}$, that do not depend on $n$ nor
$K_{0}$, such that, for all $n\geqslant N_{\eta}^{\prime\prime}$,

$\displaystyle\mathbb{P}\left(|U_{n}(a)-F(x_{0})|\geqslant\gamma_{n}\right)$
$\displaystyle\leqslant$ $\displaystyle\eta+K_{8}\sum_{k\geqslant
0}\frac{\gamma_{n}2^{j}u\left(n\right)^{-1}}{(\gamma_{n}2^{j})^{4}}$
$\displaystyle\leqslant$
$\displaystyle\eta+K_{8}\gamma_{n}^{-3}u\left(n\right)^{-1}\sum_{j\geqslant
0}2^{-3j}.$

The sum on the last line is finite, so there exists $K>0$, independent of $n$
and $K_{0}$, such that for $n$ bigger than $N_{\eta}^{\prime\prime}$

$\mathbb{P}\left(|U_{n}(a)-F(x_{0})|\geqslant\gamma_{n}\right)\leqslant\eta+K\gamma_{n}^{-3}u\left(n\right)^{-1}=\eta+\frac{K}{K_{0}^{3}}.$
(66)

The above probability can be made smaller than $2\eta$ by setting (54) for
some sufficiently large $K_{0}$ independent of $n$. This proves (53) and
completes the proof of (37).
Now, we turn to the proof of (38). It follows from (30) combined to (32) and
Lemma 5.3 that

$\widehat{f}_{n}^{-1}(f_{0}(x_{0}))=F^{-1}\circ\widehat{U}_{n}(f_{0}(x_{0}))+T_{n}(C)^{-1/2}O_{P}\left(1\right).$

Hence, by Lemma 6.7 we have

$\widehat{f}_{n}^{-1}(f_{0}(x_{0}))=F^{-1}\circ\widehat{U}_{n}(f_{0}(x_{0}))+O_{P}\left(u\left(n\right)^{-1/2}\right).$

Now, it follows from the assumption (B2) that $F^{-1}$ has a bounded
derivative in the neighborhood of $F(x_{0})$, to which
$\widehat{U}_{n}(f_{0}(x_{0}))$ belongs with probability that tends to one.
Hence, it follows from Taylor’s expansion that

$\displaystyle\widehat{f}_{n}^{-1}(f_{0}(x_{0}))$ $\displaystyle=F^{-1}\circ
F(x_{0})+O\left(|\widehat{U}_{n}(f_{0}(x_{0}))-F(x_{0})|\right)+O_{P}\left(u\left(n\right)^{-1/2}\right)$
$\displaystyle=x+O_{P}({u\left(n\right)^{-1/3}})+O_{P}\left(u\left(n\right)^{-1/2}\right),$

where we used (37) for the last equality. This proves (38) and completes the
proof of Lemma 6.8.

Supplement of “Harris recurrent Markov chains and nonlinear monotone
cointegrated models” This is the supplementary material associated with the
present article.

##  1 Markov chain theory and notation

This section extends Section 2 of the main paper, presenting a more detailed
exposition of the Markov chain theory required in the proofs. For further
details, we refer the reader to [29, 27, 12].

Let $\textbf{X}=X_{0},X_{1},X_{2},\ldots$ be a time-homogeneous Markov Chain
defined on a probability space $\left(E,\mathcal{E},\mathbb{P}\right)$ where
$\mathcal{E}$ is countably generated. Let $P\left(x,A\right)$ denote its
transition kernel, i.e. for $x\in E$ , $A\in\mathcal{E}$ we have

$P\left({x,A}\right)=\mathbb{P}\left({{X_{i+1}}\in
A\left|{{X_{i}}=x}\right.}\right),\quad i=0,1,\ldots$

Let $P^{n}(x,A)$ denote the $n$-step transition probability, i.e.

${P^{n}}\left({x,A}\right)=\mathbb{P}\left({{X_{i+n}}\in
A\left|{{X_{i}}=x}\right.}\right)\quad\forall i.$

If $\lambda$ is a probability measure in $\left(E,\mathcal{E}\right)$ such
that $\mathcal{L}\left(X_{0}\right)=\lambda$, then $\lambda$ is called the
initial measure of the chain X. A homogeneous Markov chain is uniquely
identified by its kernel and initial measure.

When the initial measure of the chain is given, we will write
$\mathbb{P}_{\lambda}$ (and $\mathbb{E}_{\lambda}$) for the probability (and
the expectation) conditioned on $\mathcal{L}\left(X_{0}\right)=\lambda$. When
$\lambda=\delta_{x}$ for some $x\in E$ we will simply write $\mathbb{P}_{x}$
and $\mathbb{E}_{x}$.

An homogeneous Markov chain is irreducible if there exists a $\sigma$-finite
measure $\phi$ on $\left(E,\mathcal{E}\right)$ such that for all $x\in E$ and
all $A\in\mathcal{E}$ with $\phi(A)>0$ we have $P^{n}(x,A)>0$ for some
$n\geqslant 1$. In this case, there exists a maximal irreducibility measure
$\psi$ (all other irreducibility measures are absolutely continuous with
respect to $\psi$). In the following, all Markov chains are supposed to be
irreducible with maximal irreducibility measure $\psi$.

For any set $C\in\mathcal{E}$, we will denote by $\sigma_{C}$ and $\tau_{C}$,
respectively, the times of first visit and first return of the chain to the
set $C$, i.e. $\tau_{C}=\operatorname{inf}\left\\{n\geqslant 1:X_{n}\in
C\right\\}$ and $\sigma_{C}=\operatorname{inf}\left\\{n\geqslant 0:X_{n}\in
C\right\\}$. The subsequent visit and return times
$\sigma_{C},\tau_{C}\left(k\right)$, $k\geqslant 1$ are defined inductively as
follows:

$\displaystyle\tau_{C}\left(1\right)=\tau_{C}\quad$
$\displaystyle,\quad\tau_{C}\left(k\right)=\operatorname{min}\left\\{n>\tau_{C}\left(k-1\right):X_{n}\in
C\right\\},$ (67) $\displaystyle\sigma_{C}\left(1\right)=\sigma_{C}\quad$
$\displaystyle,\quad\sigma_{C}\left(k\right)=\operatorname{min}\left\\{n>\sigma_{C}\left(k-1\right):X_{n}\in
C\right\\}.$ (68)

Given that our methods will only deal with the values of X in a fixed set $C$,
if $A$ is a measurable set, we will write $\mathbb{I}_{C}\\{X_{t}\in A\\}$
instead of $\mathbb{I}\\{X_{t}\in A\cap C\\}$ and if $A=E$, then we will
simply write $\mathbb{I}_{C}\left(X_{t}\right)$.

We will use $T_{n}\left(C\right)$ to denote the random variable that counts
the number of times the chain has visited the set $C$ up to time $n$, that is
$T_{n}\left(C\right)=\sum_{t=0}^{n}{\mathbb{I}_{C}\left(X_{t}\right)}$.
Similarly, we will write $T\left(C\right)$ for the total of numbers of visits
the chain X to $C$. The set $C$ is called recurrent if
$\mathbb{E}_{x}T\left(C\right)=+\infty$ for all $x\in C$. The chain X is
considered recurrent if every set $A\in\mathcal{E}$, such that
$\psi\left(A\right)>0$, is recurrent.

Although recurrent chains possess many interesting properties, a stronger type
of recurrence is required in our analysis. An irreducible Markov chain is
Harris recurrent if for all $x\in E$ and all $A\in\mathcal{E}$ with
$\psi(A)>0$ we have

$\mathbb{P}\left({{X_{n}}\in A\;\text{infinitely
often}\left|{{X_{0}}=x}\right.}\right)=1.$

An irreducible chain possesses an accessible atom, if there is a set
$\boldsymbol{\alpha}\in\mathcal{E}$ such that for all $x,y$ in
$\boldsymbol{\alpha}$: $P(x,.)=P(y,.)$ and $\psi(\boldsymbol{\alpha})>0$.
Denote by $\mathbb{P}_{\boldsymbol{\alpha}}$ and
$\mathbb{E}_{\boldsymbol{\alpha}}{\left(.\right)}$ the probability and the
expectation conditionally to $X_{0}\in\boldsymbol{\alpha}$. If X possesses an
accessible atom and is Harris recurrent, the probability of returning
infinitely often to the atom $\boldsymbol{\alpha}$ is equal to one, no matter
the starting point, i.e. $\forall x\in
E,{\mathbb{P}}_{x}\left(\tau_{\boldsymbol{\alpha}}<\infty\right)=1.$ Moreover,
it follows from the strong Markov property that the sample paths may be
divided into independent blocks of random length corresponding to consecutive
visits to $\boldsymbol{\alpha}$:

$\displaystyle\mathcal{B}_{0}$
$\displaystyle=\left(X_{0},X_{1},\ldots,X_{\tau_{\boldsymbol{\alpha}}\left(1\right)}\right)$
$\displaystyle\mathcal{B}_{1}$
$\displaystyle=\left(X_{\tau_{\boldsymbol{\alpha}}\left(1\right)+1},\ldots,X_{\tau_{\boldsymbol{\alpha}}\left(2\right)}\right)$
$\displaystyle\ldots$ $\displaystyle\mathcal{B}_{n}$
$\displaystyle=\left(X_{\tau_{\boldsymbol{\alpha}}\left(n\right)+1},\ldots,X_{\tau_{\boldsymbol{\alpha}}\left(n+1\right)}\right)$
$\displaystyle\ldots$

taking their values in the torus $\mathbb{T}=\cup_{n=1}^{\infty}E^{n}$. Notice
that the distribution of $\mathcal{B}_{0}$ depends on the initial measure,
therefore it does not have the same distribution as $\mathcal{B}_{j}$ for
$j\geqslant 1$. The sequence
$\left\\{\tau_{\boldsymbol{\alpha}}(j)\right\\}_{j\geqslant 1}$ defines
successive times at which the chain forgets its past, called regeneration
times. Similarly, the sequence of i.i.d. blocks
$\\{\mathcal{B}_{j}\\}_{j\geqslant 1}$ are named regeneration blocks. The
random variable $T\left({n}\right)=T_{n}\left(\boldsymbol{\alpha}\right)-1$
counts the number of i.i.d. blocks up to time $n$. This term is called number
of regenerations up to time $n$.

Notice that for any function defined on $E$, we can write
$\sum_{t=0}^{n}{f\left(X_{t}\right)}$ as a sum of independent random variables
as follows:

$\sum_{t=0}^{n}{f\left(X_{t}\right)}=f\left(\mathcal{B}_{0}\right)+\sum_{j=1}^{T\left(n\right)}{f\left(\mathcal{B}_{j}\right)}+{f\left(\mathcal{B}_{(n)}\right)},$
(69)

where,
$f\left(\mathcal{B}_{0}\right)=\sum_{t=0}^{\tau_{\alpha}}{f\left(X_{t}\right)}$,
$f\left(\mathcal{B}_{j}\right)=\sum_{t=\tau_{\boldsymbol{\alpha}}\left(j\right)+1}^{\tau_{\alpha}\left(j+1\right)}{f\left(X_{t}\right)}$
for $j=1,\ldots,T\left({n}\right)$ and ${f\left(\mathcal{B}_{(n)}\right)}$
denotes the last incomplete block, i.e.
${f\left(\mathcal{B}_{(n)}\right)}=\sum_{t=\tau_{\boldsymbol{\alpha}}\left(T\left(n\right)+1\right)+1}^{n}{f\left(X_{t}\right)}$.

When an accessible atom exists, the stochastic stability properties of X
amount to properties concerning the speed of return time to the atom only. For
instance, the measure $\pi_{\boldsymbol{\alpha}}$ given by:

$\pi_{\boldsymbol{\alpha}}\left(B\right)=\mathbb{E}_{\boldsymbol{\alpha}}\left(\sum_{n=1}^{\tau_{\boldsymbol{\alpha}}}\mathbb{I}{\left\\{X_{i}\in
B\right\\}}\right),\quad\forall B\in\mathcal{E}$ (70)

is invariant, i.e.

$\pi_{\boldsymbol{\alpha}}\left(B\right)=\int{P\left({x,B}\right)d\pi_{\boldsymbol{\alpha}}\left(x\right)}.$

Denote by $\mathcal{E}^{+}$ the class of nonnegative measurable functions with
positive $\psi$ support. A function $s\in\mathcal{E}^{+}$ is called small if
there exists an integer $m_{0}\geqslant 1$ and a measure
$\nu\in\mathscr{M}{\left({\mathcal{E}}\right)_{+}}$ such that

${P^{m_{0}}}\left({x,A}\right)\geqslant
s\left(x\right)\nu\left(A\right)\quad\forall x\in E,A\in\mathcal{E}.$ (71)

When a chain possesses a small function $s$, we say that it satisfies the
minorization inequality $M\left(m_{0},s,\nu\right)$. As pointed out in [29],
there is no loss of generality in assuming that $0\leqslant
s\left(x\right)\leqslant 1$ and $\int_{E}{s(x)d\nu(x)}>0$.

A set $A\in\mathcal{E}$ is said to be small if the function $\mathbb{I}_{A}$
is small. Similarly, a measure $\nu$ is small if there exist $m_{0}$, and $s$
that satisfy (71). By Theorem 2.1 in [29], every irreducible Markov chain
possesses a small function and Proposition 2.6 of the same book shows that
every measurable set $A$ with $\psi\left(A\right)>0$ contains a small set. In
practice, finding such a set consists in most cases in exhibiting an
accessible set, for which the probability that the chain returns to it in $m$
steps is uniformly bounded below. Moreover, under quite wide conditions a
compact set will be small, see [15].

If X does not possess an atom but is Harris recurrent (and therefore satisfies
a minorization inequality $M\left(m_{0},s,\nu\right)$), a splitting technique,
introduced in [28, 29], allows us to extend in some sense the probabilistic
structure of X in order to artificially construct an atom. The general idea
behind this construction is to expand the sample space so as to define a
sequence $(Y_{n})_{n\in\mathbb{N}}$ of Bernoulli r.v.’s and a bivariate chain
${\check{\textbf{X}}}=\left\\{{\left({{X_{n}},{Y_{n}}}\right)}\right\\}_{n=0}^{+\infty}$,
named split chain, such that the set
$\check{\boldsymbol{\alpha}}=\left(E,1\right)$ is an atom of this chain. A
detailed description of this construction can be found in [29].

The whole point of this construction consists in the fact that
${\check{\textbf{X}}}$ inherits all the communication and stochastic stability
properties from X (irreducibility, Harris recurrence,…). In particular, the
marginal distribution of the first coordinate process of
${\check{\textbf{X}}}$ and the distribution of the original X are identical.
Hence, the splitting method enables us to establish all the results known for
atomic chains to general Harris chains, for example, the existence of an
invariant measure which is unique up to multiplicative constant (see
Proposition 10.4.2 in [27]).

The invariant measure is finite if and only if
$\mathbb{E}_{\check{\boldsymbol{\alpha}}}{\tau_{\check{\boldsymbol{\alpha}}}}<+\infty$,
in this case we say the chain is positive recurrent, otherwise, we say the
chain is null recurrent. A null recurrent chain is called $\beta$-null
recurrent (c.f. Definition 3.2 in [24]) if there exists a small nonnegative
function $h$, a probability measure $\lambda$, a constant
$\beta\in\left({0,1}\right)$ and a slowly varying function $L_{h}$ such that

${\mathbb{E}_{\lambda}}\left({\sum\limits_{t=0}^{n}{h\left({{X_{t}}}\right)}}\right)\sim\frac{1}{{\Gamma\left({1+\beta}\right)}}{n^{\beta}}{L_{h}}\left(n\right)\quad\textnormal{as}\;n\to\infty.$

As argued in [24], is not a too severe restriction to assume $m_{0}=1$.
Therefore, throughout this paper we assume that X satisfies the minorization
inequality $M(1,s,\nu)$, i.e, there exist a measurable function $s$ and a
probability measure $\nu$ such that $0\leqslant s\left(x\right)\leqslant 1$,
$\int_{E}{s(x)d\nu(x)}>0$ and

${P}\left({x,A}\right)\geqslant s\left(x\right)\nu\left(A\right).$ (72)

A measurable and positive function $L$, defined in $[a,+\infty)$ for some
$a\geqslant 0$, is called slowly varying at $+\infty$ if it satisfies
$\lim_{x\to+\infty}{\frac{L\left(xt\right)}{L\left(x\right)}}=1$ for all
$t\geqslant a$. See [6] for a detailed compendium of these types of functions.

It was shown in Theorem 3.1 of [24] that if the chain satisfies the
minorization condition (5), then it is $\beta$-null recurrent if and only if

$\mathbb{P}\left({{\tau_{\check{\boldsymbol{\alpha}}}}>n}\right)\sim\frac{1}{{{\Gamma\left(1-\beta\right)n^{\beta}}L\left(n\right)}},$
(73)

where $L$ is a slowly varying function.

The following theorem is a compendium of the main properties of Harris’s
recurrent Markov chains that will be used throughout the proofs. Among other
things, it shows that the asymptotic behavior of $T\left({n}\right)$ is
similar to the function $u\left(n\right)$ defined as

$u\left(n\right)=\begin{cases}n,&\text{if }\textbf{X}\text{ is positive
recurrent}\\\ n^{\beta}L\left(n\right),&\text{if }\textbf{X}\text{ is
}\beta\text{-null recurrent}\end{cases}.$ (74)

The Mittag-Leffler distribution with index $\beta$ is a non-negative
continuous distribution, whose moments are given by

$\mathbb{E}\left(M^{m}_{\beta}\left(1\right)\right)=\frac{m!}{\Gamma\left(1+m\beta\right)}\;\;m\geqslant
0.$

See (3.39) in [24] for more details.

######  Theorem 1.1.

Suppose X is a Harris recurrent, irreducible Markov chain, with initial
measure $\lambda$, that satisfies the minorization condition (72). Let
$T\left({n}\right)$ be the number of complete regenerations until time $n$ of
the split chain ${\check{\textbf{X}}}$ , let $C\in\mathcal{E}$ be a small set
and $\pi$ be an invariant measure for X. Then,

  1. 1.

$0<\pi\left(C\right)<+\infty$.

  2. 2.

For any function $f$, defined on $E$, the decomposition (69) holds. Moreover,
there is a constant $K_{\pi}$, that only depends on $\pi$, such that if $f\in
L^{1}\left(E,{\pi}\right)$, then
$\mathbb{E}_{\lambda}{f\left(\mathcal{B}_{1}\right)}=K_{\pi}\int_{E}fd\pi$.

  3. 3.

$\frac{T\left({n}\right)}{T_{n}(C)}$ converges almost surely to a positive
constant.

  4. 4.

$\frac{T\left({n}\right)}{u\left(n\right)}$ converges almost surely to a
positive constant if X is positive recurrent and converges in distribution to
a Mittag-Leffler random variable with index $\beta$ if X is $\beta$-null
recurrent.

######  Remark 1.1.

The Mittag-Leffler distribution with index $\beta$ is a non-negative
continuous distribution, whose moments are given by

$\mathbb{E}\left(M^{m}_{\beta}\left(1\right)\right)=\frac{m!}{\Gamma\left(1+m\beta\right)}\;\;m\geqslant
0.$

See (3.39) in [24] for more details.

######  Remark 1.2.

Part 1 of Theorem 1.1 is Proposition 5.6.ii of [29], part 2 is equation (3.23)
of [24] and part 3 is an application of the Ratio Limit Theorem (Theorem
17.2.1 of [27]). For the positive recurrent case, part 4 also follows by the
aforementioned Ratio Limit Theorem while the claim for the null recurrent case
appears as Theorem 3.2 in [24].

##  2 Technical proofs for Section 5

Proof of Lemma 5.1. Equation (12) follows from Corollary 2 in [2] and part 2
of Theorem 1.1.

Now, we turn to the proof of (13). To do this, we adapt some of the ideas
presented in the proof of Lemma 21.2 in [33].

Let $V$ be a normal random variable independent of the $X_{i}$’s, and $\Phi$
its distribution function. it follows from (12) that conditionally on the
$X_{t}$’s, $F_{n}(V)$ converges almost surely to $F(V)$. Thus, denoting by
$\mathbb{P}_{X}$ the conditional probability given the $X_{t}$’s, it follows
from (29) that $\Phi(F_{n}^{-1}(u))=\mathbb{P}_{X}(F_{n}(V)<u)$ converges
almost surely to $\mathbb{P}_{X}(F(V)<u)=\Phi(F^{-1}(u))$ at every $u$ at
which the limit function is continuous . Since $F$ is strictly increasing in
$C$, one can find $\varepsilon>0$ such that $F^{-1}$ is continuous on
$[F(x_{0})-\varepsilon,F(x_{0})+\varepsilon]$, so the above limit function is
continuous at every $u\in[F(x_{0})-\varepsilon,F(x_{0})+\varepsilon]$. By
continuity of $\Phi^{-1}$ on $(0,1)$, $F_{n}^{-1}(u)$ converges almost surely
to $F^{-1}(u)$ for every such $u$. By monotonicity, the convergence is
uniform, hence

$\sup_{|p-F(x_{0})|\leqslant\varepsilon}|F_{n}^{-1}(p)-F^{-1}(p)|=o(1)\quad
a.s.$

as $n\to\infty$.

Proof of Lemma 5.2. This proof is an adaptation to the localized case of the
proof of Lemma 2 in [5]. Let $f^{\prime}_{C}\in\mathcal{F}^{\prime}_{C}$,
i.e., there exists $f\in\mathcal{F}$ such that $f_{C}^{\prime}(B)=\int
f(y)\,M_{C}(B,\mathrm{d}y)$. By Cauchy–Schwarz inequality,

${\left({\int f(y){\mkern
1.0mu}{M_{C}}(B,dy)}\right)^{2}}\leqslant{\ell_{C}}\left(B\right)\left({\int{{f^{2}}{M_{C}}(B,dy)}}\right),$

then

$\mathbb{E}_{Q^{\prime}}(f^{\prime
2}_{C})\leqslant\mathbb{E}_{Q^{\prime}}\left(\ell_{C}(B)\left(\int
f(y)^{2}\,M_{C}(B,dy)\right)\right)=\mathbb{E}_{Q_{C}}(f^{2})E_{Q^{\prime}}(\ell_{C}^{2}),$

where the last equality follows from (16). Applying this to the function

$\displaystyle
f_{C}^{\prime}(B)-f_{k}^{\prime}(B)=\int(f(y)-f_{k}(y))\,M_{C}(B,dy),$

when each $f_{k}$ is the center of an $\varepsilon$-cover of the space
$\mathcal{F}$ and $\|f-f_{k}\|_{L_{2}(Q_{C})}\leqslant\varepsilon$ gives the
first assertion of the lemma. To obtain the second assertion, note that
$U_{C}^{\prime}=U\ell_{C}$ is an envelope for $\mathcal{F}_{C}^{\prime}$. In
addition, we have that

$\displaystyle\|U_{C}^{\prime}\|_{L_{2}(Q^{\prime})}=U\|\ell_{C}\|_{L_{2}(Q^{\prime})}.$

From this, we derive that, for every $0<\varepsilon<1$,

$\displaystyle\mathcal{N}(\varepsilon\|U_{C}^{\prime}\|_{L_{2}(Q^{\prime})},\,\mathcal{U}_{C}^{\prime},\,L_{2}(Q^{\prime}))=\mathcal{N}(\varepsilon
U\|\ell_{C}\|_{L_{2}(Q^{\prime})},\,\mathcal{U}^{\prime},\,L_{2}(Q^{\prime})).$

Then using the first assertion of the lemma, we obtain for every
$0<\varepsilon<1$,

$\displaystyle\mathcal{N}(\varepsilon\|U_{C}^{\prime}\|_{L_{2}(Q^{\prime})},\,\mathcal{F}_{C}^{\prime},\,L_{2}(Q^{\prime}))\leqslant\mathcal{N}(\varepsilon
U,\,\mathcal{F},\,L_{2}(Q_{C})),$

which implies the second assertion of the Lemmaz whenever the class
$\mathcal{F}$ is VC with envelope $U$.

Proof of Lemma 5.3. Let $B\in\widehat{E}$ and
$g:E\times\mathbb{R}\to\mathbb{R}_{+}$. For each $y\in\mathbb{R}$ we define
$g_{y}\left(x\right)=g\left(x,y\right)$, then, using the notation of section
6.2 we will have $\widehat{g}_{y}\left(B\right)=\sum_{x\in B\cap
C}g\left(x,y\right)$. Finally, for any function $h:\mathbb{R}\to\mathbb{R}$,
we define

$\widetilde{g}^{h}_{y}\left(B\right)=(\widehat{g_{y}-h(y)})(B)=\sum_{x\in
B\cap
C}\left(g\left(x,y\right)-h\left(y\right)\right)=\widehat{g}_{y}\left(B\right)-\ell_{C}\left(B\right)h\left(y\right).$

Let $g\left(x,y\right)=\mathbb{I}\left\\{x\leqslant y\right\\}$, and $h=F$ as
defined in (8). Then, $\widehat{g}_{y}\left(B\right)=\sum_{x\in
B}\mathbb{I}_{C}{\left\\{x\leqslant y\right\\}}$ and

$\widetilde{g}_{y}^{F}\left(B\right)=\sum_{x\in B\cap
C}\left(\mathbb{I}\left\\{x\leqslant
y\right\\}-F\left(y\right)\right)=\widehat{g}_{y}\left(B\right)-\ell_{C}\left(B\right)F\left(y\right).$

From now on, we’ll remove the superindex from $\widetilde{g}_{y}^{F}$ to ease
the notation.

By the definition of $F_{n}$ and $F$ ((7) and (8)), we have that

$\displaystyle F_{n}(y)-F(y)$
$\displaystyle=\frac{1}{T_{n}(C)}\sum_{i=1}^{T_{n}(C)}{\left(\mathbb{I}{\\{X_{\sigma_{C}\left(i\right)}\leqslant
y\\}}-F(y)\right)}$
$\displaystyle=\frac{1}{T_{n}(C)}\sum_{i=0}^{n}{\left(\mathbb{I}_{C}{\\{X_{t}\leqslant
y\\}}-\mathbb{I}_{C}{\\{X_{i}\\}}F(y)\right)}$
$\displaystyle=\frac{1}{T_{n}(C)}\left(\widetilde{g}_{y}\left(\mathcal{B}_{0}\right)+\sum_{i=1}^{T\left(n\right)}{\widetilde{g}_{y}\left(\mathcal{B}_{i}\right)}+\widetilde{g}_{y}\left(\mathcal{B}_{(n)}\right)\right),$

therefore,

$\sqrt{T_{n}(C)}\Bigl{(}F_{n}(y)-F(y)\Bigr{)}=\frac{\widetilde{g}_{y}\left(\mathcal{B}_{0}\right)}{\sqrt{T_{n}(C)}}+\frac{\sum_{i=1}^{T\left(n\right)}{\widetilde{g}_{y}\left(\mathcal{B}_{i}\right)}}{\sqrt{T_{n}(C)}}+\frac{\widetilde{g}_{y}\left(\mathcal{B}_{(n)}\right)}{\sqrt{T_{n}(C)}}.$

Notice that
$\left|\widetilde{g}_{y}\left(\mathcal{B}_{0}\right)\right|\leqslant
2\ell_{C}(\mathcal{B}_{0})<+\infty$ and $T_{n}(C)\to+\infty$ almost surely,
therefore, the first term in the last equation converges almost surely to 0
uniformly in $y$. For the last term, we have that

$\frac{\left|\widetilde{g}_{y}\left(\mathcal{B}_{(n)}\right)\right|}{\sqrt{T_{n}(C)}}\leq\frac{2\ell_{C}(\mathcal{B}_{T\left(n\right)})}{\sqrt{T_{n}(C)}}=2\sqrt{\frac{T\left(n\right)}{T_{n}(C)}}\frac{\ell_{C}(\mathcal{B}_{T\left(n\right)})}{\sqrt{T\left(n\right)}},$

by (B4), the expectation of $\ell^{2}_{C}(\mathcal{B}_{1})$ is finite, then,
Lemma 1 in [2] shows that $\frac{\ell^{2}_{C}(\mathcal{B}_{n})}{n}\to 0$ a.s.
which implies that $\frac{\ell_{C}(\mathcal{B}_{n})}{\sqrt{n}}$ also converges
to 0 a.s. Since $T\left(n\right)\to+\infty$ a.s., by Theorem 6.8.1 in [19] we
have
$\frac{\ell_{C}(\mathcal{B}_{T\left(n\right)})}{\sqrt{T\left(n\right)}}\to 0$
almost surely. Joining this with the almost sure convergence of
$\frac{T(n)}{T_{n}(C)}$ to a positive constant (see Theorem 1.1) we obtain
that
$\frac{\left|\widetilde{g}_{y}\left(\mathcal{B}_{(n)}\right)\right|}{\sqrt{T_{n}(C)}}$
converges almost surely to 0 uniformly in $y$. Therefore,

$\sqrt{T_{n}(C)}\Bigl{(}F_{n}(y)-F(y)\Bigr{)}=\frac{\sum_{i=1}^{T\left(n\right)}{\widetilde{g}_{y}\left(\mathcal{B}_{i}\right)}}{\sqrt{T(n)}}+o_{P}\left(1\right).$
(75)

where we have used that $\frac{T_{n}(C)}{T(n)}$ converges almost surely to a
positive constant to use $T(n)$ instead of $T_{n}(C)$.

Then, (17) will be proved if we show that, for $\varepsilon$ small enough

$\mathop{\sup}\limits_{|y-x_{0}|\leqslant\varepsilon}{\frac{\left|\sum_{i=1}^{T\left(n\right)}{\widetilde{g}_{y}\left(\mathcal{B}_{i}\right)}\right|}{\sqrt{T(n)}}}={O_{p}}\left(1\right).$
(76)

Fix $\eta>0$ arbitrarily. By Lemma 6.7 and Slutsky’s theorem, we can find
positive numbers $\underline{a}_{\eta},\overline{a}_{\eta}$ and an integer
$N_{\eta}$ such that $P\left(\mathcal{E}_{n}\right)\geqslant 1-\frac{\eta}{2}$
for all $n\geqslant N_{\eta}$, where

$\mathcal{E}_{n}=\left\\{\underline{a}_{\eta}u\left(n\right)\leqslant
T(n)\leqslant\overline{a}_{\eta}u\left(n\right)\right\\}.$ (77)

Define
$W_{n}(\varepsilon)=\mathop{\sup}\limits_{|y-x_{0}|\leqslant\varepsilon}{\left|\sum_{i=1}^{n}{\widetilde{g}_{y}\left(\mathcal{B}_{i}\right)}\right|}$
and let $M_{\eta}$ be a fixed positive number. Then, for all $n\geqslant
N_{\eta}$

$\displaystyle\mathbb{P}\left(\frac{1}{\sqrt{T(n)}}W_{T(n)}>M_{\eta}\right)$
$\displaystyle<\mathbb{P}\left(\left\\{\frac{1}{\sqrt{T(n)}}W_{T(n)}>M_{\eta}\right\\}\cap\mathcal{E}_{n}\right)+1-\mathbb{P}\left(\mathcal{E}_{n}\right)$
$\displaystyle<\mathbb{P}\left(\left\\{\frac{1}{\sqrt{T(n)}}W_{T(n)}>M_{\eta}\right\\}\cap\mathcal{E}_{n}\right)+\frac{\eta}{2}.$
(78)

On $\mathcal{E}_{n}$, $\underline{a}_{\eta}u\left(n\right)\leqslant
T(n)\leqslant\overline{a}_{\eta}u\left(n\right)$, therefore for all
$n\geqslant N_{\eta}$

$\displaystyle\mathbb{P}\left(\left\\{\frac{1}{\sqrt{T(n)}}W_{T(n)}>M_{\eta}\right\\}\cap\mathcal{E}_{n}\right)$
$\displaystyle<\mathbb{P}\left(\left\\{\frac{1}{\sqrt{\underline{a}_{\eta}u\left(n\right)}}\max_{1\leqslant
k\leqslant\overline{a}_{\eta}u\left(n\right)}{W_{k}}>M_{\eta}\right\\}\cap\mathcal{E}_{n}\right),$
$\displaystyle<\mathbb{P}\left(\frac{1}{\sqrt{\underline{a}_{\eta}u\left(n\right)}}\max_{1\leqslant
k\leqslant\overline{a}_{\eta}u\left(n\right)}{W_{k}}>M_{\eta}\right).$ (79)

The random variables
$\left\\{\widetilde{g}_{(\cdot)}\left(\mathcal{B}_{k}\right)\right\\}_{k=1}^{\overline{a}_{\eta}u\left(n\right)}$
are i.i.d., therefore, by Montgomery-Smith’s inequality (Lemma 4 in [1]),
there exists a universal constant $K$ such that for all $n\geqslant N_{\eta}$,

$\displaystyle\mathbb{P}\left(\frac{1}{\sqrt{\underline{a}_{\eta}u\left(n\right)}}\max_{1\leqslant
k\leqslant\overline{a}_{\eta}u\left(n\right)}{W_{k}}>M_{\eta}\right)$
$\displaystyle<K\mathbb{P}\left(\frac{1}{\sqrt{\underline{a}_{\eta}u\left(n\right)}}{W_{\overline{a}_{\eta}u\left(n\right)}}>\frac{M_{\eta}}{K}\right),$
$\displaystyle<K\mathbb{P}\left(\frac{1}{\sqrt{\underline{a}_{\eta}u\left(n\right)}}{\mathop{\sup}\limits_{|y-x_{0}|\leqslant\varepsilon}{\left|\sum_{i=1}^{\overline{a}_{\eta}u\left(n\right)}{\widetilde{g}_{y}\left(\mathcal{B}_{i}\right)}\right|}}>\frac{M_{\eta}}{K}\right).$
(80)

For an arbitrary set $T$, let $\ell^{+\infty}(T)$ be the space of all
uniformly bounded, real functions on $T$, equipped with the uniform norm. Weak
convergence to a tight process in this space is characterized by asymptotic
tightness plus convergence of marginals (see Chapter 1.5 in [34]).

The class of functions
$\mathcal{G}-F=\left\\{g_{y}(\cdot)-F\left(y\right)\right\\}_{y\in\mathbb{R}}$
is VC with constant envelope $2$, hence, by Lemma 5.2, the class of functions
$\widehat{\mathcal{G}-F}$ is also VC and has $2\ell_{C}$ as envelope.
$\mathbb{E}\ell_{C}^{2}(\mathcal{B}_{1})$ is finite (by (B4)), therefore, by
Theorem 2.5 in [25], $\widehat{\mathcal{G}-F}$ is Donsker. Then, the process
$\frac{1}{\sqrt{\underline{a}_{\eta}u\left(n\right)}}{\left|\sum_{i=1}^{\overline{a}_{\eta}u\left(n\right)}{\widetilde{g}_{y}\left(\mathcal{B}_{i}\right)}\right|}$
converges weakly in $\ell^{\infty}\left[\widehat{\mathcal{G}-F}\right]$ to a
tight process $Z$. The map $y\mapsto\left\|y\right\|_{\infty}$ from
$\ell^{\infty}\left[\widehat{\mathcal{G}-F}\right]$ to $\mathbb{R}$ is
continuous with respect to the supremum norm (cf. pp 278 of [33]), therefore,
$\frac{1}{\sqrt{\underline{a}_{\eta}u\left(n\right)}}{\mathop{\sup}\limits_{|y-x_{0}|\leqslant\varepsilon}{\left|\sum_{i=1}^{\overline{a}_{\eta}u\left(n\right)}{\widetilde{g}_{y}\left(\mathcal{B}_{i}\right)}\right|}}$
converges in distribution to
${\mathop{\sup}\limits_{|y-x_{0}|\leqslant\varepsilon}}{Z\left(y\right)}$,
hence, we can find $V_{\eta}$ and $N^{\prime}_{\eta}$ such that

$\mathbb{P}\left(\frac{1}{\sqrt{\underline{a}_{\eta}u\left(n\right)}}{\mathop{\sup}\limits_{|y-x_{0}|\leqslant\varepsilon}{\left|\sum_{i=1}^{\overline{a}_{\eta}u\left(n\right)}{\widetilde{g}_{y}\left(\mathcal{B}_{i}\right)}\right|}}>V_{\eta}\right)<\frac{\eta}{2K},\quad\forall
n>N^{\prime}_{\eta}.$ (81)

Choosing $M_{\eta}=KV_{\eta}$ in 81 and joining (80), (79) and (78), completes
the proof of (17).

Now we proceed to prove (18). Let $\eta$ be fixed, by (17) and Lemma 6.7, we
can find $\varepsilon^{\prime}$, $M_{\eta}^{\prime}$ and $N_{\eta}^{\prime}$
such that

$\displaystyle\mathbb{P}\left(\sqrt{T_{n}(C)}\mathop{\sup}\limits_{|y-x_{0}|\leqslant\varepsilon^{\prime}}{\left|{F_{n}(y)-F(y)}\right|>M_{\eta}^{\prime}}\right)$
$\displaystyle<\frac{\eta}{4}\quad\forall n\geqslant N_{\eta}^{\prime}$ (82)
$\displaystyle\mathbb{P}\left(\mathcal{D}_{n}\right)$ $\displaystyle\geqslant
1-\frac{\eta}{2}\quad\forall n\geqslant N_{\eta}^{\prime}$ (83)

where $\mathcal{D}_{n}=\left\\{\underline{a}_{\eta}u\left(n\right)\leqslant
T_{n}(C)\leqslant\overline{a}_{\eta}u\left(n\right)\right\\}$. Define the sets

$\displaystyle U_{n}$
$\displaystyle=\left\\{\sqrt{T_{n}(C)}\mathop{\sup}\limits_{|p-F(x_{0})|\leqslant\varepsilon}{\left|{F_{n}^{-1}(p)-{F^{-1}}(p)}\right|}>M_{\eta}\right\\},$
$\displaystyle U_{n}^{1}$ $\displaystyle=\left\\{\exists
p\in\left[F(x_{0})-\varepsilon,F(x_{0})+\varepsilon\right]:F_{n}^{-1}(p)-F^{-1}(p)>\frac{M_{\eta}}{\sqrt{T_{n}(C)}}\right\\},$
$\displaystyle U_{n}^{2}$ $\displaystyle=\left\\{\exists
p\in\left[F(x_{0})-\varepsilon,F(x_{0})+\varepsilon\right]:F^{-1}(p)-F_{n}^{-1}(p)>\frac{M_{\eta}}{\sqrt{T_{n}(C)}}\right\\}.$

where $\varepsilon$ and $M_{\eta}$ are constants that will be specified later.

On $U_{n}^{1}\cap\mathcal{D}_{n}$,
$F_{n}^{-1}\left(p\right)>\frac{M_{\eta}}{\sqrt{T_{n}(C)}}+F^{-1}\left(p\right)>\frac{M_{\eta}}{\sqrt{\overline{a}_{\eta}u\left(n\right)}}+F^{-1}\left(p\right)$,
hence,

$\displaystyle
F_{n}\left(\frac{M_{\eta}}{\sqrt{\overline{a}_{\eta}u\left(n\right)}}+F^{-1}\left(p\right)\right)$
$\displaystyle\leqslant F_{n}\left(F_{n}^{-1}\left(p\right)\right)\leqslant
p+\frac{1}{T_{n}(C)}$ $\displaystyle\leqslant
p+\frac{1}{\underline{a}_{\eta}u\left(n\right)}.$ (84)

Assumption (B2) indicates that $F$ has bounded derivative in $C$, take $K_{1}$
as the maximum value of this derivative in $C$, then, the Mean Value Theorem
implies that

$\displaystyle p$
$\displaystyle=F\left(F^{-1}\left(p\right)\right)=F\left(\frac{M_{\eta}}{\sqrt{\overline{a}_{\eta}u\left(n\right)}}+F^{-1}\left(p\right)\right)-\frac{F^{\prime}\left(\theta_{p}\right)M_{\eta}}{\sqrt{\overline{a}_{\eta}u\left(n\right)}}$
$\displaystyle\leqslant
F\left(\frac{M_{\eta}}{\sqrt{\overline{a}_{\eta}u\left(n\right)}}+F^{-1}\left(p\right)\right)-\frac{K_{1}M_{\eta}}{\sqrt{\overline{a}_{\eta}u\left(n\right)}}.$

After plugging this into (2) we get

$F\left(\frac{M_{\eta}}{\sqrt{\overline{a}_{\eta}u\left(n\right)}}+F^{-1}\left(p\right)\right)-F_{n}\left(\frac{M_{\eta}}{\sqrt{\overline{a}_{\eta}u\left(n\right)}}+F^{-1}\left(p\right)\right)\geqslant\frac{K_{1}M_{\eta}}{\sqrt{\overline{a}_{\eta}u\left(n\right)}}-\frac{1}{\underline{a}_{\eta}u\left(n\right)}.$

Because $u\left(n\right)\to+\infty$, we can find $N_{1}$ such that
$\sqrt{\frac{\overline{a}_{\eta}}{u\left(n\right)}}\frac{1}{\underline{a}_{\eta}K_{1}}<1$
for all $n\geqslant N_{1}$, taking $M_{\eta}$ bigger than
$\frac{M_{\eta}^{\prime}}{K_{1}}\sqrt{\frac{\overline{a}_{\eta}}{\underline{a}_{\eta}}}+1$
and using that $T_{n}(C)\leqslant\underline{a}_{\eta}u\left(n\right)$ on
$\mathcal{D}_{n}$, we obtain, for all $n\geqslant N_{1}$

$F\left(\frac{M_{\eta}}{\sqrt{\overline{a}_{\eta}u\left(n\right)}}+F^{-1}\left(p\right)\right)-F_{n}\left(\frac{M_{\eta}}{\sqrt{\overline{a}_{\eta}u\left(n\right)}}+F^{-1}\left(p\right)\right)>\frac{M_{\eta}^{\prime}}{\sqrt{T_{n}(C)}}.$
(85)

Let $N_{2,\eta}$ be such that
$\frac{M_{\eta}}{\sqrt{\overline{a}_{\eta}u\left(n\right)}}<\frac{\varepsilon^{\prime}}{2}$
for $n\geqslant N_{2,\eta}$. By the continuity of $F^{-1}$ in $F(x_{0})$ there
exists $\varepsilon>0$ such that
$\left|F^{-1}\left(p\right)-x_{0}\right|\leqslant\frac{\varepsilon^{\prime}}{2}$
for all $p$ in $\left[F(x_{0})-\varepsilon,F(x_{0})+\varepsilon\right]$,
therefore, the triangular inequality implies that
$\frac{M_{\eta}}{\sqrt{\overline{a}_{\eta}u\left(n\right)}}+F^{-1}\left(p\right)$
lies in the interval
$\left[x_{0}-\varepsilon^{\prime},x_{0}+\varepsilon^{\prime}\right]$ for all
$n\geqslant N_{\eta}=\text{max}\left(N_{1},N_{2,\eta}\right)$. This, alongside
(85), shows that for all $n\geqslant N_{\eta}$

$\displaystyle U_{n}^{1}\cap\mathcal{D}_{n}$
$\displaystyle\subseteq\left\\{\exists
y\in\left[x_{0}-\varepsilon^{\prime},x_{0}+\varepsilon^{\prime}\right]:F(y)-F_{n}(y)>\frac{M^{\prime}_{\eta}}{\sqrt{T_{n}(C)}}\right\\}$
$\displaystyle\subseteq\left\\{\sqrt{T_{n}(C)}\mathop{\sup}\limits_{|y-x_{0}|\leqslant\varepsilon^{\prime}}{\left|{F_{n}(y)-F(y)}\right|>M_{\eta}^{\prime}}\right\\}.$

By a similar argument, it can be shown that

$U_{n}^{2}\cap\mathcal{D}_{n}\subseteq\left\\{\exists
y\in\left[x_{0}-\varepsilon^{\prime},x_{0}+\varepsilon^{\prime}\right]:F_{n}(y)-F(y)>\frac{M^{\prime}_{\eta}}{\sqrt{T_{n}(C)}}\right\\}\quad\forall
n\geqslant N_{\eta}.$

Using (82) and $U_{n}=U_{n}^{1}\cup U_{n}^{2}$ we obtain that
$\mathbb{P}\left(U_{n}\cap\mathcal{D}_{n}\right)\leqslant\frac{\eta}{2}$ for
all $n\geqslant N_{\eta}$. Equation (18) now follows by (83).

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