[S-]SupplementMaterial[SupplementMaterial.pdf]

# Generalized regression operator estimation for continuous time functional
data processes with missing at random response

Mohamed Chaouch1 and Naâmane Laïb2
1 Department of Mathematics, Statistics, and Physics.
Qatar University, Qatar
e-mail<EMAIL_ADDRESS>
2 CY Cergy Paris Univeristé, Laboratoire AGM, UMR 8088 du CNRS
F-95000 Cergy, France.
e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

In this paper, we are interested in nonparametric kernel estimation of a
generalized regression function, including conditional cumulative distribution
and conditional quantile functions, based on an incomplete sample
$(X_{t},Y_{t},\zeta_{t})_{t\in\mathbb{R}^{+}}$ copies of a continuous-time
stationary ergodic process $(X,Y,\zeta)$. The predictor $X$ is valued in some
infinite-dimensional space, whereas the real-valued process $Y$ is observed
when $\zeta=1$ and missing whenever $\zeta=0$. Pointwise and uniform
consistency (with rates) of these estimators as well as a central limit
theorem are established. Conditional bias and asymptotic quadratic error are
also provided. Asymptotic and bootstrap-based confidence intervals for the
generalized regression function are also discussed. A first simulation study
is performed to compare the discrete-time to the continuous-time estimations.
A second simulation is also conducted to discuss the selection of the optimal
sampling mesh in the continuous-time case. Finally, it is worth noting that
our results are stated under ergodic assumption without assuming any classical
mixing conditions.

Keywords: Asymptotic quadratic error, continuous time ergodic processes,
confidence intervals, exchangeable bootstrap, functional data, generalized
regression, missing at random.

Subject Classifications: 60F10, 62G07, 62F05, 62H15.

##  1 Introduction

Let $({\cal E},d)$ be an infinite-dimensional space equipped with a semi-
metric $d(\cdot,\cdot)$. We consider some ${\cal
E}\times\mathbb{R}\times\\{0,1\\}$-valued stationary and ergodic continuous
time process $\left\\{Z_{t}=(X_{t},Y_{t},\zeta_{t}),\
{t\in\mathbb{R}^{+}}\right\\}$ defined on a probability space $(\Omega,{\cal
A},\mathbb{P})$ and observed at any time $t\in[0,T]$, where $\zeta$ is an
indicator process taking values zero or one at any instant $t.$ For any $y$ in
a compact set $S\subset\mathbb{R}$, let $\psi_{y}(\cdot)$ be a real valued
Borel function defined on $S\times\mathbb{R}$. The generalized regression
function of $\psi_{y}(Y)$ given $X=x$ is defined by
$m_{\psi}(x,y):=\mathbb{E}(\psi_{y}(Y_{t})|X_{t}=x)=\mathbb{E}(\psi(y,Y_{t})|X_{t}=x)$
which is supposed to exist for any $x\in{\cal E}$ and independent of $t$.

Here, the process $Z$ has the same distribution as $(X,Y,\zeta)$, where $Y$ is
a real integrable process and $X$ is a continuous time process valued in the
functional space ${\cal E}$. This means that, for any fixed time $t=t_{0}$,
$X_{t_{0}}\in{\cal E}$. Therefore, if, for instance, we take ${\cal
E}:=L^{2}[0,1]$ the space of square integrable functions defined on $[0,1]$,
then for any fixed $t_{0}$, the process $X_{0}:=\\{X_{t_{0}}(s):s\in[0,1]\\}$
describes a curve.

In this paper, we assume that the response variable $Y$ is subject to the
missing at random mechanism (MAR). This means that for an available observed
sample $(X_{t},Y_{t},\zeta_{t})_{0\leq t\leq T}$, $X_{t}$ is completely
observed, whereas $\zeta_{t}=1$ if $Y_{t}$ is observed at time $t$ and
$\zeta_{t}=0$ otherwise. The random variables $\zeta$ and $Y$ are supposed to
be conditionally independent given $X$, that is
$\mathbb{P}(\zeta=1|X=x,Y=y)=\mathbb{P}(\zeta=1|X=x):=p(x)$ almost surely
(a.s.). The MAR phenomena of the response variable may occur in several
situations. For instance, in survey sampling studies the non-response is an
increasingly common problem, where the missing response reaches rates of
$25\%$ to $30\%$ or even higher (see, e.g., Sikov (2018)). In such case the
missing data become a real source of bias in survey sampling estimation.
Another example where the response may be subject to the AMR phenomena is the
household electricity consumption monitoring. Indeed, the real time collection
of intraday electricity consumption is now possible after the deployment of
smart meters at the household level. The transmission of the information from
the smart meter towards the information system goes usually through WIFI or
optical fiber networks which are significantly dependent on the weather
conditions. Therefore, a response variable such as the daily total electricity
consumption might be subject to missing at random mechanism due to bad weather
conditions.

The statistical analysis involving missing data, based on an incomplete
discrete sample $(X_{i},Y_{i},\zeta_{i})_{1\leq i\leq n}$ where the data are
assumed independent and identically distributed (i.i.d), has been considered
by several authors, see for instance Ling et al. (2015), Cheng (1994), Little
and Rubin (2002), Nittner (2003), Tsiatis (2006), Liang et al. (2007),
Efromovich (2011), Chaouch et al. (2017), and Chaouch et al. (2009). However,
less attention has been given to the case when the covariate is infinite
dimensional and the response is missing at random. One can cite, for instance,
Ferraty et al. (2013), where the authors considered the estimation of the
regression function based on an i.i.d. random sample, and Ling et al. (2015)
extended their work to discrete time ergodic processes.

The literature on the estimation of the regression function based on a
completely observed sample $(X_{t},Y_{t})_{t\geq 0}$ copies of a strongly
mixing continuous time stationary process
$(X,Y)\in\mathbb{R}^{d}\times\mathbb{R}^{d^{\prime}}$ is very extensive. One
may refer to the monograph by Bosq (1998) and the references theirin. Some of
these results are extended by Didi and Louani (2014) and Bouzebda and Didi
(2017) to the case where the underlying real-valued process is stationary and
ergodic. Chaouch and Laïb (2019) have obtained an explicit upper bound of the
asymptotic mean square error (AMSE) of kernel regression estimator when the
data are sampled from a real-valued continuous time process with a missing at
random response. Some of these results are extended to the case where $X$ is
valued in an infinite-dimensional space and $Y$ is a real-valued completely
observed. Under the $\alpha$-mixing condition Maillot (2008) established the
convergence with rates of the regression operator, whereas Chesneau and
Maillot (2014) obtained a superoptimal mean square convergence rate of the MSE
of regression function for continuous functional time process with irregular
paths.

This paper aims to extend Maillot (2008) and Chesneau and Maillot (2014) work
at several levels. First, we suppose that the continuous time process
satisfies an ergodic assumption rather than an $\alpha$-mixing one. Therefore,
the dependence condition we consider is more general and involves several
processes which do not satisfy the mixing property. On the other hand, this
paper extends results established in Ling et al. (2015) in the context of
discrete time functional data processes to the continuous time framework.
Indeed, given $(X_{t},Y_{t},\zeta_{t})_{t\in\mathbb{R}^{+}}$ copies of a
continuous-time stationary ergodic process $(X,Y,\zeta)$, we estimate the
general operator $m_{\psi}(x,y)$ which including conditional distribution
function and conditional quantiles. It is worth noting that such extension is
not obvious since it requires an appropriate definition of $\sigma$-fields
adapted to continuous time context. Such adaptation is crucial when using
martingale difference tools to establish asymptotic properties of the
estimator. Second, the response variable considered here is affected by the
MAR mechanism and therefore is not completely observed as in Maillot (2008).
Moreover, in contrast to Maillot (2008) and Chesneau and Maillot (2014), we do
not limit our study to the mean square convergence but we provide a more
exhaustive inference on the regression operator estimator including pointwise
and uniform almost sure convergence rate, identification of the limiting
distribution of our estimator and provide two methods to build confidence
intervals (central limit theorem and the bootstrap procedure).

The rest of this paper is organised as follows. In Section 2 we present the
framework adapted to continuous time ergodic processes and introduce
assumptions needed for establishing asymptotic results. The main asymptotic
properties of the estimator are discussed in Section 3. An illustration of the
performance of the proposed estimator is discussed through simulated data in
Section 4. Section 5 discusses an application to conditional quantiles.
Finally technical proofs are given in Section 6.

##  2 Framework and assumptions

To define the framework of our study, we need to introduce some definitions.
Let $X=(X_{t})_{t\in[0,\infty)}$ be a continuous time process defined on a
measurable probability space $(\Omega,{\cal F},\mathbb{P})$ and observed at
any time $t\in[0,T]$.

###### Definition 2.1

Let $\tau$ be a nonnegative real number, a measurable set $A$ is
$\tau$-invariant if $T^{\tau}(A)=A$ for any $\tau$-shift transformation
$T^{\tau}$, i.e. $(T^{\tau}(x))_{\lambda}=x_{s+\lambda}$. The process $X$ is
said to be $\tau$-ergodic if for any $\tau$-invariant set $A$,
$\mathbb{P}(A)=\mathbb{P}^{2}(A)$. It has the ergodic property if there exists
an invariant distribution $F(\cdot)$, such that for any measurable function
$h(\cdot)$ and a random variable $\xi$ distributed as $F(\cdot)$ satisfying
$\mathbb{E}(|h(\xi)|)<\infty$, we have

$\displaystyle\lim_{T\to\infty}\frac{1}{T}\int_{-\infty}^{\infty}h(X_{t})dt=\mathbb{E}(h(\xi))\quad\mbox{almost
surely}\ \mbox{(a.s.).}$ (2.1)

From now on, we will be working on the filtered probability space
$\left(\Omega,{\cal F},\\{{\cal F}_{t},t\geq 0\\},\mathbb{P}\right)$. For a
positive real number $\delta$ such that $n=\frac{T}{\delta}\in\mathbb{N}$ and
$j\in\mathbb{N}\cap[1,n]$, consider the $\delta$\- partition
$(T_{j}=j\delta)_{1\leq j\leq n}$ of the interval $[0,T]$. Furthermore, for
$t>0$ and $1\leq j\leq n$, we define the following $\sigma$-fields:

$\displaystyle\mathcal{F}_{t-\delta}$ $\displaystyle:=$
$\displaystyle\sigma((X_{s},Y_{s},\zeta_{s}):0\leq
s<t-\delta),\quad\mathcal{F}_{j}=\sigma((X_{s},Y_{s},\zeta_{s}),0\leq
s<T_{j}),$ $\displaystyle{\cal S}^{\ell}_{t,\delta}$ $\displaystyle:=$
$\displaystyle\sigma((X_{s},Y_{s},\zeta_{s});(X_{r},(1-\ell)Y_{r}):0\leq
s<t,\quad t\leq r\leq t+\delta,\;\ell\in\\{0,1\\}).$

Whenever $s<0$, $\mathcal{F}_{s}$ stands for the trivial $\sigma$-field. Here,
$\zeta_{s}$ is a standard Bernoulli process (see Section below). Notice that,
for any $\delta>0$ and $t>0$, ${\cal S}^{0}_{t,\delta}\subset{\cal
S}^{1}_{t,\delta}$, ${\cal F}_{t-\delta}\subset{\cal
S}^{\ell}_{t-\delta,\delta}\subset{\cal S}^{\ell}_{t,\delta},$ $\text{and for
any}\ j\geq 2\ \text{such that}\ T_{j-1}\leq t\leq T_{j},\quad{\cal
F}_{j-2}\subseteq{\cal F}_{t-\delta}\subset{\cal S}^{\ell}_{t,\delta}$.

The estimator $\widehat{m}_{\psi,T}(x,y)$ adapted to MAR response may be
defined, for any $\zeta\neq 0$ and $\int_{0}^{T}\zeta_{t}\Delta_{t}(x)dt\neq
0$, by

$\displaystyle\widehat{m}_{\psi,T}(x,y)=\frac{\int_{0}^{T}\zeta_{t}\psi_{y}(Y_{t})\Delta_{t}(x)dt}{\int_{0}^{T}\zeta_{t}\Delta_{t}(x)dt},$
(2.2)

where $\Delta_{t}(x)=K\left(\frac{d(x,X_{t})}{h_{T}}\right)$, $K(\cdot)$ is a
kernel density function, $h_{T}$ is the smoothing parameter tending to zero as
$T$ goes to infinity. Let $Z_{1}(x):=\int_{0}^{\delta}\Delta_{t}(x)dt$ and
define the conditional bias as

$\displaystyle
B_{T}(x,y):=\frac{\overline{m}_{\psi,T,2}(x,y)}{\overline{m}_{\psi,T,1}(x,y)}-m_{\psi}(x,y):=C_{T}(x,y)-m_{\psi}(x,y).$
(2.3)

where, for $j=1,2$,

$\displaystyle\overline{m}_{\psi,T,i}(x,y):=\frac{1}{n\mathbb{E}(Z_{1}(x))}\int_{0}^{T}\mathbb{E}\left\\{\zeta_{t}(\psi_{y}(Y_{t}))^{i-1}\Delta_{t}(x)|\mathcal{F}_{t-\delta}\right\\}dt.$
(2.4)

Let us now introduce the assumptions under which we establish our asymptotic
results.

  * (A1)

(Assumptions on the kernel function). Let $K$ is a nonnegative bounded kernel
of class $\mathcal{C}^{1}$ over its support $[0,1]$ such that $K(1)>0$. The
derivative $K^{\prime}$ exists on $[0,1]$ and satisfies the condition
$K^{\prime}(v)<0,$ for all $v\in[0,1]$ and
$|\int_{0}^{1}(K^{j})^{\prime}(v)dv|<\infty$ for $j=1,2.$

  * (A2)

(Assumptions related to the continuous time functional ergodic processes)

$(i_{0})$ Let $\alpha_{0}$ be a nonnegative real number and $x\in\cal{E}$.
Suppose, for any $0\leq s<t\leq T$ such that $t-s\leq\alpha_{0}$, there exists
a nonnegative continuous random function $f_{t,s}(x)$ a.s. bounded by a
deterministic function $b_{t,\alpha_{0}}(x)$.

Moreover, let $g_{t,s,x}(\cdot)$ be a random function defined on $\mathbb{R}$,
$f(x)$ is a deterministic nonnegative bounded function and $\phi(\cdot)$ a
nonnegative real function tending to zero (as its argument tends to 0), and
assume that.

    * (i)

$F_{x}(u):=\mathbb{P}\left(d(x,X_{t})\leq u\right)=\phi(u)f(x)+o(\phi(u))$ as
$u\rightarrow 0.$

    * (ii)

For any $0\leq s\leq t$, $F_{x}^{{\cal F}_{s}}(u):=\mathbb{P}^{{\cal
F}_{s}}(d(x,X_{t})\leq u)=\mathbb{P}(d(x,X_{t})\leq
u|\mathcal{F}_{s})=\phi(u)f_{t,s}(x)+g_{t,s,x}(u)$ with
$g_{t,s,x}(u)=o_{a.s.}(\phi(u))$ as $u\rightarrow 0,$ $g_{t,s,x}(u)/\phi(u)$
a.s. bounded and

$T^{-1}\int_{0}^{T}g_{t,t-\delta,x}(u)dt=o_{a.s.}(\phi(u))$ as
$T\rightarrow\infty$ and $u\rightarrow 0.$

    * (iii)

For any $x\in\cal{E}$:
$\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}f_{t,t-\delta}(x)dt=f(x)$,
a.s.

    * (iv)

There exists a nondecreasing bounded function $\tau_{0}$ such that, uniformly
in $u\in[0,1]$,

$\frac{\phi(hu)}{\phi(h)}=\tau_{0}(u)+o(1)\quad\mbox{as}\ h\downarrow
0\quad\mbox{ and}\ \ \int_{0}^{1}(K(v))^{\prime}\tau_{0}(v)dv<\infty.$

    * (v)

$T^{-1}\int_{0}^{T}b_{t,\alpha_{0}}(x)dt\rightarrow D_{\alpha_{0}}(x)$ as
$T\rightarrow\infty$ with $0<D_{\alpha_{0}}(x)<\infty$.

  * (A3)

(Local smoothness and continuity conditions)

Suppose, for any $(y,t)\in S\times[0,T]$ and $r$ such that and $t\leq r\leq
t+\delta$:

    * $(i)$

$\mathbb{E}\left(\psi_{y}(Y_{r})|\mathcal{S}_{t,\delta}^{1}\right)=\mathbb{E}\left(\psi_{y}(Y_{r})|X_{r}\right)=m(X_{r},y)$
a.s.

    * $(ii)$

$\exists\beta>0$ and a constant $c>0$ such that, for any
$(x^{\prime},x^{\prime\prime})\in{\cal E}^{2}$,

$|m(x^{\prime},y)-m_{y}(x^{\prime\prime},y)|\leq
cd^{\beta}(x^{\prime},x^{\prime\prime}).$

    * $(iii)$

For any $\kappa_{1}\geq 2$,
$\mathbb{E}\left(|\psi_{y}(Y_{r})|^{\kappa_{1}}|\mathcal{S}_{t,\delta}^{1}\right)=\mathbb{E}\left(|\psi_{y}(Y_{r})|^{\kappa_{1}}|X_{r}\right)$
a.s.

The functions
$W_{\kappa_{1}}(x,y):=\mathbb{E}\left(|\psi_{y}(Y)|^{\kappa_{1}}|X=x\right)$
and
$\overline{W}_{\kappa_{1}}(x,y):=\mathbb{E}\left(|\psi_{y}(Y)-m_{\psi}(x,y)|^{\kappa_{1}}|X=x\right)$
are continuous in the neighbourhood of $x$ and $\sup_{x\in{\cal C},y\in
S}|W_{\kappa_{1}}(x,y)|<\infty$ a.s.

    * $(iv)$

$\mathbb{E}(\zeta_{t}|{\cal
S}_{t,\delta}^{0})=\mathbb{E}(\zeta_{t}|X_{t})=p(X_{t})$ a.s.

For any $x\in\cal{E}$, $\sup_{\\{x^{\prime}:d(x,x^{\prime})\leq
u\\}}|p(x^{\prime})-p(x)|=o(1)$ a.s. as $u\rightarrow 0.$

    * $(iv^{\prime})$

For any $x\in{\cal E}$, $\kappa_{1}\geq 2$,
$\mathbb{E}(|\zeta_{t}|^{\kappa_{1}}|X_{t}=x):=U_{\kappa}(x)$, and

$\sup_{\\{x^{\prime}:d(x,x^{\prime})\leq
u\\}}|U_{\kappa}(x^{\prime})-U_{\kappa}(x)|=o(1)$ a.s. as $u\rightarrow 0.$

For $j=1,2$, define the following moments, which are independent of $x\in{\cal
E}$

$\displaystyle
M_{j}=K^{(j)}(1)-\int_{0}^{1}\left(K^{j}\right)^{\prime}(u)\tau_{0}(u)du.$
(2.5)

###  2.1  Comments on the assumptions

Condition (A1) is related to the choice of the kernel $K$, which is very usual
in nonparametric functional estimation. Condition (A2)(i)-(ii) reflects the
ergodicity property assumed on the continuous time functional process. It
plays an important role in studying the asymptotic properties of the
estimator. The functions $f_{t,s}$ and $f$ play the same role as the
conditional and unconditional densities in finite dimensional case, whereas
$\phi(u)$ characterizes the impact of the radius $u$ on the small ball
probability as $u$ goes to $0$. Several examples to satisfy these conditions
are given in Laïb and Louani (2010) for discrete time functional data process.

Some examples are also given to satisfy this condition in Didi and Louani
(2014) for the case where the observations $(X_{t},Y_{t})$ are sampled from an
ergodic continuous times process taking values in
$\mathbb{R}^{d}\times\mathbb{R}$ space.

Condition (A2)-(iii) involves the ergodic nature of the process where the
random function $f_{t,t-\delta}$ belongs to the space of continuous functions
${\cal C}^{0}$. Approaching the integral $\int_{0}^{T}f_{t,t-\delta}(x)dt$ by
its Riemann’s sum: $T^{-1}\int_{0}^{T}f_{t,t-\delta}(x)dt\simeq
n^{-1}\sum_{j=1}^{n}f_{j\delta,(j-1)\delta}(x)$, it is easy to prove that the
sequence $(f_{j\delta,(j-1)\delta}(x))_{j\geq 1}$ is stationary and ergodic
(see, Didi and Louani (2014)). (A2)-(iv) is an usual condition when dealing
with functional data. Assumption (A2)-(v) is a consequence of ergodic
assumption. Condition (A3) is a Markov-type condition and characterizes the
conditional moments of $\psi_{y}(Y)$. It is satisfied when considering, for
instance, the model $\psi_{y}(Y_{t})=m_{\psi}(X_{t})+\epsilon_{t}$ where
$(\epsilon_{t})$ is a square integrable process independent of $(X_{t})$ for
any $t\geq 0$.

(A3)(iv) assumes the continuity of the conditional probability of observing a
missing response. The moments $M_{j}$ are linked to the small probability
function $\tau_{0}$. One can refer to Ferraty et al. (2007) for a discussion
on the choice of $\tau_{0}$, the Kernel $K$ and the positivity of $M_{j}$.

##  3 Main results

In this section we investigate several asymptotic properties of the continuous
time generalized regression estimator. Some particular cases, related to
specific choices of the function $\psi_{y}(\cdot)$, including the conditional
cumulative distribution function and the conditional quantiles will also be
discussed.

###  3.1 Almost sure consistency rates

####  3.1.1 Pointwise consistency

The following theorem establishes an almost sure pointwise consistency rate of
$\widehat{m}_{\psi,T}(x,y).$

###### Theorem 3.1

(Pointwise consistency) Assume that (A1)-(A3) hold true and the following
conditions are satisfied

$\displaystyle\lim_{T\rightarrow\infty}T\phi(h_{T})=\infty\quad\quad\mbox{and}\quad\quad\lim_{T\rightarrow\infty}\frac{\log
T}{T\phi(h_{T})}=0.$ (3.1)

Then, we have for $T$ sufficiently large that

$\displaystyle\widehat{m}_{\psi,T}(x,y)-m_{\psi}(x,y)=\mathcal{O}(h_{T}^{\beta})+\mathcal{O}\left(\sqrt{\frac{\log
T}{T\phi(h_{T})}}\right).$ (3.2)

###### Remark 3.2

(i) Theorem 3.1 generalizes Theorem 1 of Laïb and Louani (2011) established in
the context of discrete time stationary ergodic process and Theorem 3.4 of
Ferraty et al. (2005) stated under a mixing assumption with completely
observed response and the support of $y$ is reduced to one point.

(iv) The function $\phi(h_{T})$ can decrease to zero at an exponential rate,
whenever $h_{T}$ goes to zero, therefore $h_{T}$ should be chosen decreases to
zero at a logarithmic rate.

####  3.1.2 Uniform consistency

To establish the uniform consistency with rate of the regression operator, we
need additional assumptions that allow to express the uniform convergence rate
as a function of the entropy number. Let ${\cal C}$ and $S$ be compact sets in
${\cal E}$ and $\mathbb{R}$, respectively. Consider, for any $\epsilon>0$, let
the $\epsilon$-covering number of the compact set ${\cal C}$, say,
$N_{\epsilon}:=\mathcal{N}(\epsilon,\mathcal{C},d)$, which measures how full
is the class $\mathcal{C}$, and defined as:

$\displaystyle N_{\epsilon}$ $\displaystyle:=$
$\displaystyle\min\left\\{n:\mbox{there
exist}\;c_{1},\dots,c_{n}\in\mathcal{C}\;\mbox{such that}\;\forall
x\in\mathcal{C}\;\mbox{we can find }\right.$
$\displaystyle\left.\quad\quad\quad\quad 1\leq i\leq n\;\mbox{such
that}\;d(x,c_{i})<\epsilon\right\\}.$

The finite set of points $c_{1},c_{2},\ldots,c_{N_{\epsilon}}$ is called an
$\epsilon$ -net of ${\cal C}$ if ${\cal
C}\subset\cup_{k=1}^{N_{\epsilon}}B(c_{k},\epsilon)$, where
$B(c_{k},\epsilon)$ is the ball, with respect to the topology induced by the
semi-metric $d(\cdot,\cdot)$, centred at $c_{k}$ and of radius $\epsilon$. The
quantity $\varphi_{\cal C}(\epsilon)=\log(N_{\epsilon})$ is called the
Kolmogorov’s $\epsilon$-entropy of the set ${\cal C}$ that may be seen as a
tool to measure the complexity of the subset ${\cal C}$, in the sense that
high entropy means that a large amount of information is needed to describe an
element of ${\cal C}$ with an accuracy $\epsilon$. Several examples of
$\varphi_{\cal C}(\epsilon)$ covering special cases of functional process are
given in Ferraty et al. (2010) and Laïb and Louani (2011).

  * (U0)

Assume that (A2) holds uniformly, in the following sense:

(i)

(A2)(i) and (A2)(ii) hold true with the remaining term $o(\phi(u))$ is uniform
in $x$,

(ii)

For any $x\in{\cal C}$, $\lim_{T\rightarrow\infty}\sup_{x\in{\cal
C}}\big{|}\frac{1}{T}\int_{0}^{T}f_{t,t-\delta}(x)dt-f(x)\big{|}=0\ a.s.$

(iii)

$T^{-1}\int_{0}^{T}b_{t,\alpha_{0}}(x)dt\rightarrow D_{\alpha_{0}}(x)$ as
$T\rightarrow\infty$ with $0<\sup_{x\in{\cal C}}D_{\alpha_{0}}(x)<\infty$.

(iv))

$b_{0}<\inf_{x\in\mathcal{C}}f(x)\leq\sup_{x\in\mathcal{C}}f(x)\ <\infty$ for
some nonegative real number $b_{0}$.

(v)

$\inf_{x\in C}p(x)>b_{1}$ for some nonegative real number $b_{1}$.

  * (U1)

The kernel function $K$ satisfies the following conditions:

    * (i)

K is a Hölder function of order one with a constant $a_{K}$,

    * (ii)

There exist two constants $a_{2}$ and $a_{3}$ such that $0<a_{2}\leq K(x)\leq
a_{3}<\infty$, for any $x\in\cal{C}.$

  * (U2)

For $1\leq\ell\leq 2$, the sequence of random variables
$(\psi^{\ell}_{y}(Y_{t}))_{t}$ is ergodic and
$\mathbb{E}\left(|\psi^{\ell}_{y}(Y_{0})|\right)<\infty.$

  * (U3)

There exist $c_{\psi}>0$ and nonnegative real number $\gamma$ such that for
any $y\in S$

$\sup_{y^{\prime}\in[y-u,y+u]\cap
S}|\psi_{y}(Y_{t})-\psi_{y^{\prime}}(Y_{t})|\leq c_{\psi}u^{\gamma}.$

  * (U4)

Let $T_{n}$ be the integer pat of $T$ and suppose for $T$ large enough that

$\frac{(\log
T)^{2}}{T\phi(h_{T})}<\varphi_{C}(\epsilon_{n})<\frac{T\phi(h_{T})}{\log
T}\qquad\text{with}\quad\epsilon_{n}=\frac{\log T_{n}}{T_{n}}.$

Conditions (U0) are standard in this context in order to get uniform
consistency rate. Condition (U1) is usually used when we deal with
nonparametric estimation for functional data, whereas (U2) requires to the
existence of up to the order two moments. Hypothesis (U3) is a regularity
condition upon the function $\psi_{y}(\cdot)$ which is necessary to obtain a
uniform result over the compact $S$. Hypothesis (U4) allows to cover the
subset ${\cal C}$ with a finite number of balls and to express the convergence
rate in terms of the Kolmogorov’s entropy of this subset. Similar condition
has been used in Ferraty et al. (2010), where the authors have pointed out
that for a radius not too large, one requires the quantity $\varphi_{C}(\log
T_{n}/T_{n})$ is not too small and not too large. This condition seems to
satisfy this exigence, since it implies that $\frac{\varphi_{C}(\log
T_{n}/T_{n})}{T\phi(h_{T})}$ goes to $0$ for sufficiently large $T$, in
addition some examples given in Ferraty et al. (2010) and Laïb and Louani
(2011) satisfy (U4).

Theorem 3.3 states uniform consistency rate of the kernel regression
estimator. It generalizes Theorems 2 in Laïb and Louani (2011) established in
the context of discrete time stationary ergodic process with completely
observed response.

###### Theorem 3.3

(Uniform consistency). Assume (A1), (U0)-(U4), (A3) hold true. Moreover,
suppose Conditions (3.1) are satisfied and

$\displaystyle\sum_{n\geq 1}n^{\gamma}\exp\\{(1-\eta)\varphi_{\cal
C}(\frac{\log n}{n})\\}<\infty\quad\text{for some}\ \eta>0\ \text{where}\
\gamma\ \text{is as in }\ (U3).$ (3.3)

Then, we have

$\displaystyle\sup_{y\in
S}\sup_{x\in\mathcal{C}}|\widehat{m}_{\psi,T}(x,y)-m_{\psi}(x,y)|=\mathcal{O}_{a.s.}\left(h_{T}^{\beta}\right)+\mathcal{O}_{a.s.}\left(\sqrt{\frac{\varphi_{\cal
C}(\epsilon_{T})}{T\phi(h_{T})}}\right)\quad\text{as}\quad T\to\infty.$ (3.4)

###  3.2 Asymptotic conditional bias and risk evaluation

Before evaluating the conditional bias, let us introduce some additional
notations. Consider for $i=1,2$, the following assumptions:

(BC1) Let $d_{t}(x)=d(X_{t},x)$ and suppose, for any $t\geq 0$, that

$\mathbb{E}\left[m_{\psi}(X_{t},y)-m_{\psi}(x,y)\ |d_{t}(x),\ {\cal
F}_{t-\delta}\right]=\mathbb{E}\left[m_{\psi}(X_{t},y)-m_{\psi}(x,y)\
|d_{t}(x)\right]:=\Psi_{y}(d_{t}(x)),$

the function $\Psi_{y}$ is differentiable at $0$ and satisfies $\Psi_{y}(0)=0$
and $\Psi_{y}^{\prime}(0)\neq 0$ for any $y\in\mathbb{R}$. This condition was
introduced in Ferraty et al. (2007) and used by Laïb and Louani (2010) to
evaluate the conditional bais. The introduction of $\psi_{y}(\cdot)$ allows to
make an integration with respect to the real random variable $d_{t}(x)$ rather
than with respect to the couple of random variables $(d_{t}(x),X_{t})$, where
$X_{t}$ being functional continuous random variable.

The following Proposition gives an asymptotic expression of the conditional
bias term, which generalizes Proposition 1 of Laïb and Louani (2010) for
discrete time estimator to our setting. Its proof is similar and therefore is
omitted.

######  Proposition 3.4 (Conditional Bias)

Under assumptions (A1)-(A3), (BC1) and conditions (3.1), we have

$\displaystyle B_{T}(x,y)$ $\displaystyle=$
$\displaystyle\frac{h_{T}\Psi_{y}^{\prime}(0)}{M_{1}}\left[K(1)-\int_{0}^{1}(sK(s))^{\prime}\tau_{0}(s)ds+o_{a.s.}(1)\right]+\mathcal{O}_{a.s.}\left(h_{T}\sqrt{\frac{\log
T}{T\phi(h_{T})}}\right).$

The next result gives an explicit expression of the asymptotic quadratic risk
of the estimator $\widehat{m}_{\psi,T}(x,y)$.

######  Theorem 3.5 (Quadratic risk )

Suppose that assumptions (A1)-(A3) and condition (2.1) hold true. Then, we
have, for a fixed $(x,y)\in{\cal E}\times\mathbb{R}$, whenever $p(x)>0$ and
$f(x)>0$, that

$\displaystyle\mbox{{MSE}}(x,y):=\mathbb{E}\left[\left(\widehat{m}_{\psi,T}(x,y)-m_{\psi}(x,y)\right)^{2}\right]=A_{1}h^{2}_{T}\left[A_{1}+\mathcal{O}_{a.s.}\left(\sqrt{\frac{\log
T}{T\phi(h_{T})}}\right)\right]+\frac{A_{2}(x,y)}{T\phi(h_{T})},$
$A_{1}=\frac{\Psi_{y}^{\prime}(0)}{M_{1}}\left[K(1)-\int_{0}^{1}(sK(s))^{\prime}\tau_{0}(s)ds+o(1)\right]\quad\text{and}\
A_{2}(x,y)=\frac{4\left(W_{2}(x,y)+(m_{\psi}(x,y))^{2}\right)M_{2}}{p(x)M_{1}^{2}f(x)}.$

###### Remark 3.6

. (i) Notice that for sufficiently large $T$, the expression of MSE becomes
$A_{1}^{2}h^{2}_{T}+\frac{A_{2}(x,y)}{T\phi(h_{T})}$. This result generalizes
the one in Chaouch and Laïb (2019) established in the framework of real-valued
continuous time processes where the bias term was of order $h_{T}^{2}$. Notice
however that the bias term obtained in Chaouch and Laïb (2019) is of order
$h_{T}^{2}$ which is smaller than $h_{T}$ given in Proposition 3.4. This
increase in the bias term is because of the infinite dimensional
characteristic of the functional space.

(ii) The mean squared error can be used as a theoretical guidance to select
the “optimal” bandwidth by minimizing the quantity
$A_{1}^{2}h^{2}_{T}+\frac{A_{2}(x,y)}{T\phi(h_{T})}$ with respect to $h_{T}$.
The terms $A_{1}$ and $A_{2}(x,y)$ have unknown explicit from; and should be
replaced by their empirical consistent estimators: $\Psi_{y,T}^{\prime}(0),\
(M_{j,T})_{j=1,2},\ \tau_{0,T},\ p_{T},W_{2,T},\ f_{T}.$ Notice that,
$\Psi_{y}^{\prime}(0)$ may be viewed as real regression function with response
variable $m_{\psi}(X,y)-m_{\psi}(x,y)$ and covariable $d(X,x)$, it may be then
estimated by a kernel regression estimate $\Psi^{\prime}_{y,T}(0)$ by
replacing $m_{\psi}(x,y)$ by its estimator $\widehat{m}_{\psi,T}(X,y)$.

###  3.3 Asymptotic normality

###### Theorem 3.7

Assume conditions (A1)-(A3) are fulfilled and

$\displaystyle\lim_{T\rightarrow\infty}T\phi(h_{T})=+\infty,\
h_{T}^{\beta}\sqrt{T\phi(h_{T})}=o(1)\quad\text{and}\quad h_{T}^{\beta}\log
T^{1/2}=o(1)\quad\text{as}\ T\rightarrow\infty.$ (3.5)

Then, we have, for any $(x,y)\in{\cal E}\times S$ such that $f(x)>0$

$\sqrt{T\phi(h_{T})}(\widehat{m}_{\psi,T}(x,y)-m_{\psi}(x,y))\quad\xrightarrow{d}\quad{\cal
N}(0,\sigma^{2}(x,y)),$

where

$\displaystyle\sigma^{2}(x,y)\leq\frac{1}{f(x)}\frac{M_{2}}{M_{1}^{2}p(x)}\overline{W}_{2}(x,y):=\frac{1}{f(x)}\widetilde{V}(x,y)\quad\text{as}\quad
T\longrightarrow\infty,$ (3.6)

and

$\overline{W}_{2}(x,y)=\mathbb{E}\left[\left(\psi_{y}(Y)-m_{\psi}(x,y)\right)^{2}\
|X=x\right].$

Note that, the statement (3.6) gives only an upper bound of the asymptotic
variance $\sigma^{2}(x,y)$. The following proposition gives an estimate of
$\widetilde{V}(x,y)$ that will be needed to construct confidence bounds for
the unknown operator $m(x,y)$.

###### Proposition 3.8

Suppose conditions of Theorem 3.7 hold and $\sigma^{2}(x,y)>0$, then

$\widetilde{V}_{T}(x,y):=\frac{\sqrt{M_{T,2}}}{M_{1,T}}\sqrt{\frac{\overline{W}_{2,T}(x,y)}{TF_{x,T}(h_{T})p_{T}(x)}},$

is an asymptotically consistent estimator for $\widetilde{V}(x,y)$. The
quantities $M_{1,T}$, $M_{2,T}$, $\overline{W}_{2,T}$, $p_{T}(x)$ and
$F_{x,T}$ are empirical versions of $M_{2}$, $M_{1}$, $\overline{W}_{2}$,
$p(x)$ and $F_{x}$ respectively.

The estimators $M_{1,T}$ and $M_{2,T}$ depend on the unknown quantity
$\tau_{0}$ given in (A2)(iv) that may be estimated by:

$\tau_{0,T}(u)=\frac{F_{x,T}(uh_{T})}{F_{x,T}(h_{T})}\quad\text{with}\quad
F_{x,T}(u)=\frac{1}{T}\int_{0}^{T}1\kern-1.99997pt{\rm
l}_{\left\\{d(x,X_{t})\leq u\right\\}}dt,$

whereas $\overline{W}_{2,T}$ can be deduced from a nonparametric estimator for
the conditional variance function $\sigma^{2}_{T}$ (see Laïb and Louani
(2010)).

###  3.4 Continuous time confidence intervals

####  3.4.1 Asymptotic confidence intervals

Using the non-decreasing property of the cumulative standard Gaussian
distribution function and the estimator $\widetilde{V}_{T}(x,y)$ we can then,
with the help of Proposition 3.8 and Theorem 3.7, constructed confidence bands
for the regression function $m_{\psi}(x,y)$, which are similar to the those
obtained in the discrete time case. This is the subject of the following
corollary.

###### Corollary 3.9

Assume conditions of Theorem (3.7) are fulfilled and conditions in (3.5) are
replaced by

$\displaystyle\lim_{T\rightarrow\infty}TF_{x,T}(h_{T})=+\infty\quad\text{and}\quad\lim_{T\rightarrow\infty}h_{T}^{\beta}\sqrt{TF_{x,T}(h_{T})}=0.$
(3.7)

Then, for any $0<\alpha<1$, the $(1-\alpha)100\%$ confidence bands for
$m_{\psi}(x,y)$ are given by

$\displaystyle\widehat{m}_{\psi,T}(x,y)\pm
c_{1-\alpha/2}\frac{M_{T,1}}{\sqrt{M_{T,2}}}\sqrt{\frac{\overline{W}_{T,2}(x,y)}{TF_{x,T}(h_{T})p_{T}(x)}},\quad\text{as}\
T\to\infty,$ (3.8)

where $c_{1-\alpha/2}$ is the quantile of standard Gaussian distribution.

####  3.4.2 Exchangeable bootstrap-based confidence intervals

In a variety of statistical problems, the bootstrap provides a simple method
for circumventing technical difficulties due to the intractable distribution
theory (this is the case in Theorem 3.7, where the limiting law depends
crucially on the unknown variance). Resampling techniques become a powerful
tool, e.g., for setting confidence bands, that we will illustrate in the
sequel. The key idea behind the bootstrap is that if a sample is
representative of the underlying population, then one can make inferences
about the population characteristics by resampling from the current sample. To
apply this approach for constructing confidence bands we have to define a
bootstrap version that has the same asymptotic distribution as
$\sqrt{T\phi(h_{T})}(\widehat{m}_{\psi,T}(x,y)-m_{\psi}(x,y))$. Inspection of
the proof of Theorem 3.7 shows that the principal term leading to establish
the asymptotic normality is given by
$\frac{\sqrt{T\phi(h_{T})}Q_{T}(x,y)}{\widehat{m}_{\psi,T,1}(x,y)}$, where
$Q_{T}$ is defined in (6.1). Therefore, a bootstrap version for this quantity
will be sufficient for estimating the true quantile $c_{1-\alpha/2}$ for a
given $0<\alpha<1$. This may be done by combining Theorem 3.7, Theorem 2.1. of
Pauly (2011) for martingale difference arrays sequences and Theorem 3 of
Bouzebda et al. (2016). In this respect, define

$\displaystyle S_{T}(x,y):=\sqrt{T\phi(h_{T})}Q_{T}(x,y)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\xi_{T,i}(x,y),$

where

$\xi_{T,i}(x,y)=\eta_{T,i}(x,y)-\mathbb{E}\left[\eta_{T,i}(x,y)\ \Big{|}{\cal
F}_{t-\delta}\right],$

and

$\displaystyle\eta_{T,i}(x,y)$ $\displaystyle=$
$\displaystyle\frac{1}{\mathbb{E}Z_{1}}\sqrt{\frac{\phi_{T}(h)}{n}}\int_{T_{i-1}}^{T_{i}}\zeta_{t}\Delta_{t}(x)\left[\psi_{y}(Y_{t})-m_{\psi}(x,y)\right]dt.$
(3.9)

For every fixed $n\geq 1$, the sequence $(\xi_{T,i})_{1\leq i\leq n}$ is a
centred martingale difference array with respect ${\cal F}_{i-1}$(see the
proof of Lemma 6.7).

Define the resampling version of the quantity $S_{T}(x,y)$ as

$\displaystyle S^{\star}_{T}(x,y)$ $\displaystyle=$
$\displaystyle\sqrt{n}\sum_{i=1}^{n}W_{T,i}\xi_{T,i}(x,y),$ (3.10)

where $W_{T,i}$ are weight functions satisfying conditions $W.1-W.4$ in
Bouzebda et al. (2016). The following proposition, which is a consequence of
Theorem 3.7, Lemma 6.4 and the Slutsky’s lemma, states the asymptotic
normality of the statistic $S^{\star}_{T}$. Its proof may be obtained
similarly to that of Theorem 3 of Bouzebda et al. (2016).

###### Proposition 3.10

Assume that the above conditions related to bootstrap weights hold and the
sequences of random variables $\left\\{W_{T,i}:\ 1\leq i\leq n\right\\}$ and
$\left\\{\xi_{T,i}:\ 1\leq i\leq n\right\\}$ are independent. Then under the
assumptions of Theorem 3.7 and for any $x\in\cal E$ such that $f(x)>0$, we
have

$\displaystyle S^{\star}_{T}(x,y)\quad\xrightarrow{d}\quad{\cal
N}(0,\sigma^{2}(x,y))\ \text{as}\quad T\to\infty,\ where\ \sigma^{2}(x,y)\
\text{ is as in Theorem}.\ \ref{AsymNormality}$ (3.11)

###### Remark 3.11

. Observe that $S^{\star}_{T}$ depends on the unknown function
$m_{\psi}(x,y)$. Substitute $m_{\psi}(x,y)$ by $\widehat{m}_{\psi,T}(x,y)$ and
$S^{\star}_{T}$ by its estimate version $\widehat{S^{\star}_{T}}$, it follows
from Lemma 6.1 and Theorem 3.1 that
$\widehat{S^{\star}_{T}}-S^{\star}_{T}=o(1)$ as $T$ goes to infinity.
Therefore the statement (3.11) still holds if we replace $S^{\star}_{T}$ by
$\widehat{S^{\star}_{T}}$.

Proposition 3.10 turns out to be useful for obtaining confidence bands for the
regression function $m_{\psi}(x,y)$. Indeed, let $0<\alpha<1$ be given, then,
under conditions of Theorem 3.7, we have

$\displaystyle\lim_{T\rightarrow\infty}\mathbb{P}\left(\widehat{m}_{\psi,T}(x,y)-c_{1-\alpha/2}(T\phi(h_{T}))^{-1/2}\leq
m_{\psi}(x,y)\leq\widehat{m}_{\psi,T}(x,y)+c_{1-\alpha/2}(T\phi(h_{T}))^{-1/2}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}=\mathbb{P}\left(|{\cal
N}(0,{\sigma}^{2}(x,y))|\leq c_{1-\alpha/2}\right),$ (3.12)

where $c_{1-\alpha/2}$ is the $\alpha$-quantile of ${\cal
N}(0,\sigma^{2}(x,y))$. In practice the true quantile $c_{1-\alpha/2}$ cannot
be computed since the variance $\sigma^{2}(x,y)$ is unknown. Nevertheless,
from Proposition 3.10, we can approximate the true quantile $c_{1-\alpha/2}$
by means of the quantiles of the bootstrapped distribution of
$\widehat{S^{\star}_{T}}$ (see Subsection 3.5 for details).

###  3.5 Sampling schemes and computation of the confidence intervals

In the previous section, the process was supposed to be observable over
$[0,T]$. However, in practice the data are often collected according to a
sampling scheme since it is difficult to observe a path continuously at any
time $t$ over the interval $[0,T]$.

Hereafter, we briefly discuss the effect of a sampling scheme on the
construction of confidence intervals for the regression function
$m_{\psi}(\cdot,\cdot)$. Data sampling from a continuous time process at
instants $(t_{k})_{k=1,\dots,n}$ can be made regularly, irregularly or even
randomly. For a sake of simplicity, we consider here the case where the
instants $(t_{k})$ are irregularly spaced, that is $\inf_{1\leq k\leq
n}|t_{k+1}-t_{k}|=\delta>0.$ Now, for $\ell\in\\{0,1\\}$ and $k\in\\{1,\ldots
n\\}$, we define the following increasing families of $\sigma$-algebra:

$\displaystyle\mathcal{F}_{k}:=\mathcal{F}_{t_{k}}=\sigma\left((X_{t_{1}},Y_{t_{1}},\zeta_{t_{1}}),\dots,(X_{t_{k}},Y_{t_{k}},\zeta_{t_{k}})\right),$

and

$\displaystyle{\cal G}^{\ell}_{k}:={\cal
G}^{\ell}_{t_{k}}=\sigma\left(\left(X_{t_{1}},Y_{t_{1}},\zeta_{t_{1}}\right),\dots,\left(X_{t_{k}},Y_{t_{k}},\zeta_{t_{k}}\right);X_{t_{k+1}},(1-\ell)\zeta_{t_{k+1}}\right).$

The purpose then consists in estimating $m_{\psi}(\cdot,\cdot)$ given the
discrete time ergodic stationary process
$\left(X_{t_{k}},Y_{t_{k}},\zeta_{t_{k}}\right)_{k=1,\dots,n}$ sampled from
the underlying continuous time process $\left\\{(X_{t},Y_{t},\zeta_{t}):0\leq
t\leq T\right\\}$. In case of a regular sampling scheme, that is $T=n\delta$,
the estimator $\widehat{m}_{\psi,T}(x,y)$ defined in (2.2) becomes

$\displaystyle\widehat{m}_{\psi,n}(x,y)=\frac{\sum_{k=1}^{n}\zeta_{t_{k}}\psi_{y}(Y_{t_{k}})K\left(\frac{d(x,X_{t_{k}})}{h_{n}}\right)}{\sum_{k=1}^{n}\zeta_{t_{k}}K\left(\frac{d(x,X_{t_{k}})}{h_{n}}\right)},\quad\quad
t_{k}=k\delta.$ (3.13)

a) Asymptotic confidence intervals

Notice that Theorem 3.5 sill holds for the estimate
$\widehat{m}_{\psi,n}(x,y)$ when replacing $T$ by $n\delta$. The limiting law
is a Gaussian random variable with mean zero and variance function
$\sigma^{2}(x,y)=\frac{1}{f(x)}\frac{M_{2}}{M_{1}^{2}p(x)}\overline{W}_{2}(x,y).$
Making use of Corollary 3.9 and considering similar steps as in Laïb and
Louani (2010), it follows that, for any $0<\alpha<1$, the asymptotic
confidence intervals of $m_{\psi}(x,y)$ is given as:

$\displaystyle\widehat{m}_{\psi,n}(x,y)\pm
c_{1-\alpha/2}\frac{M_{n,1}}{\sqrt{M_{n,2}}}\sqrt{\frac{\overline{W}_{n,2}(x,y)}{nF_{x,n}(x)p_{n}(x)}},\quad\text{as}\
n\to\infty,$

where $c_{1-\alpha/2}$ is the quantile of standard Gaussian distribution.

b) Exchangeable bootstrap-based confidence intervals

The discrete-time resampling version of the quantity
$\widehat{S^{\star}_{T}}(x,y)$ is defined for any fixed $x\in\cal{E}$ as

$\widehat{S^{\star}_{n}}(x,y)=\sqrt{n}\left\\{\sum_{k=1}^{n}\left(W_{n,t_{k}}-\overline{W}_{n}\right)\widehat{\xi}_{n,t_{k}}^{\star}(x,y)\right\\},$

where $\overline{W}_{n}:=n^{-1}\sum_{k=1}^{n}W_{n,t_{k}}$,

$\displaystyle\widehat{\xi}_{n,t_{k}}^{\star}(x,y)$ $\displaystyle:=$
$\displaystyle\sqrt{\frac{F_{x,n}(h)}{n}}\,\zeta_{t_{k}}\left[Y_{t_{k}}-\widehat{m}_{\psi,n}(x,y)\right]\frac{\Delta_{t_{k}}(x)}{\mathbb{E}(\Delta_{t_{1}}(x))}$
$\displaystyle-\mathbb{E}\left\\{\sqrt{\frac{F_{x,n}(h)}{n}}\,\zeta_{t_{k}}\left[Y_{t_{k}}-\widehat{m}_{\psi,n}(x,y)\right]\frac{\Delta_{t_{k}}(x)}{\mathbb{E}(\Delta_{t_{1}}(x))}\mid{\cal
F}_{k-1}\right\\},$

and $\\{W_{n,t_{k}}:k=1,\dots,n\\}$ the bootstrap weights generated at the
instants $(t_{k})_{k=1,\dots,n}$, such that, for $n\geq 1$, the sequences of
random variables $\\{\widehat{\xi}_{n,t_{k}}(x,y):k=1,\dots,n\\}$ and
$\\{W_{n,t_{k}}:k=1,\dots,n\\}$ are independent.

In practice the bootstrap procedure can be summarized as follows:

  * •

Suppose
$\left(W^{(1)}_{n,t_{1}},\ldots,W^{(1)}_{n,t_{n}}\right),\ldots,\left(W^{(B)}_{n,t_{1}},\ldots,W^{(B)}_{n,t_{n}}\right)$
are $B$ independent vectors, sampled independently from the sample
$\\{\widehat{\xi}_{n,t_{k}}^{*}(x,y):k=1,\dots,n\\}$.

  * •

Consider the random variables

$\widehat{S^{\star}_{n}}^{(\ell)}(x,y)=\left|\sqrt{n}\left\\{\sum_{k=1}^{n}\left(W_{n,t_{k}}^{(\ell)}-\overline{W}_{n}^{(\ell)}\right)\widehat{\xi}_{n,t_{k}}^{*}(x,y)\right\\}\right|,~{}~{}\ell=1,\ldots,B.$

  * •

Using Proposition 3.10 and Remark 3.11, one can use the smallest $z>0$ as an
approximation of the unknown true quantile $c_{1-\alpha/2}$, such that

$\frac{1}{B}\sum_{\ell=1}^{B}1\kern-1.99997pt{\rm
l}_{\left\\{\widehat{S^{\star}_{n}}^{(\ell)}(x,y)\leq z\right\\}}\geq
1-\alpha.$

##  4 Simulation study

This section aims to discuss numerically some aspects related to continuous
time processes that might affect the quality of estimation of the operator
$m_{\psi}(\cdot,y)$. In this section we consider $\psi_{y}(Y_{t})=Y_{t}$ and
therefore $m_{\psi}(x,y)$ being the conditional expectation $m(x)$ of $Y_{t}$
given $X_{t}=x.$ The first simulation aims to compare the quality of
estimation of $m(x)$ based on a continuous time and discrete time processes.
In simulation 2 we discuss the choice of the “optimal” sampling mesh $\delta$
in the case of continuous time processes and assess its sensitivity towards
the missing at random mechanism.

###  4.1 Simulation 1: continus-time versus discrete-time estimators

In this first simulation we try to compare the estimation of the regression
operator when discrete and continuous time processes are considered. We want
whether or not considering a continuous time processes may improve the quality
of the predictions. We suppose that the functional space ${\cal
E}=L^{2}([-1,1])$ endowed with its natural norm. The generation of continuous
time processes $(\\{X_{t}(s):s\in[-1,1]\\},Y_{t})_{t\in[0,T]}$ is obtained by
considering the following steps:

  1. 1.

First, we simulate an Ornstein-Uhlenbeck (OU) process $(Z_{t})_{t\geq 0}$
solution of the following stochastic differential equation

$\displaystyle dZ_{t}=2(5-Z_{t})dt+7d\mbox{W}_{t},$ (4.1)

where $\mbox{W}_{t}$ denotes a Wiener process. Here, we take $dt=0.005.$

  2. 2.

Let $\Gamma(\cdot)$ be the operator mapping $\mathbb{R}$ into $L^{2}([-1,1])$
defined, for any $z\in\mathbb{R}$, as follows:

$\Gamma(z):=(1+\lfloor z\rfloor-z)\mbox{P}_{\mbox{num}(\lfloor
z\rfloor)}+(z-\lfloor z\rfloor)\mbox{P}_{\mbox{num}(\lfloor z+1\rfloor)},$

where $P_{j}$ is the Legendre polynomials of degree $j$ and
$\mbox{num}(z):=1+2\,z\,\mbox{sign}(z)-\mbox{sign}(z)(1+\mbox{sign}(z))/2$ and
$\lfloor\cdot\rfloor$ denotes the floor function.

  3. 3.

We consider that curves are sampled at 400 equispaced values in [-1,1] and
defined, for any $t\in[0,T]$, as

$X_{t}(s)=\Gamma(Z_{t})(s),\quad\quad s\in[-1,1].$

  4. 4.

To generate the real-valued process $(Y_{t})_{t\in[0,T]}$, the following
nonlinear functional regression model is considered:

$\displaystyle Y_{t}=m(X_{t})+\varepsilon_{t},$ (4.2)

where $m(x):=\int_{-1}^{1}x^{2}(t^{\prime})dt^{\prime}$ and
$\varepsilon_{t}=U_{t}-U_{t-1}$ where $U_{t}$ is a Wiener process independent
of $X_{t}$.

Observe that the OU process $\\{Z_{t}:t\in[0,T]\\}$ is a real-valued
continuous time process (since $dt$ tends to zero). The operator
$\Gamma(\cdot)$ has a role to transform each observation in the process
$Z_{t}$ into a curve through the Legendre polynomials. In such way the
functional variable $X$ is generated continuously as is the process $(Z_{t})$.
Moreover, notice that steps 1, 2 and 3 are devoted to simulate the continuous
time functional process $\\{X_{t}(s):s\in[-1,1]\\}_{t\in[0,T]}$, whereas in
step 4 the real-valued continuous time process $(Y_{t})_{t\in[0,T]}$ is
generated. A sample of 20 simulated curves is displayed in Figure 1 and an
example of the real-valued process $(Y_{t})$ is given in Figure 2.

Figure 1:  A sample of 20 simulated curves $\\{X_{t}(s):s\in[-1,1]\\}$. Figure
2:  A realisation of the process $(Y_{t})_{t\in[0,200]}$.

Now, our purpose is to compare, in terms of estimation accuracy, the
continuous-time estimator with the discrete-time one for different values of
$T=50,200,1000$ and several missing at random rates. It is worth noting that
the continuous time process $(X_{t},Y_{t})$ is observed at every instant
$t=\delta,2\delta,\dots,n\delta$, where $\delta=0.005$ and $n=T/\delta$.
However, the discret-time process is observed only at the instants
$t=1,2,\dots,n.$

As in Ferraty et al. (2013) and Ling et al. (2015) we consider that the
missing at random mechanism is led by the following probability distribution:

$\displaystyle
p(x)=\mathbb{P}(\zeta_{t}=1|X_{t}=x)=\mbox{expit}\left(\int_{-1}^{1}x^{2}(s)ds\right),$
(4.3)

where $\mbox{expit}(u)=e^{u}/(1+e^{u})$, for $u\in\mathbb{R}.$ Now, we specify
the tuning parameters on which depends our estimation given in (2.2). We
choose the quadratic kernel defined as
$K(u)=\frac{3}{4}(1-u^{2})1\kern-1.99997pt{\rm l}_{[0,1]}(u)$ and because
curves are smooth enough we choose as semi-metric the $L_{2}$-norm of the
second derivatives of the curves, that is for $t_{1}\neq t_{2}$,
$d(X_{t_{1}},X_{t_{2}})=\left(\int_{-1}^{1}[X_{t_{1}}^{(2)}(s)-X_{t_{2}}^{(2)}(s)]^{2}ds\right)^{1/2}.$
We used the local cross-validation method on the $\kappa$-nearest neighbours
introduced in Ferraty and Vieu (2006) page 116 to select the optimal bandwidth
for both discrete and continuous time regression estimators. The accuracy of
the discrete and the continuous time regression estimators is evaluated on
$M=500$ replications. The accuracy is measured, at each replication
$j=1,\dots,M$, by using the squared errors
$\mbox{SE}_{T}^{j}:=\left(\widehat{m}_{T}^{j}(x)-m(x)\right)^{2}$ and
$\mbox{SE}_{n}^{j}:=\left(\widehat{m}_{n}^{j}(x)-m(x)\right)^{2}$ for the
continuous-time and the discrete-time estimators, respectively. Observe that
the discrete time estimator of the regression operator is defined as:

$\widehat{m}_{n}(x):=\dfrac{\sum_{t=1}^{n}\zeta_{t}Y_{t}\Delta_{t}(x)}{\sum_{t=1}^{n}\zeta_{t}\Delta_{t}(x)}.$

Table 1: Summary statistics of $(\mbox{SE}^{j})_{j=1,\dots,500}$ for discrete and continuous time estimators of the regression function. MAR rate |  |  | Continuous ($\times 10^{-2}$) |  |  | Discrete ($\times 10^{-2}$) |
---|---|---|---|---|---|---|---
| $T$ | 50 | 200 | 1000 | 50 | 200 | 1000
| $Q_{25\%}$ | 0.56 | 0.54 | 0.18 | 1.6 | 1.3 | 0.7
$p=20\%$ | Median | 2.4 | 2.58 | 1.25 | 7.5 | 3.7 | 2.1
| Mean | 5.39 | 4.62 | 4 | 15.11 | 11.8 | 9.1
| $Q_{75\%}$ | 6.21 | 7.13 | 4.7 | 16.06 | 10.9 | 4.7
| $Q_{25\%}$ | 0.49 | 0.6 | 0.3 | 1.7 | 1 | 0.5
$p=40\%$ | Median | 4.77 | 2.7 | 2.2 | 4.9 | 6.1 | 2
| Mean | 6.93 | 8.8 | 4.6 | 12.4 | 9.9 | 8.3
| $Q_{75\%}$ | 9.39 | 10.6 | 4.5 | 16.05 | 11.3 | 11.1

Table 1 shows that continuous time regression estimator is more accurate than
the discrete time one. Moreover, when $T$ increases the squared errors
decrease much more quickly when working with the continuous time process.

###  4.2 Simulation 2: optimal sampling mesh selection

The purpose of this simulation is to investigate another aspect related to
continuous time processes. The selection of the “optimal” sampling mesh is one
of the most important topics in continuous time processes.

First of all we generate a continuous-time functional data process according
to the following equation:

$X_{t}(s)=Z_{t}\left(1-\sin(s-\pi/3)\right),\quad\quad
s\in[0,\pi/3]\quad\mbox{and}\quad t\in[0,T],$

where $Z_{t}$ is an OU process solution of the stochastic differential
equation (4.1) and practically observed at the instants
$t=\delta,2\delta,\dots,n\delta$ with $n=200$ fixed. Here, we take different
values of sampling mesh $\delta$, and we study the accuracy of the estimator
to identify the optimal mesh, say $\delta^{\star}$, that minimises the Mean
Integrated Square Error (MISE). Observe that each curve observed at the
instant $t$ is discretized in 100 equidistant points over the interval
$[0,\pi/3].$ The response variable is obtained following the hereafter
nonlinear functional regression model:

$Y_{t}=m(X_{t})+\varepsilon_{t},$

where the operator $m(\cdot)$ is defined as
$m(X_{t})=\left(\int_{0}^{\pi/3}X_{t}^{\prime}(s)ds\right)^{2}$ and
$\varepsilon_{t}\sim N(0,0.075).$

Moreover, the missing at random mechanism in this simulation is also supposed
to be the same as described in the first simulation as per equation (4.3). For
the tuning parameters used to build the estimator, we considered the quadratic
kernel and given the shape of the true regression operator, which depends on
the first derivative of the functional covariate, the Euclidean distance
between the first order derivatives of the curves is adopted as a semi-metric.
Finally the bandwidth is selected according to the local cross-validation
method based on the $\kappa$-nearest neighbours as detailed in Ferraty and
Vieu (2006) page 116.

For each value of sampling mesh $\delta$, the regression operator $m(\cdot)$
is estimated over a grid of 50 different fixed curves $x$ and the whole
procedure is repeated over $N=500$ replications. Finally the MISE is
calculated, for each value of $\delta$, according to the following equation

$MISE(\delta):=\dfrac{1}{N}\sum_{k=1}^{N}\dfrac{1}{50}\sum_{j=1}^{50}\left(m(X_{j}^{(k)})-\widehat{m}_{\delta}(X_{j}^{(k)})\right)^{2}.$

Observe that $\widehat{m}_{\delta}(\cdot)$ the estimator of $m(\cdot)$ depends
on the sampling mesh $\delta$, so is the MISE.

Figure 3:  The $MISE(\delta)$ obtained for different values of sampling mesh
$\delta$ and several missing at random rates.

Figure 3 displays the values of MISE obtained for different values of sampling
mesh $\delta$ and a missing at random rate of 10%, 50% and 0% (complete data),
respectively. One can observe that higher is the missing at random rate,
higher will be errors in estimating the regression operator.

| $\delta^{\star}$ | MISE($\delta^{\star}$) | $\overline{MISE(\delta)}$ | $Q_{25\%}$ | $Q_{50\%}$ | $Q_{75\%}$
---|---|---|---|---|---|---
Complete data | 0.30 | 0.0476 | 0.0562 | 0.0510 | 0.0524 | 0.0550
MAR=10% | 0.36 | 0.0555 | 0.0643 | 0.0587 | 0.0612 | 0.0635
MAR=50% | 0.38 | 0.0635 | 0.0767 | 0.0714 | 0.0750 | 0.0770
Table 2: The optimal sampling mesh, $\delta^{\star}$, obtained for different
MAR rates and some summary statistics of the MISE($\delta$).

Table 2 shows that higher is the missing at random rate longer we need to
observe the underlying process to collect the $n=200$ observations to be able
to reasonably estimate the regression operator. Indeed, when the data is
complete the optimal time interval $T^{\star}=n\delta^{\star}=200\times
0.3=60$. However, when $MAR=10\%$ (resp. 50%) the optimal time interval is
equal to $T^{\star}=200\times 0.36=72$ (resp. $T^{\star}=200\times 0.38=76$).
Consequently, it can be concluded that when the missing at random mechanism is
heavily affecting the response variable, we need to collect data over a longer
period of time. This allows to get sufficient information about the dynamic of
the underlying continuous time process and therefore get a better estimate of
the regression operator.

##  5 Application to conditional quantiles

Let $x\in{\cal E}$ be fixed and $y\in\mathbb{R}$, then if
$\psi_{y}(Y)=1\kern-1.99997pt{\rm l}_{\left\\{]-\infty,\ y]\right\\}}(Y)$ the
operator $m_{\psi}(x,y)$ is no more but the conditional cumulative
distribution function (df) of $Y$ given $X=x$, namely $\
F(y|x)=\mathbb{P}(Y\leq y|X=x)$ which may be estimated by
$\widehat{F}_{T}(y|x):=\widehat{m}_{\psi,T}(x,y).$ For a given
$\alpha\in(0,1),$ the $\alpha^{th}$-order conditional quantile of the
distribution of $Y$ given $X=x$ is defined as
$q_{\alpha}(x)=\inf\\{y\in\mathbb{R}:\ F(y|x)\geq\alpha\\}.$

Notice that, whenever $F(\cdot|x)$ is strictly increasing and continuous in a
neighbourhood of $q_{\alpha}(x)$, the function $F(\cdot|x)$ has a unique
quantile of order $\alpha$ at a point $q_{\alpha}(x)$, that is
$F(q_{\alpha}(x)|x)=\alpha.$ In such case

$q_{\alpha}(x)=F^{-1}(\alpha|x)=\inf\\{y\in\mathbb{R}:\ F(y|x)\geq\alpha\\},$

which may be estimated uniquely by
$\widehat{q}_{T,\alpha}(x)=\widehat{F}_{T}^{-1}(\alpha|x)$. Conditional
quantiles has been widely studied in the literature when the predictor $X$ is
of finite dimension, see for instance, Gannoun et al. (2003) and Ferraty et
al. (2005) for dependent functional data.

  * $(a)$

Almost sure pointwise and uniform convergence

Under the same conditions of Theorem 3.1, the statement (3.2) still holds for
the estimator of the cumulative conditional distribution function
$\widehat{F}_{T}(y|x)$. That is $\widehat{F}_{T}(\alpha|x)$ converges, almost
surely, towards $F(y|x)$ with a rate
$\mathcal{O}(h_{T}^{\beta})+\mathcal{O}(\sqrt{\log T/(T\phi(h_{T}))}).$

Consequently, since
$F(q_{\alpha}(x)|x)=\alpha=\widehat{F}_{T}(\widehat{q}_{T,\alpha}(x)|x)$ and
$\widehat{F}_{T}(\cdot|x)$ is continuous and strictly increasing, then we have
$\forall\epsilon>0,\ \exists\eta(\epsilon)>0,\ \forall
y,\quad\big{|}\widehat{F}_{T}(y|x)-\widehat{F}_{T}(q_{\alpha}(x)|x)\big{|}\leq\eta(\epsilon)\quad\Rightarrow\quad|y-q_{\alpha}(x)|\leq\epsilon$
which implies that, $\forall\epsilon>0,\ \exists\eta(\epsilon)>0,$

$\displaystyle\mathbb{P}\left(\left|\widehat{q}_{T,\alpha}(x)-q_{\alpha}(x)\right|\geq\eta(\epsilon)\right)\leq\mathbb{P}\left(\left|\widehat{F}_{T}(\widehat{q}_{T,\alpha}(x)|x)-\widehat{F}_{T}(q_{\alpha}(x)|x)\right|\geq\eta(\epsilon)\right)$
$\displaystyle=\mathbb{P}\left(|F(q_{\alpha}(x)|x)-\widehat{F}_{T}(q_{\alpha}(x)|x)\geq\eta(\epsilon)\right).$
(5.1)

Therefore, the statement (3.2) still holds for the conditional quantile
estimator $\widehat{q}_{T,\alpha}(x)$ whenever conditions of Theorem 3.1 are
satisfied. Ferraty et al. (2005) derived similar pointwise convergence rate by
inverting the estimator of the conditional cumulative distribution function.
Their result has been obtained under mixing condition and additional
assumptions on the joint distribution, and the Lipschitz condition upon
$F(y|x)$ and its derivatives with respect $y$.

Regarding the almost sure uniform convergence, observe that under conditions
of Theorem 3.3, the statement (3.4) still holds true for the $\sup_{y\in
S}\sup_{x\in{\cal C}}|\widehat{F}_{T}(y(x)-F_{T}(y(x)|$, when $\psi_{y}(Y)$ is
replaced by $\mathbf{1}_{\\{]-\infty,\ y]\\}}(Y)$. Moreover, assume that, for
fixed $x_{0}\in{\cal C}$, $F(y|x_{0})$ is differentiable at
$q_{\alpha}(x_{0})$ with $\frac{\partial}{\partial
y}F(y|x_{0})|_{y=q_{\alpha}(x_{0})}:=g(q_{\alpha}(x_{0})|x_{0})>\nu>0$, and
$g(\cdot|x)$ is uniformly continuous for all $x\in{\cal C}$. Knowing that
$\widehat{F}_{T}(\widehat{q}_{T,\alpha}(x)|x)=F(q_{\alpha}(x)|x)=\alpha$ and
making use of a Taylor’s expansion of the function
$F(\widehat{q}_{T,\alpha}(x)|x)$ around $q_{\alpha}(x)$, we can write

$\displaystyle
F(\widehat{q}_{T,\alpha}(x)|x)-F(q_{\alpha}(x)|x)=\left(\widehat{q}_{T,\alpha}(x)-q_{\alpha}(x)\right)g\left(q^{*}_{T,\alpha}(x)|x\right)$
(5.2)

where $q^{*}_{T,\alpha}(x)$ lies between $q_{\alpha}(x)$ and
$\widehat{q}_{T,\alpha}(x)$. It follows then from (5.2) that the inequality
(5) still hods true uniformly in $x$ and $y$. Moreover, the fact that
$\widehat{q}_{T,\alpha}(x)$ converges (a.s.) towards $q_{\alpha}(x)$ as $T$
goes to infinity, combined with the uniformly continuity of $g(\cdot|x)$,
allow to write that

$\displaystyle\sup_{x\in{\cal
C}}\big{|}\widehat{q}_{T,\alpha}(x)-q_{\alpha}(x)\big{|}\sup_{x\in{\cal
C}}\big{|}g\left(q_{\alpha}(x)|x\right)\big{|}=O_{a.s.}\left(\sup_{y\in
S}\sup_{x\in{\cal C}}|\widehat{F}_{T}(y|x)-F(y|x)|\right).$ (5.3)

Since $g(q_{\alpha}(x)|x)$ is uniformly bounded from below, we can then clame
that the estimator $\widehat{q}_{T,\alpha}(x)$ converges uniformly towards
$q_{\alpha}(x)$ with the same convergence rate given in (3.4), as $T$ goes to
infinity.

  * $(b)$

Confidence intervals

Confidence bounds for the conditional quantiles $q_{\alpha}(x)$ may be
obtained according to the following steps. First, consider a Taylor’s
expansion of $\widehat{F}_{T}(\cdot|x)$ around $q_{\alpha}(x)$ and making use
of the fact that $\widehat{q}_{T,\alpha}(x)$ converges (a.s.) towards
$q_{\alpha}(x)$ as $T$ goes to infinity, one gets

$\displaystyle\widehat{q}_{T,\alpha}(x)-q_{\alpha}(x)=-\frac{1}{\widehat{g}_{T}\left(q_{\alpha}(x)|x\right)}\left(\widehat{F}_{T}(q_{\alpha}(x)|x)-F(q_{\alpha}(x)|x)\right),$
(5.4)

where $\widehat{g}_{T}(\cdot|x)$ is a consistent estimator of $g(\cdot|x)$.
Then, replacing $\psi_{y}(Y)$ by the indicator function, we get under
conditions of Corollary 3.9, the following $(1-\alpha)100\%$ confidence bands
for $q_{\alpha}(x)$

$\displaystyle\widehat{q}_{T,\alpha}(x)\ \pm
c_{1-\alpha/2}\frac{M_{T,1}\widehat{g}_{T}\left(\widehat{q}_{T,\alpha}(x)|x\right)}{\sqrt{M_{T,2}}}\sqrt{\frac{\alpha(1-\alpha)}{TF_{x,T}(h_{T})p_{T}(x)}},\quad\text{as}\
T\to\infty.$ (5.5)

##  6 Proofs

In this section and for sake of simplification the subscript $\psi$ will be
omitted. Consider the following quantities

$\displaystyle Q_{T}(x,y)$ $\displaystyle:=$
$\displaystyle(\widehat{m}_{T,2}(x,y)-\overline{m}_{T,2}(x,y))-m(x,y)(\widehat{m}_{T,1}(x,y)-\overline{m}_{T,1}(x,y)),$
(6.1) $\displaystyle R_{T}(x,y)$ $\displaystyle:=$ $\displaystyle-
B_{T}(x,y)(\widehat{m}_{T,1}(x,y)-\overline{m}_{T,1}(x,y)).$ (6.2)

We have then

$\displaystyle\widehat{m}_{T}(x,y)-m(x,y)=\widehat{m}_{T}(x,y)-C_{T}(x,y)+B_{T}(x,y)=B_{T}(x,y)+\frac{Q_{T}(x,y)+R_{T}(x,y)}{\widehat{m}_{T,1}(x,y)},$
(6.3)

We start first by stating some technical lemmas that will be used later.

###### Lemma 6.1

Assume that assumptions (A1)-(A2) are satisfied, then we have for any $j\geq
1$ and $\ell\geq 1$

  * $(i)$

$\frac{1}{\phi(h_{T})}\mathbb{E}\left(\Delta_{t}^{\ell}(x)|\mathcal{F}_{j-2}\right)=M_{\ell}f_{t,T_{j-2}}(x)+\mathcal{O}_{a.s.}\left(\frac{g_{t,T_{j-2},x}(h_{T})}{\phi(h_{T})}\right),$

  * $(ii)$

$\frac{1}{\phi(h_{T})}\mathbb{E}\left(\Delta_{t}^{\ell}(x)\right)=M_{\ell}f(x)+o(1)$.

Proof. The proof is similar to the proof of Lemma of Laïb and Louani (2010).

###### Lemma 6.2

Let $(Z_{n})_{n\geq 1}$ be a sequence of real martingale differences with
respect to the sequence of $\sigma$-fields
$\left(\mathcal{F}_{n}=\sigma(Z_{1},\dots Z_{n})\right)_{n\geq 1}$ where
$\sigma(Z_{1},\dots,Z_{n})$ is the $sigma$-field generated by the random
variable $Z_{1},\dots,Z_{n}$. Set $S_{n}=\sum_{i=1}^{n}Z_{i}.$ For any $p\geq
2$ and any $n\geq 1$, assume that there exist some nonnegative constants $C$
and $d_{n}$ such that $\mathbb{E}\left(Z_{n}^{p}|\mathcal{F}_{n-1}\right)\leq
C^{p-2}p!d_{n}^{2}$ almost surely. Then, for any $\epsilon>0,$ we have

$\mathbb{P}\left(|S_{n}|>\epsilon\right)\leq
2\exp\left\\{-\frac{\epsilon^{2}}{2(D_{n}+C\epsilon)}\right\\},$

where $D_{n}=\sum_{i=1}^{n}d_{i}^{2}.$

Proof. See Theorem 8.2.2 of de la Pena and Giné (1999).

Proof of Theorem 3.1. The proof of Theorem 3.1 is a consequence of
decomposition (6.1) and Lemmas 6.3-6.4-6.5 established below. $\hfil\Box$

###### Lemma 6.3

Assume (A1)-(A3) tougher with conditions (3.1) hold true. Then, we have for
$T$ sufficiently large

$\displaystyle\widehat{m}_{T,j}(x,y)-\overline{m}_{T,j}(x,y)=\mathcal{O}_{a.s.}\left(\sqrt{\frac{\log
T}{T\phi(h_{T})}}\right)\quad\text{with}\quad j\in\\{1,2\\}.$

Proof of Lemma 6.3 . Consider the case where $j=2$ and define the process

$\displaystyle L_{T}:=\widehat{m}_{T,2}(x,y)-\overline{m}_{T,2}(x,y)$
$\displaystyle=$
$\displaystyle\frac{1}{n\mathbb{E}(Z_{1}(x))}\sum_{j=1}^{n}\int_{T_{j-1}}^{T_{j}}\left[\zeta_{t}\psi_{y}(Y_{t})\Delta_{t}(x)-\mathbb{E}\left(\zeta_{t}\psi_{y}(Y_{t})\Delta_{t}(x)|\mathcal{F}_{t-\delta}\right)\right]dt$
$\displaystyle=:$
$\displaystyle\frac{1}{n\mathbb{E}(Z_{1}(x))}\sum_{j=1}^{n}\mathcal{L}_{T,j}(x,y),$

Observe that, for any $j\geq 1$ and $t\in[T_{j-1},T_{j}]$, ${\cal
F}_{j-2}\subset{\cal F}_{t-\delta}\subset{\cal F}_{j-1}$, therefore
$L_{T,j}(x,y)$ is ${\cal F}_{j-1}$-measurable,
$\mathbb{E}\left(|L_{T,j}(x,y)|\right)<\infty$ provided
$\mathbb{E}(\zeta^{2}_{t})<\infty$ and $\mathbb{E}(X^{2}_{t})<\infty$ (in view
of Cauchy-schwartz inequality). Moreover, letting
$\eta_{j}:=\int_{T_{j-1}}^{T_{j}}\left[\zeta_{t}(\psi_{y}(Y_{t})\Delta_{t}(x)\right]dt,$
then $L_{T,j}(x,y)=\eta_{j}-\mathbb{E}[\eta_{j}|{\cal F}_{t-\delta}]$ and

$\mathbb{E}\left\\{L_{T,j}(x,y)|{\cal
F}_{j-2}\right\\}=\mathbb{E}\left\\{\mathbb{E}\left[\eta_{j}|{\cal
F}_{t-\delta}\right]|{\cal
F}_{j-2}\right\\}-\mathbb{E}\left\\{\mathbb{E}\left[\eta_{j}|{\cal
F}_{t-\delta}\right]|{\cal F}_{j-2}\right\\}=0.$

Hence $(L_{T,j}(x,y))_{j\geq 1}$ is a sequence of martingale differences with
respect to the family $({\cal F}_{j-1})_{j\geq 1}$. To be able to apply Lemma
6.2 we need first to check its conditions. Applying Jensen and Minkowski
inequalities, we get, for any $\kappa\geq 2$, that

$\displaystyle|\mathbb{E}\left[\mathcal{L}^{\kappa}_{T,j}(x,y)|\mathcal{F}_{j-2}\right]|$
$\displaystyle\leq$ $\displaystyle
2^{\kappa}\int_{T_{j-1}}^{T_{j}}\mathbb{E}\left[|\zeta_{t}\psi_{y}(Y_{t})\Delta_{t}(x)|^{\kappa}|\mathcal{F}_{j-2}\right]dt.$

Using successfully a double conditioning with respect to the $\sigma$-fields
$\mathcal{S}_{t,\delta}^{0}$ and $\mathcal{S}_{t,\delta}^{1}$ combined with
(A3)(i), (A3)(iii), (A3)(iv), (A3)(i$v^{\prime}$) and the fact that $\zeta$
and $Y$ anr conditionally independent given $X$, one gets for any
$t\in[T_{j-1},\ T_{j}]$ and any $p\geq 2$, that

$\displaystyle\mathbb{E}\left[|\zeta_{t}\psi_{y}(Y_{t})\Delta_{t}(x)|^{p}|\mathcal{F}_{j-2}\right]=\mathbb{E}\left\\{\Delta^{p}_{t}(x)U_{p}(X_{t})W_{p}(X_{t},y)\Big{|}\mathcal{F}_{j-2}\right\\}\leq
C_{W,U}\;\mathbb{E}\left(\Delta_{t}^{p}(x)|\mathcal{F}_{j-2}\right),$

where $C_{W,P}=\max(\sup_{x}|U_{p}(x)|,\sup_{x,y}|W_{p}(x,y)|)$ which is a
finite positive constant independent on $(x,y)$ in view of conditions
(A3)(iii) and (A3)(i$v^{\prime}$) .

Since the kernel $K$ and the function $\tau_{0}$ are bounded from above by a
positive constants $a_{1}$ and $c_{0}$ respectively, and the function
$f_{t,T_{j-2}}(x)$ is bounded by the deterministic function
$b_{t,2\delta}(x)$; using then assumption (A2)(ii) combined with Lemma 6.1 to
get

$\displaystyle\left|\mathbb{E}\left[\mathcal{L}^{\kappa}_{T,j}(x,y)|\mathcal{F}_{j-2}\right]\right|$
$\displaystyle=$ $\displaystyle
2^{\kappa}\int_{T_{j-1}}^{T_{j}}C_{W,U}\phi(h_{T})\left[M_{\kappa}f_{t,T_{j-2}}(x)+\mathcal{O}\left(\frac{g_{t,T_{j-2},x}(h_{T})}{\phi_{y}(h_{T})}\right)\right]dt$
(6.4) $\displaystyle\leq$
$\displaystyle\kappa!(2a_{1})^{\kappa-2}\phi(h_{T})\left[(2a_{1})^{2}C_{U,W}\int_{T_{j-1}}^{T_{j}}b_{t,2\delta}(x)dt+o(1)\right].$

Letting
$d_{j-2}^{2}:=\phi_{y}(h_{T})\left[(2a_{1})^{2}C_{U,W}\int_{T_{j-1}}^{T_{j}}b_{t,2\delta}(x)dt+o(1)\right]$
and define

$D_{n}=\frac{1}{n}\sum_{j=2}^{n}d_{j-2}^{2}=\phi(h_{T})\left\\{(2a_{1})^{2}C_{W,U}\sum_{j=2}^{n}\int_{T_{j-1}}^{T_{j}}b_{t,2\delta}(x)dt+o(1)\right\\}$.
Since $T=n\delta$, it follows from (A2)(v) that
$D_{n}=T\phi(h_{T})\left[(2a_{1})^{2}C_{W,U}D_{2\delta}(x)+o(1)\right].$

Moreover, we have from the statement (ii) of Lemma 6.1 that
$n\mathbb{E}\left(Z_{1}(x)\right)=\mathcal{O}(T\phi(h_{T}))$. Lemma 6.2
combined with condition (3.1) allow to write, for any $\epsilon_{0}>0$, that

$\displaystyle\mathbb{P}\left(\Big{|}\widehat{m}_{T,2}(x,y)-\overline{m}_{T,2}(x,y)\Big{|}>\epsilon_{0}\sqrt{\frac{\log
T}{T\phi(h_{T})}}\right)$ $\displaystyle=$
$\displaystyle\mathbb{P}\left(\Big{|}\sum_{j-1}^{n}{\cal
L}_{T,j}(x,y)\Big{|}>n\mathbb{E}(Z_{1}(x))\epsilon_{0}\left(\frac{\log
T}{T\phi(h_{T})}\right)^{1/2}\right)$ (6.5) $\displaystyle\leq$ $\displaystyle
2\exp\left\\{-c\epsilon_{0}^{2}\log
T\right\\}=\frac{2}{T^{c\epsilon_{0}^{2}}}$

where $c$ is a positive constant. Let $T_{n}:=[T]$ be the integer part of $T$
and choosing $\epsilon_{0}$ large enough, and using condition (3.1) with
Borel-Cantelli Lemma to conclude that (3.2) is valid since $\sum_{n\geq
1}\frac{1}{T_{n}^{c\epsilon_{0}}}<\infty$. $\hfill\Box$.

###### Lemma 6.4

Under assumption (A1)-(A3), we have

$\widehat{m}_{T,1}(x)\longrightarrow
p(x),\quad\mbox{a.s.}\quad\quad\mbox{as}\quad T\rightarrow+\infty.$

Proof. Let us introduce the following decomposition

$\displaystyle\widehat{m}_{T,1}(x)-p(x)=\left(\widehat{m}_{T,1}(x)-\overline{m}_{T,1}(x)\right)+\left(\overline{m}_{T,1}(x)-p(x)\right)=:{\cal
M}_{T,1}(x)+{\cal M}_{T,2}(x)$ (6.6)

We have from Lemma 6.3 that

$\displaystyle{\cal M}_{T,1}=\mathcal{O}_{a.s.}\left(\left(\frac{\log
T}{T\phi(h_{T})}\right)^{1/2}\right)=o_{a.s}(1).$ (6.7)

Let us now focus on the second term ${\cal M}_{T,2}$. Using a double
conditioning with respect to the $\sigma$-field ${\cal
S}_{t-\delta,\delta}^{1}$, assumptions (A2)(ii)-(iii),
(A3)(iv)-(i$v^{\prime}$) and Lemma 6.1 one gets

$\displaystyle\overline{m}_{T,1}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{n\mathbb{E}(Z_{1}(x))}\int_{0}^{T}\mathbb{E}\left[\Delta_{t}(x)p(X_{t})|\mathcal{F}_{t-\delta}\right]dt$
(6.8) $\displaystyle=$
$\displaystyle\frac{1}{n\mathbb{E}(Z_{1}(x))}\left(p(x)+o(1)\right)T\phi(h_{T})\left\\{M_{1}\frac{1}{T}\int_{0}^{T}f_{t,t-\delta}(x)dt+\frac{1}{T}\int_{0}^{T}\mathcal{O}\left(\frac{g_{t,t-\delta,x}(h_{T})}{\phi(h_{T})}\right)dt\right\\}$
$\displaystyle=$
$\displaystyle\left(p(x)+o(1)\right)\frac{1}{n\mathbb{E}(Z_{1}(x))}\;\mathcal{O}(T\phi(h_{T}))=p(x)+o(1).$

Thus $\overline{m}_{T,1}(x)$ converges almost surely towards $p(x)$ as
$T\rightarrow+\infty$, which achieve the proof of this lemma.

###### Lemma 6.5

Under hypotheses (A3)(i)-(ii), for a fixed $x\in{\cal E}$, we have

$\displaystyle B_{T}(x,y)=\mathcal{O}_{a.s.}(h_{T}^{\beta}),\quad$ (6.9)
$\displaystyle
R_{T}(x,y)=\mathcal{O}_{a.s.}\left(h_{T}^{\beta}\left(\frac{\log
T}{T\phi(h_{T})}\right)^{1/2}\right)\quad\text{and}\quad
Q_{T}(x,y)=\mathcal{O}_{a.s.}\left(\left(\frac{\log
T}{T\phi(h_{T})}\right)^{1/2}\right).$ (6.10)

Proof. Observe that

$\displaystyle
B_{T}(x,y)=\frac{\overline{m}_{T,2}(x,y)-m(x,y)\overline{m}_{T,1}(x)}{\overline{m}_{T,1}(x)}=:\frac{\widetilde{B}_{T}(x,y)}{\overline{m}_{T,1}(x)}.$
(6.11)

Using (A3)(i)-(ii) and a double conditioning with respect to the
$\sigma$-field ${\cal S}_{t-\delta,\delta}^{0}$, and Lemma 6.4, we get

$\displaystyle\left|\widetilde{B}_{T}(x,y)\right|$ $\displaystyle=$
$\displaystyle\frac{1}{n\mathbb{E}(Z_{1}(x))}\int_{0}^{T}\mathbb{E}\left(\zeta_{t}\Delta_{t}(x)\left|m(X_{t},y)-m(x,y)\right|\;|\mathcal{F}_{t-\delta}\right)dt$
$\displaystyle\leq$ $\displaystyle\sup_{u\in
B(x,h_{T})}|m(u,y)-m(x,y)|\times\frac{1}{n\mathbb{E}(Z_{1}(x))}\int_{0}^{T}\mathbb{E}\left(\zeta_{t}\Delta_{t}(x)|\mathcal{F}_{t-\delta}\right)dt$
$\displaystyle=$
$\displaystyle\mathcal{O}(h_{T}^{\beta})\times|\overline{m}_{T,1}(x)|$

which implies that $B_{T}(x,y)=\mathcal{O}(h_{T}^{\beta})$. The statements of
(6.10) follow from (6.9) and Lemma 6.3.

Proof of Theorem 3.5. We have from (6.3) and Lemma (6.4) that

$\displaystyle\textsc{MSE}(x,y)$ $\displaystyle=$
$\displaystyle\mathbb{E}\left(\widehat{m}_{T}(x,y)-m(x)\right)^{2}\simeq\mathbb{E}\left(B_{T}(x,y)+\frac{Q_{T}(x,y)+R_{T}(x,y)}{p(x)}\right)^{2}$
(6.12) $\displaystyle\simeq$
$\displaystyle\mathbb{E}\left(B^{2}_{T}(x,y)\right)+\frac{1}{p^{2}(x)}\left[\mathbb{E}\left(Q^{2}_{T}(x,y)\right)+\mathbb{E}\left(R^{2}_{T}(x,y)\right)\right],$

where the products $2\mathbb{E}\left[B_{T}(x,y)(Q_{T}(x,y)+R_{T}(x,y))\right]$
and $2\mathbb{E}\left[Q_{T}(x,y)\times R_{T}(x,y)\right]$ have been ignored
because by the Cauchz-schwarz inequality

$\displaystyle\mathbb{E}\left[B_{T}(x,y)(Q_{T}(x)+R_{T}(x,y))\right]$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\left(B^{2}_{T}(x,y)\right)^{1/2}\times\mathbb{E}\left((Q_{T}(x,y)+R_{T}(x,y)\right)^{1/2}$
$\displaystyle\leq$
$\displaystyle\max\left\\{\mathbb{E}\left(B^{2}_{T}(x,y)\right),\
\mathbb{E}\left[(Q_{T}(x,y)+R_{T}(x,y))^{2}\right]\right\\}.$

We have the same inequality for the second product. The proof of Theorem 3.5
results from Theorem 3.4 and Lemma 6.6 below, which gives an upper bound of
the expectation of $Q_{T}^{2}(x,y)$ and $R^{2}_{T}(x,y)$, respectively.

###### Lemma 6.6

Assume that assumptions (A1)-(A3) hold true, then

$\displaystyle\mathbb{E}(Q^{2}_{T}(x,y))\simeq\frac{4p(x)(W_{2}(x,y)+(m(x,y))^{2})M_{2}}{T\phi(h_{T})M_{1}^{2}f(x)},$
(6.13)

Proof. Ignoring the product term as above, one may write

$\mathbb{E}\left[Q^{2}_{T}(x,y)\right]\simeq\mathbb{E}\left[\widehat{m}_{T,2}(x,y)-\overline{m}_{T,2}(x,y)\right]^{2}+m^{2}(x,y)\
\mathbb{E}\left[\widehat{m}_{T,1}(x)-\overline{m}_{T,1}(x)\right]^{2}:=I_{T1}+m^{2}(x,y)I_{T2}.$

The terms $I_{T1}$ and $I_{T2}$ can be handled similarly. We will juste
evaluate the first one. Since $(T_{j}=j\delta)_{0\geq j\geq n}$ is a
$\delta$-partition of $[0,T]$, we have

$\displaystyle\widehat{m}_{T,2}(x,y)-\overline{m}_{T,2}(x,y)$ $\displaystyle=$
$\displaystyle\frac{1}{n\mathbb{E}(Z_{1}(x))}\sum_{j=1}^{n}\int_{T_{j-1}}^{T_{j}}\left[\zeta_{t}(\psi_{y}(Y_{t})\Delta_{t}(x)-\mathbb{E}\left\\{\zeta_{t}\psi_{y}(Y_{t})\Delta_{t}(x)|{\cal
F}_{t-\delta}\right|\\}\right]dt$ (6.14) $\displaystyle:=$
$\displaystyle\frac{1}{n\mathbb{E}(Z_{1}(x))}\sum_{j=1}^{n}L_{T,j}(x,y).$

Since $(L_{T,j}(x,y))_{j\geq 1}$ is a sequence of martingale differences with
respect to the family $({\cal F}_{j-1})_{j\geq 1}$, then
$\mathbb{E}(L_{T,j}(xy,)L_{T,k}(x,y))=0\quad\mbox{for every}\quad
j,k\in\\{1,\ldots n\\}\quad\mbox{ such that}\quad j\neq k.$ Therefore (by
ignoring the product term), we have

$\displaystyle I_{T1}$ $\displaystyle=$
$\displaystyle\mathbb{E}\left[\widehat{m}_{T,2}(x,y)-\overline{m}_{T,2}(x,y)\right]^{2}\simeq\frac{1}{n^{2}(\mathbb{E}(Z_{1}(x)))^{2}}\
\sum_{j=1}^{n}\mathbb{E}(L_{T,j}(x,y))^{2}.$ (6.15)

Using Jensen inequality and conditioning successively two times with respect
to ${\cal S}_{t-\delta,\delta}^{0}$ and ${\cal S}_{t-\delta,\delta}^{1}$
combined wit (A3)(iii)-(iv), $I_{T1}$ may bounded as

$\displaystyle I_{T1}$ $\displaystyle\leq$
$\displaystyle\frac{4}{n^{2}(\mathbb{E}(Z_{1}(x)))^{2}}\
\sum_{j=1}^{n}\int_{T_{j-1}}^{T_{j}}\mathbb{E}\left[\Delta_{t}^{2}(x)p(X_{t})W_{2}(X_{t},y)\right]dt.$
$\displaystyle=$
$\displaystyle\frac{4(p(x)+o(1))(W_{2}(x,y)+o(1))[M_{2}f(x)+o(1)]}{T\phi(h_{T})[M_{1}f(x)+o(1)]^{2}}.$

On the other side, we can easily show that

$\displaystyle
I_{T2}\leq\frac{4(p(x)+o(1))[M_{2}f(x)+o(1)]}{T\phi(h_{T})[M_{1}f(x)+o(1)]^{2}}.$

Therefore,

$\displaystyle\mathbb{E}(Q^{2}_{T}(x,y))$ $\displaystyle\simeq$ $\displaystyle
I_{T1}+m^{2}(x,y)I_{T2}=\frac{4p(x)(W_{2}(x,y)+(m(x,y))^{2})M_{2}}{T\phi(h_{T})M_{1}^{2}f^{2}(x)}$

Moreover, using the decomposition (6.2) and Theorem 3.4 and Lemma 6.3 one can
see that $\mathbb{E}(Q^{2}_{T}(x,y))$ is negligible with respect to
$\mathbb{E}(Q_{T}^{2}(x,y)).$ This completes the proof.

Proof of Theorem 3.7. The proof of Theorem 3.7 is based essentially on Lemma
6.7 established below, which gives the normality asymptotic of the principal
term in the decomposition (6.3). Indeed, using decomposition (6.3), one may
write

$\displaystyle\sqrt{T\phi(h_{T})}\left(\widehat{m}_{T}(x,y)-m(x,y)\right)=\sqrt{T\phi(h_{T})}B_{T}(x,y)+\frac{\sqrt{T\phi(h_{T})}Q_{T}(x,y)+\sqrt{T\phi(h_{T})}R_{T}(x,y)}{\widehat{m}_{T,1}(x)}.$
(6.16)

Lemma (6.4) implies, under assumption (A1)-(A3), that
$\widehat{m}_{T,1}(x)\longrightarrow p(x)\ a.s.$ as $T\rightarrow\infty.$
Moreover, using Lemma (6.5), we get under (A3)(i)-(ii) combined with
conditions (3.5) that
$\sqrt{T\phi(h_{T})}B_{T}(x,y)=\mathcal{O}_{a.s.}(h_{T}^{\beta}\sqrt{T\phi(h_{T})})=o_{a.s}(1),$
and

$\sqrt{T\phi(h_{T})}R_{T}(x,y)=\mathcal{O}_{a.s.}\left(\sqrt{T\phi(h_{T})}h_{T}^{\beta}\left(\frac{\log
T}{T\phi(h_{T})}\right)^{1/2}\right)=0\left(h_{T}^{\beta}\log
T^{1/2}\right)=o_{a.s.}(1)$

The proof may be then achieved by Lemma 6.7 and Slutsky’s Theorem. $\hfil\Box$

###### Lemma 6.7

Under conditions (A1)-(A3), we have

$\sqrt{T\phi(h_{T})}(\widehat{m}_{T}(x,y)-m(x,y))\quad\xrightarrow{d}\quad{\cal
N}(0,\tilde{\sigma}^{2}(x,y))\quad\text{where}\quad$
$\tilde{\sigma}^{2}(x,y)\leq\frac{M_{2}}{M_{1}^{2}f(x)}p(x)\overline{W}_{2}(x,y)\quad\text{as}\quad
T\longrightarrow\infty.$

Proof. We have

$\displaystyle\sqrt{T\phi(h_{T})}Q_{T}(x,y)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\xi_{T,i}(x,y),\quad\text{with}\quad\xi_{T,i}(x,y)=\eta_{T,i}(x,y)-\mathbb{E}\left[\eta_{T,i}(x,y)\
\Big{|}{\cal F}_{t-\delta}\right]\text{and}$ $\displaystyle\eta_{T,i}(x,y)$
$\displaystyle=$
$\displaystyle\frac{1}{\mathbb{E}Z_{1}}\sqrt{\frac{\phi_{T}(h)}{n}}\int_{T_{i-1}}^{T_{i}}\zeta_{t}\Delta_{t}(x)\left[\psi_{y}(Y_{t})-m(x,y)\right]dt$
(6.17)

Now observe that for any $i\geq 1$ and $t\in[T_{i-1},T_{i}]$ ${\cal
F}_{i-2}\subset{\cal F}_{t-\delta}\subset{\cal F}_{i-1}$. Therefore
$(\xi_{T,i}(x,y))_{i\geq 1}$ is ${\cal F}_{i-1}$-measurable, and
$\mathbb{E}(|\xi_{T,i}|)<\infty$ provided $\mathbb{E}(\zeta_{t}^{2})<\infty$
and $\mathbb{E}(X_{t}^{2})<\infty$. Moreover, we have for any $1\leq i\leq n$
that $\mathbb{E}\left(\xi_{T,i}\ \Big{|}{\cal
F}_{i-2}\right)=\mathbb{E}\left\\{\mathbb{E}\left[\eta_{i}\ |{\cal
F}_{t-\delta}\right]\Big{|}{\cal
F}_{i-2}\right\\}-\mathbb{E}\left\\{\mathbb{E}\left[\eta_{i}\ |{\cal
F}_{t-\delta}\right]\Big{|}{\cal F}_{i-2}\right\\}=0.$

Hence $(\xi_{T,i}(x,y))_{i\geq 1}$ is a sequence of martingale differences
with respect to the $\sigma$-fields $({\cal F}_{i-1})_{i\geq 1}$. To prove the
asymptotic normality, it suffices to check the two following conditions (see,
Corollary 3.1, p. 56, Hallqnd Heyde (1980)):

(a)
$\sum_{i=1}^{n}\mathbb{E}[\xi^{2}_{T,i}(x,y)|\mathcal{F}_{i-2}]\overset{\mathbb{P}}{\rightarrow}\tilde{\sigma}^{2}(x,y)$

and (b)
$n\mathbb{E}[\xi_{T,i}^{2}(x,y)\mathbf{I}_{\\{|\xi_{T,i}(x,y)|>\epsilon\\}}]=o(1)$
holds, for any $\epsilon>0$,

Proof of (a). Observe now that

$\Big{|}\sum_{i=1}^{n}\mathbb{E}\left[\eta^{2}_{T,i}(x,y)\ \Big{|}{\cal
F}_{i-2}\right]-\sum_{i=1}^{n}\mathbb{E}\left[\xi^{2}_{T,i}(x,y)\ \Big{|}{\cal
F}_{i-2}\right]\Big{|}\leq\Big{|}\sum_{i=1}^{n}\left(\mathbb{E}\left[\eta_{T,i}(x,y)\
\Big{|}{\cal F}_{i-2}\right]\right)^{2}\Big{|}$

Using (A1), (A3)(i)-(iv) with Lemma 6.1, and conditioning two times with
respect to the $\sigma$-field ${\cal S}^{1}_{t-\delta,\delta}$ and the fact
that $n\mathbb{E}Z_{1}(t)=O(T\phi(h))$, we have

$\displaystyle\Big{|}\mathbb{E}\left(\eta_{T,i}\Big{|}{\cal
F}_{i-2}\right)\Big{|}$ $\displaystyle=$
$\displaystyle\frac{1}{\mathbb{E}Z_{1}}\sqrt{\frac{\phi_{T}(h)}{n}}\Big{|}\int_{T_{i-1}}^{T_{i}}\mathbb{E}\left(p(X_{t})\Delta_{t}(x)\left[m(X_{t},y)-m(x,y)\right]dt\Big{|}{\cal
F}_{i-2}\right)\Big{|}$ $\displaystyle\leq$
$\displaystyle\sqrt{n\phi_{T}(h)}\frac{p(x)|}{n\mathbb{E}Z_{1}}\sup_{u\in{\cal
B}(x,h)}|m(u,y)-m(x,y)|\sup_{u\in{\cal
B}(x,h)}|p(u)-p(x)|\Big{|}\int_{T_{i-1}}^{T_{i}}\mathbb{E}\left(\Delta_{t}(x)dt\Big{|}{\cal
F}_{i-2}\right)\Big{|}$ $\displaystyle=$ $\displaystyle
O\left(\sqrt{n\phi_{T}(h)}\
h^{\beta}\right)O\left(\phi(h_{T})\int_{T_{i-1}}^{T_{i}}f_{t,T_{i-2}}(x)dt+o(1)\right)\frac{1}{n\mathbb{E}Z_{1}}$
$\displaystyle=$ $\displaystyle O\left(\sqrt{n\phi_{T}(h)}\
h^{\beta}\right)O\left(\frac{1}{T}\int_{T_{i-1}}^{T_{i}}b_{t,\alpha_{0}}(x)dt)\right).$

It follows by (A2)-(iii) and the Cauchez-Schwarz inequality that

$\sum_{i=1}^{n}\left(\mathbb{E}\left[\eta_{T,i}(x,y)\ \Big{|}{\cal
F}_{i-2}\right]\right)^{2}=O(h^{2\beta}\phi(h))=o(1).$

Thus we have only to show that:
$\sum_{i=1}^{n}\mathbb{E}[\eta^{2}_{T,i}(x,y)|\mathcal{F}_{i-2}]\overset{\mathbb{P}}{\rightarrow}\sigma^{2}(x,y)$.
Using again the Cauchez-Schwarz inequality, one may write

$\displaystyle\sum_{i=1}^{n}\mathbb{E}[\eta^{2}_{T,i}(x,y)|\mathcal{F}_{i-2}]$
$\displaystyle=$
$\displaystyle\frac{1}{(\mathbb{E}Z_{1})^{2}}\frac{\phi_{T}(h)}{n}\sum_{i=1}^{n}\mathbb{E}\left[\left(\int_{T_{i-1}}^{T_{i}}\zeta_{t}\Delta_{t}(x)[\psi_{y}(Y_{t})-m(x,y)]dt\right)^{2}\Big{|}{\cal
F}_{i-2}\right]$ (6.18) $\displaystyle\leq$
$\displaystyle\frac{\delta}{(\mathbb{E}Z_{1})^{2}}\frac{\phi_{T}(h)}{n}\sum_{i=1}^{n}\mathbb{E}\left[\int_{T_{i-1}}^{T_{i}}\zeta_{t}^{2}\Delta^{2}_{t}(x)[\psi_{y}(Y_{t})-m(x,y)]^{2}dt\Big{|}{\cal
F}_{i-2}\right]$ $\displaystyle=$
$\displaystyle\frac{\delta}{(\mathbb{E}Z_{1})^{2}}\frac{\phi_{T}(h)}{n}\sum_{i=1}^{n}\mathbb{E}\left[\int_{T_{i-1}}^{T_{i}}\zeta_{t}^{2}\Delta^{2}_{t}(x)[\psi_{y}(Y_{t})-m(X_{t},y)]^{2}dt\Big{|}{\cal
F}_{i-2}\right]$ $\displaystyle+$
$\displaystyle\frac{\delta}{(\mathbb{E}Z_{1})^{2}}\frac{\phi_{T}(h)}{n}\sum_{i=1}^{n}\mathbb{E}\left[\int_{T_{i-1}}^{T_{i}}\zeta_{t}\Delta^{2}_{t}(x)[m(X_{t},y)-m(x,y)]^{2}dt\Big{|}{\cal
F}_{i-2}\right]$ $\displaystyle=:$ $\displaystyle A_{n}+C_{n}$

Conditioning three times with respects to ${\cal F}_{t-\delta}$ and ${\cal
S}^{\ell}_{t-\delta}$, and using Conditions (A3)(ii-(iv)-(i$v^{\prime}$) and
the fact that $T=n\delta$, to get from Lemma 6.1 that

$\displaystyle A_{n}$ $\displaystyle=$
$\displaystyle\frac{\delta}{(\mathbb{E}Z_{1})^{2}}\frac{\phi_{T}(h)}{n}\sum_{i=1}^{n}\mathbb{E}\left[\int_{T_{i-1}}^{T_{i}}p(X_{t})\Delta^{2}_{t}(x)\overline{W}_{2}(X_{t},y)dt\Big{|}{\cal
F}_{i-2}\right]$ $\displaystyle\leq$
$\displaystyle\frac{\delta}{(\mathbb{E}Z_{1})^{2}}\frac{\phi_{T}(h)}{n}(p(x)+o(1))|(\overline{W}_{2}(x,y)+o(1))\sum_{i=1}^{n}\mathbb{E}\left[\int_{T_{i-1}}^{T_{i}}\Delta^{2}_{t}(x)dt\Big{|}{\cal
F}_{i-2}\right]$ $\displaystyle\leq$
$\displaystyle(\delta+o(1))p(x)\overline{W}_{2}(x,y)\frac{\phi^{2}_{T}(h)}{(\mathbb{E}Z_{1})^{2}}\
\frac{1}{n}\sum_{i=1}^{n}\int_{T_{i-1}}^{T_{i}}\mathbb{E}\left[\frac{1}{\phi_{T}(h)}\Delta^{2}_{t}(x)\Big{|}{\cal
F}_{i-2}\right]dt$ $\displaystyle\leq$
$\displaystyle\delta(\delta+o(1))p(x)\overline{W}_{2}(x)\frac{\phi^{2}_{T}(h)}{(\mathbb{E}(Z_{1})^{2}}M_{2}\left\\{\frac{1}{T}\sum_{i=1}^{n}\int_{T_{i-1}}^{T_{i}}f_{t,T_{i-2}}(x)dt+O_{a.s.}\left[\frac{1}{T}\sum_{i=1}^{n}\int_{T_{i-1}}^{T_{i}}\frac{g_{t,T_{i-2},x}(h_{T})}{\phi_{T}(h)}dt\right]\right\\}$

Using the Riemann’s sum combined with condition (A2)(iii), it follows that

$\frac{1}{T}\sum_{i=1}^{n}\int_{T_{i-1}}^{T_{i}}f_{t,T_{i-2}}(x)dt\leq\frac{1}{T}\int_{0}^{T}f_{t,T_{t-\delta}}(x)dt\longrightarrow
f(x)\quad a.s.\quad as\ T\longrightarrow\infty.$

and by (A2)(ii), which states that
$\frac{g_{t,T_{i-2},x}(h_{T})}{\phi_{T}(h)}=o(1)\quad\text{as}\quad
T\longrightarrow\infty.$ Thus,

$\displaystyle A_{n}$ $\displaystyle\leq$
$\displaystyle\delta(\delta+o(1))p(x)\overline{W}_{2}(x,y)\frac{\phi^{2}_{T}(h)}{(\delta\phi_{T}(h_{T})M_{1}f(x)+o(1))^{2}}M_{2}\left[f(x)+o(1)\right]$
(6.20) $\displaystyle=$
$\displaystyle\frac{M_{2}}{M_{1}^{2}f(x)}p(x)\overline{W}_{2}(x,y):=\tilde{\sigma}^{2}(x,y)\quad\text{as}\quad
T\longrightarrow\infty.$

Making use of the same arguments as above combined wi the fact that
$\sup_{u\in{\cal B}(x,h)}|m(x)-m(u)|=h^{2\beta}$, we get $C_{n}=o_{a.s.}(1).$

Proof of part (b). Using successively Hölder, Markov, Jensen and Minkowski
inequalities combined with conditions (A3)(iii), (A3)(i$v^{\prime}$) and Lemma
(6.1), we get , for any $\epsilon>0$, any $p$ and $q$ such that $1/p+1/q=1$,
that

$\displaystyle
n\mathbb{E}[\xi_{T,i}^{2}(x,y)\mathbf{I}_{\\{|\xi_{T,i}(x,y)|>\epsilon\\}}]$
$\displaystyle\leq$ $\displaystyle
4n(\epsilon/2)^{-2q/p}\mathbb{E}[|\eta_{T,i}|^{2q}]=O\left(T\phi(h_{T})^{-\gamma/2}\right)=o_{a.s}(1)$

by taking $2q=2+\gamma$ ($0<\gamma<1$), since $T\phi(h_{T})$ towards to
infinity as $T$ goes to infinity.

Proof of Corollary 3.9. We have

$\displaystyle\sqrt{\frac{TF_{x,T}(h_{T})}{\widetilde{V}^{2}_{T}(x,y)}}\left(\widehat{m}_{T}(x,y)-m(x,y)\right)=\sqrt{\frac{F_{x,T}(h_{T})}{\phi(h_{T})f(x)}}\sqrt{\frac{\sigma^{2}(x,y)f(x)}{\widetilde{V}^{2}_{T}(x,y)}}\sqrt{\frac{T\phi(h_{T})}{\sigma^{2}(x,y)}}\left(\widehat{m}_{T}(x,y)-m(x,y)\right)$
(6.21)

We have from the consistency of $F_{x,T}(h_{T})$ and A2(i) that
$\frac{F_{x,T}(h_{T})}{\phi(h_{T})f(x)}$ goes to $1$ a.s. as $T$ goes
$+\infty$. By Theorem 3.7 that quantity
$\sqrt{\frac{T\phi(h_{T})}{\sigma(x,y)}}\left(\widehat{m}_{T}(x,y)-m(x,y)\right)$
converges to ${\cal N}(0,1)$ as $T\rightarrow\infty$. Using then the non-
decreasing property of the cumulative standard Gaussian distribution function
$\Psi$, we get, for a given risk $0<\alpha<1$, the $(1-\alpha)-$ pseudo-
confidence bands:

$\displaystyle\sqrt{\frac{T\phi(h_{T})}{\sigma^{2}(x,y)}}\left|\widehat{m}_{T}(x,y)-m(x,y)\right|\leq\Psi^{-1}(1-\frac{\alpha}{2}).$
(6.22)

Considering now the statement (3.6) combined with Proposition 3.8, it holds
that

$\displaystyle\lim_{T\rightarrow\infty}\frac{\sigma^{2}(x,y)f(x)}{\widetilde{V}^{2}_{n}(x,y)}\leq\lim_{T\rightarrow\infty}\frac{f(x)V^{2}(x,y)}{\widetilde{V}^{2}_{n}(x,y)}=\lim_{T\rightarrow\infty}\frac{\widetilde{V}^{2}(x,y)}{\widetilde{V}^{2}_{n}(x,y)}=1\quad
a.s.,$ (6.23)

because $\widetilde{V}_{n}^{2}(x,y)$ is a consistent estimator of
$\widetilde{V}^{2}(x,y)$. The proofs follows then from the statements (6.21),
(6.22) and (6.23) $\hfill\Box$

Proof of Theorem 3.3

Letting $\lambda_{T}=\sqrt{\frac{\varphi_{\cal
C}(\epsilon_{T})}{T\phi(h_{T})}}$ with $\epsilon_{T}=\frac{\log T}{T}$ and
consider $n\delta=T$, $n=[T]\ (T\geq 1)$, consequently $1\leq\delta<2$.
Observe that

$\displaystyle\sup_{y\in
S}\sup_{x\in\mathcal{C}}|\widehat{m}_{T}(x,y)-m(x,y)|\leq$
$\displaystyle\sup_{y}\sup_{x\in\mathcal{C}}|B_{T}(x,y)|+\frac{\sup_{y\in
S}\sup_{x\in\mathcal{C}}|R_{T}(x,y)|+\sup_{y\in
S}\sup_{x\in\mathcal{C}}|Q_{T}(x,y)|}{\inf_{x\in\mathcal{C}}|\widehat{m}_{T,1}(x)|}.$
(6.24)

Since
$\inf_{x\in\mathcal{C}}\big{|}\widehat{m}_{T,1}(x)\big{|}>\inf_{x\in\mathcal{C}}\big{|}p(x)\big{|}-\sup_{x\in\mathcal{C}}\big{|}\widehat{m}_{T,1}(x)-p(x)\big{|},$
using the same steps of the proof of Lemma 6.4, one can show, under conditions
(A1), (U0), (A3)(iv)-(i$v^{\prime}$), (A3)(i)-(ii), the second term in the
above inequality equals zero. Thus, making use (U0)(v), we get for
sufficiently large $T$ that
$\inf_{x\in\mathcal{C}}\Big{|}\widehat{m}_{T,1}(x)\Big{|}>b_{1}$ a.s.

Concerning the conditional bias, inspection of the proof of the statement (6)
shows that the term $\tilde{B}_{T}(x,y)$ is bounded above by a constant which
is independent of $x$ and $y$, therefore, we have under (A3)(i)-(ii) that

$\displaystyle\sup_{y}\sup_{x\in\mathcal{C}}\big{|}B_{T}(x,y)\big{|}\leq\frac{\sup_{y}\sup_{x\in\mathcal{C}}\big{|}\widetilde{B}_{T}(x,y)\big{|}}{\inf_{x\in\mathcal{C}}\big{|}\overline{m}_{T,1}(x)\big{|}}\leq\frac{1}{b_{1}}\sup_{y}\sup_{x\in\mathcal{C}}\big{|}\widetilde{B}_{T}(x,y)\big{|}=\mathcal{O}_{a.s.}(h_{T}^{\beta}).$
(6.25)

Moreover, under the assumption of Lemma 6.8, making use of the decomposition
(6.1), it follows from (U0)(iv) that $\sup_{y\in
S}\sup_{x\in\mathcal{C}}|Q_{T}(x,y)|=\mathcal{O}_{a.s.}(\lambda_{T})$. In the
other hand conditions (6.2) and (6.25) allow to conclude $\sup_{y\in
S}\sup_{x\in\mathcal{C}}|R_{T}(x,y)|=\mathcal{O}_{a.s.}\left(\lambda_{T}h_{T}^{\beta}\right)$.
The mean task is to prove the following Lemma that allows with the statement
(6.3) to achieve the proof of the Theorem 3.3

###### Lemma 6.8

Assume that (A1), (U0)(i)-(iii), (A3)(i), (A3)(iii-iv), (A3)(i$v^{\prime}$),
(U1)-(U4) together with conditions (3.1) and (3.3) are satisfied. Then we have

$\sup_{y\in
S}\sup_{x\in\mathcal{C}}|\widehat{m}_{T,2}(x,y)-\overline{m}_{T,2}(x,y)|=\mathcal{O}_{a.s.}(\lambda_{T}).$

Proof of Lemma 6.8 .

Let $\epsilon>0$ be given and consider a covering of the class of functions
${\cal C}$ by closed balls

$B(c_{k},\epsilon)=\\{x\in\mathcal{C}:d(x,c_{k})<\epsilon\\},\ 1\leq
k\leq=\mathcal{N}(\epsilon,\mathcal{C},d):=N_{\epsilon},$ that is ${\cal
C}\subset\cup_{k=1}^{N_{\epsilon}}B(c_{k},\epsilon)$. Then we have

$\displaystyle\sup_{y\in
S}\sup_{x\in\mathcal{C}}|\widehat{m}_{T,2}(x,y)-\overline{m}_{T,2}(x,y)|$
$\displaystyle\leq$ $\displaystyle\sup_{y\in S}\max_{1\leq k\leq
N_{\epsilon}}\sup_{x\in
B(c_{k},\epsilon)}|\widehat{m}_{T,2}(x,y)-\widehat{m}_{T,2}(c_{k},y)|$ (6.26)
$\displaystyle+$ $\displaystyle\sup_{y\in S}\max_{1\leq k\leq
N_{\epsilon}}|\widehat{m}_{T,2}(c_{k},y)-\overline{m}_{T,2}(c_{k},y)|$
$\displaystyle+\sup_{y\in S}\max_{1\leq k\leq N_{\epsilon}}\sup_{x\in
B(c_{k},\epsilon)}|\overline{m}_{T,2}(x,y)-\overline{m}_{T,2}(c_{k},y)|$
$\displaystyle=:$
$\displaystyle\mathcal{H}_{T,1}+\mathcal{H}_{T,2}+\mathcal{H}_{T,3}.$

Let us now focus on the first term $\mathcal{H}_{T,1}$. We have for any $x\in
B(c_{k},\epsilon)$ and $y\in S$

$\displaystyle\widehat{m}_{T,2}(x,y)-\widehat{m}_{T,2}(c_{k},y)$
$\displaystyle=$
$\displaystyle\frac{1}{n\mathbb{E}(Z_{1}(x))}\int_{0}^{T}\zeta_{t}\psi_{y}(Y_{t})\left[\Delta_{t}(x)-\Delta_{t}(c_{k})\right]dt$
(6.27) $\displaystyle+$
$\displaystyle\frac{1}{n\mathbb{E}(Z_{1}(x))\mathbb{E}(Z_{1}(c_{k}))}\int_{0}^{T}\zeta_{t}\psi_{y}(Y_{t})\Delta_{t}(c_{k})\left[\mathbb{E}(Z_{1}(c_{k}))-\mathbb{E}(Z_{1}(x))\right]dt$
$\displaystyle:=$
$\displaystyle\mathcal{I}_{T,1}(c_{k},y)+\mathcal{I}_{T,2}(c_{k},y).$

Making use of the property of ergodicity, conditions $(U_{1})$ and $(U_{2})$
and the boundedness of $\zeta$, we get for $T$ sufficiently large and any
$(x,y)\in B(c_{k},\epsilon)\times S$ that

$\displaystyle|\mathcal{I}_{T,1}(c_{k},y)|$ $\displaystyle\leq$ $\displaystyle
a_{3}c_{\zeta}\frac{\epsilon}{h_{T}}\frac{1}{\mathbb{E}(\Delta_{0}(x))}\frac{1}{T}\int_{0}^{T}|\psi_{y}(Y_{t})|dt$
(6.28) $\displaystyle\leq$
$\displaystyle\frac{a_{3}c_{\zeta}}{a_{2}}\frac{\epsilon}{h_{T}}\mathcal{O}_{a.s.}\sup_{y\in
S}\mathbb{E}\left(|\psi_{y}(Y_{0})|\right)=\mathcal{O}_{a.s.}\left(\frac{\epsilon}{h_{T}}\right).$

On the other hand, we have under the above conditions

$\displaystyle|\mathcal{I}_{T,2}(c_{k},y)|\leq
c_{\zeta}\frac{\mathbb{E}\left(|\Delta_{0}(c_{k})-\Delta_{0}(x)|\right)}{\mathbb{E}(\Delta_{0}(x))}\frac{1}{n\mathbb{E}(Z_{1}(c_{k}))}\int_{0}^{T}|\psi_{y}(Y_{t})|\Delta_{t}(c_{k})dt$
(6.29) $\displaystyle\leq$
$\displaystyle\frac{c_{\zeta}a^{2}_{3}}{a^{2}_{2}}\frac{\epsilon}{h_{T}}\frac{1}{T}\int_{0}^{T}|\psi_{y}(Y_{t})|dt=\frac{c_{\zeta}a^{2}_{3}}{a^{2}_{2}}\frac{\epsilon}{h_{T}}\sup_{y\in
S}\mathbb{E}\left(|\psi_{y}(Y_{0})|\right)=\mathcal{O}_{a.s.}\left(\frac{\epsilon}{h_{T})}\right).$

The constants in the right hand side of (6.28) and (6.29) are independent of
$x$ and $y$, thus

$\displaystyle{\cal H}_{T,1}=\sup_{y\in S}\max_{1\leq k\leq
N_{\epsilon}}\sup_{x\in
B(c_{k},\epsilon)}|\mathcal{I}_{T,1}(c_{k},y)+\mathcal{I}_{T,2}(c_{k},y)|=O_{a.s.}\left(\frac{\epsilon_{T}}{h_{T}}\right)$
(6.30)

Similarly we get, under the above conditions, the same bound of the term
$\mathcal{H}_{T,3}$, . Using condition (U4) we conclude, for $T$ sufficiently
large enough, that

$\displaystyle\mathcal{H}_{T,1}=\mathcal{H}_{T,3}=O_{a.s.}\left(\lambda_{T}\right)\quad\text{with}\quad\epsilon_{T}=\frac{\log
T}{T}$ (6.31)

Consider now, the intermediate term $\mathcal{H}_{T,2}$. For this purpose,
cover $S$ with $\nu_{T}=[T^{\gamma}]+1$ intervals
$I_{k\prime}=[\ell_{n}-y_{k^{\prime}},\ \ell_{n}+y_{k^{\prime}}]$ of centre
$y_{k^{\prime}}\in S$ and length $2\ell_{T}\leq c/\nu_{T}$, for some
$\gamma>0$, such that $S\subset\cup_{k^{\prime}=1}^{\nu_{T}}I_{k^{\prime}}$.
Then we have

$\displaystyle{\cal H}_{T,2}$ $\displaystyle=$ $\displaystyle\sup_{y\in
S}\max_{1\leq k\leq
N_{\epsilon}}|\widehat{m}_{T,2}(c_{k},y)-\overline{m}_{T,2}(c_{k},y)|$ (6.32)
$\displaystyle\leq$ $\displaystyle\max_{1\leq k^{\prime}\leq\nu_{n}}\
\max_{1\leq k\leq N_{\epsilon}}\ \sup_{y_{k}\in I_{k^{\prime}}}\
|\widehat{m}_{T,2}(c_{k},y)-\widehat{m}_{T,2}(c_{k},y_{k^{\prime}})|$
$\displaystyle+$ $\displaystyle\max_{1\leq k^{\prime}\leq\nu_{T}}\ \max_{1\leq
k\leq N_{\epsilon}}\
|\widehat{m}_{T,2}(c_{k},y_{k^{\prime}})-\overline{m}_{T,2}(c_{k},y_{k^{\prime}})|$
$\displaystyle+$ $\displaystyle\max_{1\leq k^{\prime}\leq\nu_{T}}\ \max_{1\leq
k\leq N_{\epsilon}}\ \sup_{y_{k}\in I_{k^{\prime}}}\
|\overline{m}_{T,2}(c_{k},y_{k^{\prime}})-\overline{m}_{T,2}(c_{k},y)|$
$\displaystyle:=$ $\displaystyle{\cal R}_{1,T}+{\cal R}_{2,T}+{\cal R}_{3,T}$

Using (U1)(ii), (U2) and (U3) combined with the ergodic property and because
$\zeta$ is bounded, one may write

$\displaystyle|\widehat{m}_{T,2}(c_{k},y)-\widehat{m}_{T,2}(c_{k},y_{k^{\prime}})|$
$\displaystyle=$
$\displaystyle[T\mathbb{E}(\Delta_{0}(c_{k}))]^{-1}\int_{0}^{T}|\zeta_{t}|\Delta_{t}(c_{k})\left|\psi_{y}(Y_{t})-\psi_{y_{k^{\prime}}}(Y_{t})\right|dt$
(6.33) $\displaystyle\leq$ $\displaystyle
c_{\psi}\frac{a_{3}}{a_{2}}\frac{1}{\nu_{T}^{\gamma}}\mathbb{E}(|\zeta_{0}|)=\mathcal{O}_{a.s.}(T^{-\gamma}).$

The same bound may be obtained for the quantity
$\overline{m}_{T,2}(c_{k},y_{k^{\prime}})-\overline{m}_{T,2}(c_{k},y)$. Since
the constants in the above terms are independent of $c_{k}$ and
$y_{k^{\prime}}$, we conclude that

$\displaystyle{\cal R}_{1,T}={\cal R}_{3,T}=O_{a.s}(\lambda_{T})$ (6.34)

because $\lim_{T\to+\infty}\frac{1}{\lambda_{T}\nu_{T}^{\gamma}}=0$ in view of
condition (U4).

We turn now to the intermediate term ${\cal R}_{2,T}$. Using the fact that
$\varphi_{\cal C}(\epsilon_{n})=\log N_{\epsilon}$, $\nu_{T}=[T^{\gamma}]+1$,
the statement (6.5) which still true under conditions (A1), (U0)(i)-(iii),
(A3)(i), (A3)(iii-iv) and (A3)(i$v^{\prime}$), we get for any
$\epsilon_{0}>0$, whenever condition (3.1) and the assumption (U4) are
satisfied, that

$\displaystyle\mathbb{P}({\cal R}_{2,T}>\lambda_{T})=$
$\displaystyle\mathbb{P}\left(\max_{1\leq k^{\prime}\leq\nu_{T}}\ \max_{1\leq
k\leq N_{\epsilon}}\
|\widehat{m}_{T,2}(c_{k},y_{k^{\prime}})-\overline{m}_{T,2}(c_{k},y_{k^{\prime}})|>\lambda_{T}\right)$
(6.35) $\displaystyle\leq$
$\displaystyle\sum_{k^{\prime}=1}^{\nu_{T}}\sum_{k=1}^{N_{\epsilon}}\mathbb{P}\left(|\widehat{m}_{T,2}(c_{k})-\overline{m}_{T,2}(c_{k})|>\lambda_{T}\right)$
$\displaystyle\leq$ $\displaystyle
2\nu_{T}N_{\epsilon_{n}}\exp\left\\{-c\epsilon_{0}^{2}\varphi_{C}(\epsilon_{n})\right\\}=2\nu_{T}N_{\epsilon_{n}}^{1-c\epsilon_{0}^{2}}$
$\displaystyle\leq$ $\displaystyle
2n^{\gamma}N_{\epsilon_{n}}^{1-c\epsilon_{0}^{2}}$

because $T=n\delta$ and $1\leq\delta<2$. Choosing $c\epsilon_{0}^{2}=\eta$ and
considering condition (3.3) which is equivalent to

$\sum_{n\geq 1}n^{\gamma}N_{\epsilon_{n}}^{1-\eta}<\infty\quad\text{for some}\
\eta>0$

to conclude by Borel-Cantelli Lemma the end of the proof of Lemma 6.8
$\hfil\Box$.

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