# Inference on volatility estimation with missing data: a functional data
approach

Abdelbasset Djeniah  (A. Djeniah) Laboratory of Stochastic Models, Statistics
and Applications. University of Saida Dr. Moulay Tahar. B.P. 138, En-Nasr,
Saida, 2000 Algeria<EMAIL_ADDRESS>,  Mohamed Chaouch
(M. Chaouch) Statistics Program, Department of Mathematics, Statistics and
Physics College of Arts and Sciences, Qatar University, Doha, Qatar
<EMAIL_ADDRESS>and  Amina Angelika Bouchentouf  (A. A. Bouchentouf)
Laboratory of Mathematics, Djillali Liabes University of Sidi Bel Abbes, Sidi
Bel Abbes, Algeria<EMAIL_ADDRESS>
###### Abstract.

This paper aims to investigate nonparametric estimation of the volatility
component in a heteroscedastic scalar-on-function regression model when the
underlying discrete-time process is ergodic and affected by a missing at
random mechanism. First, we introduce a simplified estimator of the regression
and volatility operators based on observed data only. We study their
asymptotic properties, such as almost sure uniform consistency rate and
asymptotic distribution. Then, the simplified estimators are used to impute
the missing data in the original process in order to improve the estimation of
the regression and volatility components. The asymptotic properties of the
imputed estimators are also investigated. A numerical comparison between the
estimators is discussed through simulated data. Finally, a real-data analysis
is conducted to model the volatility of daily Brent crude oil returns using
intraday, 1-minute frequency, natural gas returns.

###### Key words and phrases:

Ergodic processes, Functional time series, Missing at random, Imputation,
Volatility.

###### 2010 Mathematics Subject Classification:

60F10, 62G07, 62F05

##  1\. Introduction

In the last 15 years, the capital markets have seen significant development,
introducing high-frequency trading and a shift of market towards high-
frequency and algorithm trading. It was always believed that high-frequency
trading and automated trading were source price shocks and rising of
volatility. Therefore, more interest was recently given in modeling the
volatility with high-frequency financial data. Nowadays with the progress in
technology devises we do have more access to data at a very fine time scale.
High frequency data refers to a type of data that is collected at a very
granular level and at frequent intervals of time, typically in sub-daily or
intra-day increments. It captures observations or measurements of various
variables at a high frequency, often with a time resolution of seconds,
minutes, or hours.

From a financial market analysis perspective, understanding and modeling the
volatility with high frequency financial data and if possible predict it would
be of great interest to investors to take the right decisions. Moreover,
financial firms, that trade assets on high-frequency time scale, are not just
interested in short-term forecasting of future values of financial assets, but
also in measuring the uncertainty associated to such predictions through the
volatility component. Two major models of volatility are introduced in the
literature which are Generalized Autoregressive Conditionally Heteroskedastic
(GARCH) and Stochastic Volatility (SV) models. Univariate ARCH model was
firstly introduced by Engle (1982), soon followed by the generalization to
GARCH model of Bollerslev (1987). The GARCH models and their later extensions
were quickly found to be relevant for the conditional volatility of financial
returns observed at a monthly and higher frequency, and thus to the study of
the intertemporal relation between risk and expected return. Although early
GARCH models have been and are still widely used, a viewpoint slowly emerged,
according to which these models may be too rigid for fitting return series,
especially over a long span. An alternative to GARCH-type models is the class
of SV models, which postulate that volatility is driven by its own stochastic
process. The major difference to GARCH models is that, conditional on the
information set available upon time $t-1$, volatility at time $t$ is not known
but rather an unobserved random variable. On the other hand SV models have
some advantages compared with GARCH models. For instance, SV models (see the
handbook by Andersen et al. (2009)) offer a natural economic interpretation of
volatility. They are easier to connect with continuous-time diffusion models
with SV, and are often found to be more flexible in the modeling of financial
returns.

On the other hand, nonparametric autoregressive models with ARCH-type errors
were also introduced in Laïb  (2005) and Fan and Yao (1998) to relax the
parametric assumptions on which the GARCH models depend. Despite their
flexibility in modeling nonlinear pattern in the volatility component,
nonparametric approaches suffer from the well-known curse of dimensionality.
To overcome this issue, one can assume that the functional form of the
volatility component can be semiparametrically specified such that it has both
parametric and nonparametric components. For instance, additive volatility
model, assuming the target volatility function can be written as the sum of
functions of covariates. The additive model can effectively reduce the
dimensionality of the multivariate regression model and improve the
convergence rate of the resulting volatility estimator. Alternatively, single-
index volatility model is another approach for modeling conditional variance.
For more details about the last two approaches, the reader can be referred to
Su et al. (2012).

It is worth noticing that the above mentioned approaches assume that the time
series is completely observed and the predictor as well as the response
variable are both observed at the same time frequency. In practice, despite
the modern technology, which allows to collect the data at a very fine time
scale, financial time series can still be missing. For instance, there are
some regular holidays, such as Thanksgiving day and Christmas, for which stock
price data are missing. There are many other technical reasons, such as
breakdown in devices recording data, computer’s sudden shutdown, …, that make
stretches of data missing. In the literature of financial data analysis, it is
commonly assumed that the data are completely observed which is not realistic.
Therefore, the problem of missing data arises whenever there is a disturbance
in the sequence of the series in terms of observations, hence it is necessary
to address this such problem. Moreover, in many situations, asset prices might
be recorded at different time frequency. For instance one may be interested in
assessing the effect of intraday (1-minute frequency) natural gas returns on
the daily (1-day frequency) oil price returns volatility (see Section 5 for
more details). Figure 1 shows an example where the response variable (here
daily brent oil return) is real-valued whereas the predictor (1-minute
frequency) natural gas intraday curve is a functional random variable. In such
situation models such as GARCH and SV models cannot be applied and
nonparametric functional models become a tool of great interest.

Figure 1. (left) Sample of three intraday (1-minute frequency) Natural Gas
curves. (right) The stochastic process of daily Brent Oil return and the dots
represent the corresponding three preselected days.

During the last two decades, functional data analysis attracted considerable
attention in statistical research owing to its extensive applications in
numerous practical areas such as engineering, geology, biology, medicine,
chemistry, climatology, economics, and so on. The literature on the subject
initially focused on parametric models (cf. Bosq (2000), Ramsay and Silverman
(2002, 2005)), nonparametric models (Ferraty and Vieu (2006), Geenens (2011),
Ling and Vieu (2018)), and semi-parametric models (see Goia and Vieu (2014)
and Vieu (2018) among others). More recently, nonparametric functional models
became more and more popular in solving econometric problems. For instance
Ferraty and Quintela-del-Río (2016) estimated two risk measures, namely the
value-at-risk and the expected shortfall, conditionally to a functional
variable. Müller et al. (2011) introduced a functional volatility process for
modeling volatility trajectories for high frequency observations in financial
markets. Hörmann et al. (2013) introduced a functional version of the ARCH
model to model high-resolution tick data which can be described by continuous-
time process. Wang et al. (2014) used functional principle components analysis
to find the most relevant patterns in the Shanghai stock exchange 50 index.
Caldeira and Torrent (2017) used nonparametric functional data analysis to
forecast the US term structure of interest rates. Recently, Chaouch (2019)
investigated volatility estimation in scalar-on-function heteroscedastic
regression model when data are completely observed.

This paper aims to generalize the work in Chaouch (2019) to the case when the
response variable is missing at random, extend results in Pérez-Gonzàlez et
al. (2010) to the case when the predictor is an infinite-dimensional random
variable. In contrast to Ling et al. (2015), this paper considers a
heteroscedastic functional regression model where two components have to be
estimated under missing at random assumption, namely the regression and
conditional variance operators. First of all we define the initial estimators
based on the available data. In other words missing data do not contribute in
calculating the regression and volatility estimators. Then, we make use of the
simplified estimators to impute missing data in the original process. Finally,
we re-estimate the parameters in our model based on imputed data and compare
them with the simplified estimator. The study of the asymptotic properties of
the simplified as well as imputed estimators are deeply investigated in this
paper. We established the pointwise along with the uniform consistency rate.
The asymptotic distribution is also provided and confidence intervals are
estimated.

The structure of the paper is as follows. Section 2 introduces the
heteroscedastic scalar-on-function regression model as well as the main
assumptions defining the framework of our study. In Section 3 we introduce the
simplified as well as the nonparametric imputed estimator. Then we focus on
the discussion of asymptotic properties of estimators, including the uniform
almost-sure convergence (with rates) and the asymptotic distribution of the
proposed estimators. To illustrate these asymptotic properties, both simulated
and real data analysis are conducted in Section 4 and 5, respectively. In
Section 6, we thoroughly discuss our results and offer various perspectives.
Finally, technical proofs of the main asymptotic results are detailed in the
Appendix.

###  1.1. Notations

We denote $o_{a.s.}(v)$ a real random function $z$ such that $z(v)/v$
converges to zero almost surely as $v\rightarrow 0$. Similarly, we define
$\mathcal{O}_{a.s.}(v)$ as a real random function $z$ such that $z(v)/v$ is
almost surely bounded. We denote by $C$ a positive generic constant that may
change value. We use the notation $\overset{\mathcal{D}}{\longrightarrow}$ to
denote convergence in distribution.

##  2\. Settings

In this section we introduce the heteroscedastic scalar-on-function regression
model, define the missing at random assumption and present the ergodic
assumption that we assume our functional data satisfy.

###  2.1. Model

Let $(X_{t},Y_{t})_{t=1,\dots,n}$ be a sample of discrete time ergodic random
processes taking values in $\mathcal{E}\times\mathbb{R}$ and distributed as
$(X,Y)$. Here, $Y$ represents the variable of interest, and $X$ is a
functional covariate taking values in an infinite-dimensional space
$\mathcal{E}$ equipped with a semi-metric $d(\cdot,\cdot)$111A semi-metric
(sometimes called pseudo-metric) $d(\cdot,\cdot)$ is a metric which allows
$d(x_{1},x_{2})=0$ for some $x_{1}\neq x_{2}$., defining a topology to measure
the proximity between two elements of $\mathcal{E}$ and which is disconnected
of the definition of $X$ in order to avoid measurability problems. We suppose
that the data generation process is described by the following heteroscedastic
functional regression model:

(2.1) $\displaystyle
Y_{t}=m(X_{t})+\sqrt{U(X_{t})}\;\varepsilon_{t},\quad\quad t=1,\dots,n,$

where $m(\cdot)=\mathbb{E}(Y|X=\cdot)$ is the regression operator and
$U(\cdot)=\text{var}(Y|X=\cdot)$ is the conditional variance operator which
are supposed to be unknown. Here, we assume that the sequence
$\varepsilon_{1},\varepsilon_{2},\ldots$ of random variables forms a
martingale difference sequence such that

(2.2) $\displaystyle\mathbb{E}(\varepsilon_{t}|\mathcal{G}_{t-1})=0\;\text{
a.s.}\quad\mbox{and}\quad\mbox{var}(\varepsilon_{t}|\mathcal{G}_{t-1})=1\;\;\mbox{a.s.},$

where $\mathcal{G}_{t-1}$ is the $\sigma$-field generated by
$\\{(X_{1},Y_{1}),\dots,(X_{t-1},Y_{t-1}),X_{t}\\}$. We denote by
$\mathcal{F}_{t-1}$ the $\sigma$-field generated by
$\\{(X_{1},Y_{1}),\dots,(X_{t-1},Y_{t-1})\\}.$

It is worth noting that model (2.1), where the response is real-valued and the
covariate is infinite dimensional, encompasses several interesting volatility
models studied in the literature. In the following, we discuss some particular
cases.

Case 1 - Parametric autoregressive model with ARCH errors: let
$\mathcal{E}=\mathbb{R}^{d}$, where $d\geq 1$, and consider
$X_{t-1}\equiv(Y_{t-1},\dots,Y_{t-d})^{\top}$, then model (2.1) becomes

$\displaystyle
Y_{t}=m(Y_{t-1},\dots,Y_{t-d})+U(Y_{t-1},\dots,Y_{t-d})\;\varepsilon_{t}.$

Moreover, when $m(X_{t-1})\equiv
m(X_{t-1};{\bm{\alpha}})=\sum_{j=1}^{q}\alpha_{j}Y_{t-j}$ and
$U(X_{t-1};{\bm{\beta}})=1+\sum_{j=1}^{d}\beta_{j}Y_{t-j}^{2}$, model (2.1)
becomes the AR-ARCH model introduced in Borkovec (2001).

Case 2 - Nonparametric autoregressive model with ARCH errors: in order to
avoid any miss-specification of the parametric form assumed on the regression
and volatility functions in the AR-ARCH model above, Fan and Yao (1998)
introduced a local linear estimator of the conditional variance function in a
time series regression model where $Y\in\mathbb{R}$ and
$\mathcal{E}=\mathbb{R}.$ Then, Laïb  (2005) investigated a local constant
estimator of the parameters in model (2.1) when $Y\in\mathbb{R}$ and
$\mathcal{E}=\mathbb{R}^{d}.$

Case 3 - Stochastic volatility model: another interesting reason to study
model (2.1) is that it includes continuous-time stochastic models which are
used to model diffusion processes. Several financial assets, say $X$, are
modeled using diffusion processes solution of the following stochastic
differential equation:

(2.3) $\displaystyle dX_{t}=\mu(X_{t})dt+\sigma(X_{t})dW_{t},\quad\quad t>0,$

where $W_{t}$ is a standard Brownian motion. The drift $\mu(\cdot)$ and the
diffusion $\sigma^{2}(\cdot)$ are in general unknown functions. Several well-
known models in financial econometrics can be written under the form (2.3)
with a specific form of drift and diffusion functions. In practice, a
diffusion process $\\{X_{t}\\}$ cannot be observed continuously over time. It
is rather observed at instants $\\{t=i\Delta|i=0,\dots,n\\},$ where $\Delta>0$
is very small. For instance the series could be observed hourly, daily, weekly
or monthly. High-frequency financial data are usually daily or intradaily
series. Following Euler discretization scheme, one gets a discretized version
of (2.3). That is

(2.4) $\displaystyle
X_{t+\Delta}-X_{t}=\mu(X_{t})\Delta+\sigma(X_{t})\Delta^{1/2}\varepsilon_{t},$

where $\\{\varepsilon_{t}\\}$ is a sequence of independent and identically
distributed standard normal random variables. Taking
$Y_{t}=X_{t+\Delta}-X_{t}$, $\mu(X_{t})\Delta=:m(X_{t}),$ and
$\sigma(X_{t})\Delta^{1/2}=:\sqrt{U(X_{t})},$ model (2.4) can be viewed as a
special case of model (2.1).

When the response variable as well as the predictor are completely observed,
Chaouch (2019) introduced nonparametric estimators of $m(\cdot)$ and
$U(\cdot)$. That is, for any $x\in\mathcal{E}$, we have

(2.5)
$m_{n,c}(x)=\frac{\displaystyle\sum_{t=1}^{n}Y_{t}K\left\\{\frac{d_{1}(x,X_{t})}{h_{n,1}}\right\\}}{\displaystyle\sum_{t=1}^{n}K\left\\{\frac{d_{1}(x,X_{t})}{h_{n,1}}\right\\}}\quad\mbox{and}\quad
U_{n,c}(x)=\frac{\displaystyle\sum_{t=1}^{n}\big{\\{}Y_{t}-m_{n}(X_{t})\big{\\}}^{2}W\left\\{\frac{d_{2}(x,X_{t})}{h_{n,2}}\right\\}}{\displaystyle\sum_{t=1}^{n}W\left\\{\frac{d_{2}(x,X_{t})}{h_{n,2}}\right\\}},$

where $K$ and $W$ are kernel functions. Sequences $h_{n,1}=h_{1}$ and
$h_{n,2}=h_{2}$ consist of positive real numbers and decrease to zero as
$n\to\infty$. Here, we consider two different semi-metrics $d_{1}$ and
$d_{2}$, respectively for the regression and the conditional variance
estimators, to measure the similarity between two functional random variables
$X_{t}$ and $X_{s}$ in $\mathcal{E}$, for $t\neq s.$

In practice, the response variable may be missing at random due to several
reasons, such as data loss, non-response, or data entry errors. Ling et al.
(2015) investigated the estimation of the regression operator $m(\cdot)$ when
the response variable is missing at random, the predictor is a completely
observed functional random variable and the error term in the model has a
constant variance. Crambes and Henchiri (2019) studied the estimation of the
regression operator under homoscedatic functional linear model. To the best of
our knowledge nothing has been done for the estimation of the variance
operator when the response is missing at random. In finite-dimensional case
(i.e. $\mathcal{E}=\mathbb{R}^{d}$, where $d\geq 1$) Pérez-Gonzàlez et al.
(2010) discussed nonparametric estimation of the conditional variance function
when response is missing at random and completely observed fixed design
predictor.

###  2.2. Missing at random assumption

In practice it is very frequent that data are not completely observed. Here,
we allow the response variable $Y_{t}$ to be missing at random at any instant
$t=1,\dots,n$, whereas the predictor $X$ is completely observed. In order to
check whether an observation is complete or missing, an indicator function
$\delta$ is introduced. Thus, $\delta_{t}=1$ if $Y_{t}$ is observed, and zero
if $Y_{t}$ is missing, for any $t=1,\dots,n.$ We suppose that the Bernoulli
random variable $\delta$ satisfies

(2.6)
$\displaystyle\mathbb{P}(\delta=1|X=x,Y=y)=\mathbb{P}(\delta=1|X=x)=:\pi(x).$

Here, $\pi:\mathcal{E}\rightarrow[0,1]$ is the conditional probability of
observing the response variable and is usually unknown. This assumption allows
to conclude that $\delta$ and $Y$ are conditionally independent given $X.$
Note that assumption (2.6) says that the response variable does not provide
additional information, on top of that given by the explanatory variable, to
predict whether an individual will present a missing response.

###  2.3. Ergodicity condition

Let $\\{Z_{n},n\in\mathbb{Z}\\}$ be a stationary sequence. Consider the
backward field $\mathcal{B}_{n}=\sigma(Z_{k};k\leq n)$ and the forward field
$\mathcal{A}_{m}=\sigma(Z_{k};k\geq m)$. The sequence is said strongly mixing
if $\sup_{A\in\mathcal{B}_{0},B\in\mathcal{A}_{n}}\Big{|}\mathbb{P}(A\cap
B)-\mathbb{P}(A)\mathbb{P}(B)\Big{|}=\varphi(n)\to 0\quad\mbox{as}\quad
n\to\infty.$ The sequence is called ergodic if
$\lim_{n\to\infty}\frac{1}{n}\sum_{t=0}^{n-1}\Big{|}\mathbb{P}\left(A\cap\tau^{-t}B\right)-\mathbb{P}(A)\mathbb{P}(B)\Big{|}=0,$
where $\tau$ is the time-evolution or shift transformation. The naming of
strong mixing in the above definition is a more stringent condition than what
is ordinarily referred (when using the vocabulary of measure-preserving
dynamical systems) to as strong mixing, namely that
$\lim_{n\to\infty}\mathbb{P}(A\cap\tau^{-n}B)=\mathbb{P}(A)\mathbb{P}(B),$ for
any two measurable sets $A,B$ (see Rosenblatt (1972)). Hence, strong mixing
implies ergodicity. However, the converse is not true: there exist ergodic
sequences which are not strong mixing. The ergodicity condition is then a
natural condition and less restrictive than any type of mixing for which usual
nonparametric estimators (density, regression, …) are convergent. It seems to
be a condition of obtaining large numbers of law, since it is well known from
the ergodic theorem that, for a stationary ergodic process $Z$, we have
$\lim_{n\to\infty}\frac{1}{n}\sum_{t=1}^{n}Z_{t}=\mathbb{E}(Z_{1}),\
\mbox{almost surely (a.s.)}.$We refer to the book of Krengel (1985) for an
account of details and results on the ergodic theory.

##  3\. Main results

In this section we define the simplified as well as the imputed estimator of
the regression and conditional variance operators. Then, we investigate their
asymptotic properties.

###  3.1. Simplified estimator

Observe that multiplying (2.1) by $\delta_{t}$ one gets
$\delta_{t}Y_{t}=\delta_{t}m(X_{t})+\delta_{t}U^{1/2}(X_{t})\varepsilon_{t}.$
Taking the conditional expectation, given $X_{t}=x$, from both sides and
making use of (2.6) one gets

$\displaystyle\mathbb{E}\bigg{\\{}\delta_{t}Y_{t}|X_{t}=x\bigg{\\}}$
$\displaystyle=$
$\displaystyle\mathbb{E}\bigg{\\{}\delta_{t}m(X_{t})|X_{t}=x\bigg{\\}}+\mathbb{E}\bigg{\\{}\delta_{t}U^{1/2}(X_{t})\varepsilon_{t}|X_{t}=x\bigg{\\}}$
$\displaystyle=$ $\displaystyle m(x)\mathbb{E}(\delta_{t}|X_{t}=x).$

Therefore, the regression operator when the response variable is missing at
random can be written, for any $x\in\mathcal{E},$ as

(3.1) $\displaystyle
m(x)=\dfrac{\mathbb{E}\left(\delta_{t}Y_{t}|X_{t}=x\right)}{\mathbb{E}(\delta_{t}|X_{t}=x)}.$

Similarly, one can write
$\delta_{t}\left(Y_{t}-m(X_{t})\right)^{2}=\delta_{t}U(X_{t})\,\varepsilon_{t}^{2}.$
Then, taking the conditional expectation given $X_{t}=x,$ for both sides of
the above equality one gets

$\mathbb{E}\left(\delta_{t}\left(Y_{t}-m(X_{t})\right)^{2}|X_{t}=x\right)=U(x)\,\mathbb{E}(\delta_{t}|X_{t}=x).$

Consequently, the variance operator is defined, for any $x\in\mathcal{E},$ as

(3.2) $\displaystyle
U(x)=\dfrac{\mathbb{E}\left(\delta_{t}\left(Y_{t}-m(X_{t})\right)^{2}|X_{t}=x\right)}{\mathbb{E}(\delta_{t}|X_{t}=x)}.$

Given a random sample $(X_{t},Y_{t},\delta_{t})_{t=1,\dots,n}$ we define a
simplified estimator of $m(x)$ and $U(x)$ defined in (3.1) and (3.2),
respectively, as:

$\displaystyle
m_{n,0}(x)=\frac{\displaystyle\sum_{t=1}^{n}\delta_{t}\,Y_{t}K\left\\{\frac{d_{1}(x,X_{t})}{h_{n,1}}\right\\}}{\displaystyle\sum_{t=1}^{n}\delta_{t}\,K\left\\{\frac{d_{1}(x,X_{t})}{h_{n,1}}\right\\}}\quad\mbox{and}\quad
U_{n,0}(x)=\frac{\displaystyle\sum_{t=1}^{n}\delta_{t}\,\big{(}Y_{t}-m_{n,0}(X_{t})\big{)}^{2}W\left\\{\frac{d_{2}(x,X_{t})}{h_{n,2}}\right\\}}{\displaystyle\sum_{t=1}^{n}\delta_{t}\,W\left\\{\frac{d_{2}(x,X_{t})}{h_{n,2}}\right\\}}.$

###  3.2. Nonparametric imputed estimator

To build a nonparametric imputed estimator of the regression operator we first
need to impute the missing values in the original response process using the
initial estimator of the regression. That is
$\widehat{Y}_{t}=\delta_{t}Y_{t}+(1-\delta_{t})m_{n,0}(X_{t}).$ Based on the
sample $(X_{t},\widehat{Y}_{t})_{t=1,\dots,n},$ a nonparametric imputed
estimator of $m(x)$ is obtained by replacing, in (2.5), $Y_{t}$ by
$\widehat{Y}_{t}.$ Thus, one gets, for any fixed $x\in\mathcal{E},$

(3.3) $\displaystyle
m_{n,1}(x)=\frac{\displaystyle\sum_{t=1}^{n}\widehat{Y}_{t}K\left\\{\frac{d_{1}(x,X_{t})}{h_{n,1}}\right\\}}{\displaystyle\sum_{t=1}^{n}K\left\\{\frac{d_{1}(x,X_{t})}{h_{n,1}}\right\\}}.$

On the other side, an imputed estimator of the conditional variance could be
obtained in two steps: first the missing residuals are imputed
nonparametrically such that
$\widehat{r}_{t}=\delta_{t}r_{t}+(1-\delta_{t})U_{n,0}(X_{t})$, where
$r_{t}=(Y_{t}-m_{n,0}(X_{t}))^{2}$ if $\delta_{t}=1$ and is not observed if
$\delta_{t}=0$. Then, based on the sample
$(X_{t},\widehat{r}_{t})_{t=1,\dots,n}$, we defined nonparametric imputed
estimator of $U(x)$, for any $x\in\mathcal{E},$ as follows:

(3.4) $\displaystyle
U_{n,1}(x)=\frac{\displaystyle\sum_{t=1}^{n}\widehat{r}_{t}{W}\left\\{\frac{d_{2}(x,X_{t})}{h_{n,2}}\right\\}}{\displaystyle\sum_{t=1}^{n}W\left\\{\frac{d_{2}(x,X_{t})}{h_{n,2}}\right\\}}.$

For $k\in\\{1,2\\}$ , let
$F_{x,k\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}(u)=\mathbb{P}(d_{k\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}(x,X)\leq
u)=\mathbb{P}(X\in$ $B(x,u))$ and
$F_{x,k}^{\mathcal{F}_{t-1}}(u)=\mathbb{P}\left(d_{k}(x,X)\leq
u\mid\mathcal{F}_{t-1}\right)=\mathbb{P}\left(X\in
B(x,u)\mid\mathcal{F}_{t-1}\right)$ be, respectively, the marginal
distribution and the conditional marginal distribution of $X$ given the
$\sigma$-field $\mathcal{F}_{t-1}.$

Our main results consist in establishing the almost sure uniform consistency
with rate of the simplified and the imputed estimator of the regression
operator and the conditional variance. For this purpose, let us denote by
$\mathcal{C}$ a class of elements in the functional space $\mathcal{E}$ and
consider, for any $\eta>0$,

$\begin{array}[]{ll}N\left(\eta,\mathcal{C},d_{\mathcal{C}}\right)=\min\\{n:&\text{there
exist}\quad c_{1},\ldots,c_{n}\in\mathcal{C}\quad\text{such that}\quad\forall
x\in\mathcal{C}\\\ &\text{there exists}\quad
k\in\\{1,\ldots.,n\\}\quad\text{such that}\quad
d_{\mathcal{C}}\left(x,c_{k}\right)<\eta\\},\end{array}$

a number which measures how full is the class $\mathcal{C}.$ Our results are
established under the following assumptions listed below. In the sequel, we
use $\mathcal{K}$ to either denote the kernel $K$ or $W.$

  * (A1)

    * (i)

$\mathcal{K}$ is a nonnegative bounded kernel of class $\mathcal{C}^{1}$ over
its support $[0,1]$ with $\mathcal{K}(1)>0$. The derivative
$\mathcal{K}^{\prime}$ exists on $[0,1)$ and satisfies the condition
$\mathcal{K}^{\prime}(v)<0\text{ for all }v\in[0,1)\text{ and }$

$\left|\displaystyle\int_{0}^{1}\left(\mathcal{K}^{j}\right)^{\prime}(v)dv\right|<\infty\text{
for }j=1,2.$

    * (ii)

$\mathcal{K}$ is a Hölder function of order $\gamma$ with a constant $a_{0}$.

    * (iii)

There exist constants $a_{1}$ and $a_{2}$ such that
$0<a_{1}\leq\mathcal{K}(v)\leq a_{2}<\infty$ for all $v\in\mathcal{C}$.

  * (A2)

For $x\in\mathcal{E}$, there exists a sequence of nonnegative random variables
$(f_{t,1})_{i\geq 1}$ almost surely bounded by a sequence of deterministic
quantities ($b_{t}(x))_{t\geq 1}$, a sequence of random functions
$(\psi_{t,x})_{t\geq 1}$, a deterministic nonnegative bounded function $f_{1}$
and a nonnegative real function $\phi(.)$ that tend to zero, as its argument
tends to 0,

    * (i)

$F_{x,k}(u)=\phi_{k}(u)f_{1}(x)+o(\phi_{k}(u))\text{ as }u\rightarrow 0$ and
$k\in\\{1,2\\},$ where $o(\phi_{k}(u))$ is uniform in $x$.

    * (ii)

For $k\in\\{1,2\\}$ and $\forall t\in
N,F_{x,k}^{\mathcal{F}_{t-1}}(u)=\phi_{k}(u)f_{t,1}(x)+\psi_{t,x}(u)$ with
$\psi_{t,x}(u)=o_{a.s}(\phi_{k}(u))$ as $u\rightarrow
0,\frac{\psi_{t,x}(u)}{\phi_{k}(u)}$ almost surely bounded for any
$x\in\mathcal{C}$ and
$n^{-1}\displaystyle\sum_{t=1}^{n}\psi_{t.x}(u)=o_{a.s}\left(\phi_{k}(u)\right)$
as $n\rightarrow\infty$ and $u\rightarrow 0$. where
$o_{a.s}\left(\phi_{k}^{j}(u)\right)$ is uniform in $x$.

    * (iii)

$\displaystyle{\lim_{n\rightarrow\infty}\sup_{x\in\mathcal{C}}}\left|f_{1}(x)-n^{-1}\sum_{t=1}^{n}f_{t,1}(x)\right|=0$
almost surely.

    * (iv)

For $k\in\\{1,2\\}$, there exists a nondecreasing bounded function
$\tau_{0,k}$ such that, uniformly in $s\in[0,1]$,
$\frac{\phi_{k}(hu)}{\phi_{k}(h)}=\tau_{0,k}(u)+o(1)$, as $h\downarrow 0$, and
we have

$\displaystyle\int_{0}^{1}\left({\mathcal{K}}^{j}(u)\right)^{\prime}\tau_{0,k}(u)dt<\infty$
for $j\geq 1$.

    * (v)

$n^{-1}\sum_{t=1}^{n}b_{t}(x)\rightarrow D(x)<\infty$ as $n\rightarrow\infty$,
and $0<\displaystyle{\sup_{x\in\mathcal{C}}}D(x)<\infty$.

    * (vi)

$\displaystyle{0<\theta_{0}\leq\inf_{x\in\mathcal{C}}f_{1}(x)\leq\sup_{x\in\mathcal{C}}f_{1}(x)<\infty}$
for some nonnegative real number $\theta_{0}$.

    * (vii)

$\displaystyle{\inf_{x\in\mathcal{C}}}\pi(x)>\theta_{1}$ for some positive
real number $\theta_{1}\in[0,1].$

  * (A3)

    * (i)

There exists $\rho>1$ such that
$\mathbb{E}\\{U(X_{t})^{\rho^{2}/2(\rho-1)}\\}<\infty$ and $\underset{1\leq
t\leq n}{\max}\mathbb{E}(|\varepsilon_{t}|^{\rho}|\mathcal{G}_{t-1})<\infty$.

    * (ii)

$\mathbb{E}\\{(\varepsilon^{2}_{t}-1)^{2}|X_{t})=\mathbb{E}\\{(\varepsilon^{2}_{t}-1)^{2}|\mathcal{G}_{t-1}\\}=\omega(X_{t})$
is continuous in a neighborhood of $x$ as $h\rightarrow 0,$ that is

$\sup_{\\{u:d(x,u)\leq h\\}}|\omega(u)-\omega(x)|=o(1).$

    * (iii)

For some $\kappa>0$, $\underset{1\leq t\leq
n}{\max}\mathbb{E}\\{(\varepsilon^{2}_{t}-1)^{2+\kappa}|\mathcal{G}_{t-1}\\}<\infty$.

    * (iv)

$\mathbb{E}\\{|U(X_{t})|^{\rho^{2}/(\rho-1)}\\}<\infty$ and $\underset{1\leq
t\leq
n}{\max}\mathbb{E}(|\varepsilon_{t}^{2}-1|^{\rho\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}|\mathcal{G}_{t-1})<\infty$.

    * (v)

For any $t\geq 1$,
$\mathbb{E}(\delta_{t}|\mathcal{G}_{t-1})=\mathbb{E}(\delta_{t}|X_{t})=\pi(X_{t})$
is continuous in a neighborhood of $x$ as $h\rightarrow 0,$ that is

$\sup_{\\{u:d(x,u)\leq h\\}}|\pi(u)-\pi(x)|=o(1),\text{where o(1) is uniformly
in $x$}.$

  * (A4)

    * (i)

For any $(u,v)\in\mathcal{E}^{2}$, $|m(u)-m(v)|<c_{1}d_{1}^{\alpha}(u,v)$, for
some constant $c_{1}>0$ and $\alpha>0,$

    * (ii)

For any $(u,v)\in\mathcal{E}^{2}$, $|U(u)-U(v)|<c_{2}d_{2}^{\beta}(u,v)$, for
some constant $c_{2}>0$ and $\beta>0.$

    * (iii)

$U^{\kappa+2}(.)$ is continuous in a neighborhood of $x$ as $h\rightarrow 0$,
that is

$\sup_{\\{u:d(x,u)\leq h\\}}|U^{\kappa+2}(u)-U^{\kappa+2}(x)|=o(1).$

Assumption (A1)(i) is related to the choice of the kernel $\mathcal{K}$, which
is very usual in nonparametric functional estimation. Notice that Parzen
symmetric kernel is not adequate in this context since the random process
$d(x,X_{t})$ is positive, therefore we consider $\mathcal{K}$ with support [0,
1]. This is a natural generalization of the assumption usually made on the
kernel in the multivariate case where $\mathcal{K}$ is supposed to be a
spherically symmetric density function. The assumption $\mathcal{K}(1)>0$ and
$\mathcal{K}^{\prime}<0$ guarantee that $M_{1,W,2}>0$ for all limit functions
$\tau_{0}.$ The condition $\mathcal{K}(1)>0$ is needed to define the moments
$M_{j,W,2}$ which are, in this case, determined by the value $\mathcal{K}(1).$
Assumption (A1)(ii) is a Hölder-type that requires a certain smoothness of the
kernels and assumption (A1)(iii) requires that the kernel $\mathcal{K}$ has to
be bounded away from 0 which is relatively usual in nonparametric functional
data estimation.

Conditions (A2)(i)-(ii) reflect the ergodicity property assumed on the
discrte-time functional process. It plays an important role in studying the
asymptotic properties of the estimator. The functions $f_{t,1}$ and $f_{1}$
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 of processes
to satisfy these conditions are given in Laïb and Louani (2010). Conditions
(A2)(iii) and (A2)(v) are basically established to meet the ergodic Theorem.
Assumptions (A3)(i), (A3)(iii) and (A3)(iv) are more technical and require
boundness of high order moments of the errors and conditional variance.
(A3)(ii) and (A3)(v) assume the continuity of the operators $\pi(\cdot)$ and
$\omega(\cdot).$ Finally, assumption (A4) imposes some smoothness of the
regression and conditional variance operators.

###  3.3. Asymptotic properties of the simplified estimator

In this subsection we investigate asymptotic properties of the simplified
estimators. That includes a uniform consistency rate, asymptotic distribution
and confidence intervals.

####  3.3.1. Uniform consistency

######  Theorem 3.1.

Suppose that assumptions (A1), (A2), (A3)(i),(v), (A4)(i) hold true and the
following conditions

(3.5) $\lim_{n\rightarrow\infty}n\phi_{1}\left(h_{1}\right)=\infty\,\
\hbox{and}\,\
\lim_{n\rightarrow\infty}\left\\{n\phi_{1}\left(h_{1}\right)\right\\}^{-1}\log
n=0$

are satisfied. In addition, for a sequence of positive real numbers
$\lambda_{n}$ tending to zero, as $n\rightarrow\infty$ and
$\eta=\eta_{n}=o(h_{1})$, assume that, we have

(3.6) $\displaystyle\lim_{n\rightarrow\infty}\frac{\log
N\left(\eta,\mathcal{C},d_{\mathcal{C}}\right)}{n\lambda_{n}^{2}\phi_{1}\left(h_{1}\right)\ell_{n}^{-2}}=0$
and $\displaystyle\sum_{n\geq
1}\exp\left[-\lambda_{n}^{2}\ell_{n}^{-2}\mathcal{O}\left\\{n\phi_{1}\left(h_{1}\right)\right\\}\right]<\infty\text{,}$

where $\ell_{n}$ is a sequence of positive numbers that tends to infinity as
$n\rightarrow\infty$ defined by
$\ell_{n}=\left(\frac{\log
n}{\lambda_{n}\phi_{1}(h_{1})^{(\rho-1)/\rho}}\right)^{1/(\rho-1)},\rho$ is
given in (A3)(i). Then, one gets

$\sup_{x\in\mathcal{C}}\left|m_{n,0}(x)-m(x)\right|=\mathcal{O}_{a.s.}\left(h_{1}^{\alpha}\right)+\mathcal{O}_{a\cdot
s.}\left(\lambda_{n}\right),$

where $\alpha$ is given in (A4)(i).

######  Remark 3.1.

The uniform consistency rate for the regression operator is identical to the
one derived in Chaouch (2019) when the response variable is completely
observed. In addition, the same rate has been achieved in Laïb and Louani
(2010) for the homoscedastic scalar-on-function regression model. Furthermore,
note that if $\lambda_{n}=\mathcal{O}\left(\sqrt{\log
n/(n\phi_{1}(h_{1})})\right)$, condition (3.6) is met, and hence the uniform
consistency rate for $m_{n,0}(x)$ is
$\mathcal{O}(h_{1}^{\alpha})+\mathcal{O}\left(\sqrt{\log
n/(n\phi_{1}(h_{1})})\right)$ which is the same rate derived in Ferraty and
Vieu (2006) when the data are independent.

######  Theorem 3.2.

Assume that assumptions (A1), (A2), (A3)(i),(iv)-(v), (A4)(i)-(ii), conditions
(3.5)-(3.6), and the following conditions are satisfied

(3.7) $\lim_{n\rightarrow\infty}n\phi_{2}\left(h_{2}\right)=\infty\,\
\hbox{and}\,\
\lim_{n\rightarrow\infty}\left\\{n\phi_{2}\left(h_{2}\right)\right\\}^{-1}\log
n=0.$

Moreover, for a sequence of positive real numbers $\lambda_{n}^{\prime}$
tending to zero as $n\rightarrow\infty$, suppose that we have

(3.8) $\displaystyle\lim_{n\rightarrow\infty}\frac{\log
N\left(\eta,\mathcal{C},d_{\mathcal{C}}\right)}{n\left(\lambda_{n}^{\prime}\right)^{2}\phi_{2}\left(h_{2}\right)\left(\ell_{n}^{\prime}\right)^{-2}}=0$
and $\displaystyle\sum_{n\geq
1}\exp\left[-\left(\lambda_{n}^{\prime}\right)^{2}\left(\ell_{n}^{\prime}\right)^{-2}\mathcal{O}\left\\{n\phi_{2}\left(h_{2}\right)\right\\}\right]<\infty,$

where $\ell_{n}^{\prime}$ is a sequence of positive numbers that goes to
$\infty$, as $n\rightarrow\infty$ defined by
$\ell_{n}^{\prime}=\left(\frac{\log
n}{\lambda_{n}^{\prime}\phi_{2}(h_{2})^{(\rho-1)/\rho}}\right)^{1/(\rho-1)},\rho$
is given in (A3)(i). Then, one has

$\sup_{x\in\mathcal{C}}\left|U_{n,0}(x)-U(x)\right|=\mathcal{O}_{a.s.}\left(h_{1}^{2\alpha}+h_{2}^{\beta}\right)+\mathcal{O}_{a\cdot
s.}\left(\lambda_{n}^{\prime}+\lambda_{n}^{2}\right),$

where $\beta$ is given in (A4)(ii).

######  Remark 3.2.

The initial estimator $U_{n,0}(x)$ has the same uniform almost sure
consistency rate obtained in Chaouch (2019) where response variable is
completely observed. Moreover, note that by considering $h_{1}=h_{2}=h$,
$d_{1}(\cdot,\cdot)=d_{2}(\cdot,\cdot)$, $\alpha=\beta$,
$\phi_{1}(\cdot)=\phi_{2}(\cdot)=\phi(\cdot)$, and
$\lambda_{n}=\lambda_{n}^{\prime}=\mathcal{O}\left(\sqrt{\log
n/(n\phi(h)})\right)$, condition (3.7) is satisfied. Therefore, the uniform
convergence rate of $U_{n,0}(x)$ is
$\mathcal{O}(h^{\beta})+\mathcal{O}\left(\sqrt{\log n/(n\phi(h)})\right)$.
Furthermore, in the finite-dimensional case (i.e.
$\mathcal{E}=\mathbb{R}^{d}$), the rate becomes
$\mathcal{O}(h^{\beta})+\mathcal{O}\left(\sqrt{\log n/nh^{d}}\right)$ which
matches with the rate obtained in Laïb  (2005) for a nonlinear autoregressive
model with ARCH errors.

####  3.3.2. Asymptotic distribution

######  Theorem 3.3.

Assume that assumptions (A1)-(A4) and conditions (3.5)-(3.8) hold true then

$\sqrt{n\phi_{2}\left(h_{2}\right)}\left\\{U_{n,0}(x)-U(x)\right\\}\overset{\mathcal{D}}{\longrightarrow}\mathcal{N}\left(0,\sigma_{0}^{2}(x)\right),$

where $\mathcal{N}(\cdot,\cdot)$ denotes the normal distribution and
$\sigma_{0}^{2}(x)=\frac{M_{2,W,2}U^{2}(x)\omega(x)}{M_{1,W,2}^{2}\pi(x)f_{1}(x)}$

with

(3.9)
$M_{j,W,2}=W^{j}(1)-\int_{0}^{1}\left(W^{j}\right)^{\prime}(u)\tau_{0,2}(u)du,\,\
for\,\ j=1,2.$

Theorem 3.3 represents an extension of the asymptotic distribution established
in Theorem 3 in Chaouch (2019) where data are completely observed. Note that
the asymptotic conditional variance depends on the conditional probability of
observing data $\pi(x)$. Higher (resp. smaller) is the MAR rate (i.e. smaller
(resp. higher) is $\pi(x)$) higher (resp. smaller) will be
$\sigma_{0}^{2}(x)$. Therefore, less (resp. more) efficient will be the
initial estimator. If the data are completely observed (i.e. $\pi(x)=1,\forall
x\in\mathcal{E}$), then $\sigma_{0}^{2}(x)$ coincides with the asymptotic
conditional variance obtained in Chaouch (2019).

####  3.3.3. Asymptotic confidence intervals

Our purpose here is to build asymptotic confidence intervals for $U(x)$ for
any fixed curve $x\in\mathcal{E}$ based on a normal approximation. Note that
the asymptotic variance in Theorem 3.3 contains the unknown quantities which
are $\pi(\cdot),$ $\omega(\cdot),$ $U(\cdot),$ $M_{1,W,2},$ $M_{2,W,2}$ and
$\tau_{0,2}(u).$ Each of these parameters is replaced by its empirical
version. We replace $U(x)$ by $U_{n,0}(x)$ and $\omega(x)$ and $\pi(x)$ are
estimated as follows:

$\omega_{n}(x)=\frac{\displaystyle\sum_{t=1}^{n}(\varepsilon^{2}_{n,t}-1)^{2}H\left(\frac{d_{3}(x,X_{t})}{h_{3}}\right)}{\displaystyle\sum_{t=1}^{n}H\left(\frac{d_{3}(x,X_{t})}{h_{3}}\right)},$

where $\varepsilon_{n,t}=[Y_{t}-m_{n,0}(X_{t})]/[U_{n,0}(X_{t})]^{1/2}$ and

$\pi_{n}(x)=\frac{\displaystyle\sum_{t=1}^{n}\delta_{t}\tilde{H}\left(\frac{d_{4}(x,X_{t})}{h_{4}}\right)}{\displaystyle\sum_{t=1}^{n}\tilde{H}\left(\frac{d_{4}(x,X_{t})}{h_{4}}\right)},$

where $H$ and $\tilde{H}$ are kernel functions, $d_{3}(\cdot,\cdot)$ and
$d_{4}(\cdot,\cdot)$ are semi-metrics, and $h_{3}$ and $h_{4}$ are bandwidths
used to estimate $\omega(x)$ and $\pi(x)$. Moreover, by assumptions (A2)(i)
and (A2)(iv), we can define the estimator of $\tau_{0,2}$ as
$\widehat{\tau}_{0,2}(u)=\widehat{F}_{x,2}(uh_{2})/\widehat{F}_{x,2}(u),$
where
$\widehat{F}_{x,2}(u)=n^{-1}\sum_{t=1}^{n}\mathds{1}_{\\{d_{2}(x,X_{t})\leq
u\\}}.$ Further, we replace $\tau_{0,2}$ by its estimator
$\widehat{\tau}_{0,2}(u)$ in (3.9) to obtain a plug-in estimator
$\widehat{M}_{j,W,2}$ of $M_{j,W,2}$, for $j=\\{1,2\\}.$

######  Corollary 3.1.

Under the conditions of Theorem 3.3, we have

(3.10)
$\left\\{\frac{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widehat{M}_{1,W,2}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sqrt{\widehat{M}_{2,W,2}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}}\sqrt{\frac{n\widehat{F}_{x,2}(h)\pi_{n}(x)}{\omega_{n}(x)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}U^{2}_{n,0}(x)}}\right\\}\left\\{U_{n,0}(x)-U(x)\right\\}\overset{\mathcal{D}}{\longrightarrow}\mathcal{N}\left(0,1\right),\,\
\hbox{as}\,\ n\rightarrow\infty.$

This corollary plays a key role in estimating confidence intervals of $U(x)$.
More precisely, equation (3.10) leads to obtain the following asymptotic
$100(1-\nu)\%$ confidence interval for the conditional variance $U(x)$

(3.11) $\mathrm{CI}^{S}_{\nu}=U_{n,0}(x)\left\\{1\pm
q_{\nu/2}\frac{\sqrt{\widehat{M}_{2,W,2}}}{\widehat{M}_{1,W,2}}\sqrt{\frac{\omega_{n}(x)}{n\widehat{F}_{x,2}(h)\pi_{n}(x)}}\right\\}\text{,
}$

where $q_{\nu/2}$ is the upper $\nu/2$ quantile of the standard normal
distribution.

The confidence interval displayed in (3.11) reveals that higher is the missing
rate ($\pi(x)\rightarrow 0$) wider will be the confidence interval and less
accurate is the interval estimation.

###  3.4. Asymptotic properties of the nonparametric imputed estimator

In this subsection we investigate asymptotic properties of the nonparametric
imputed estimator of the regression and conditional variance operators.

####  3.4.1. Uniform consistency

######  Theorem 3.4.

Under the same conditions of Theorem 3.1, we have

$\sup_{x\in\mathcal{C}}\left|m_{n,1}(x)-m(x)\right|=\mathcal{O}_{a.s.}\left(h_{1}^{\alpha}\right)+\mathcal{O}_{a\cdot
s.}\left(\lambda_{n}\right),$

where $\alpha$ is given in (A4)(i).

######  Theorem 3.5.

Under the same conditions of Theorem 3.2, we obtain

$\sup_{x\in\mathcal{C}}\left|U_{n,1}(x)-U(x)\right|=\mathcal{O}_{a.s.}\left(h_{1}^{2\alpha}+h_{2}^{\beta}\right)+\mathcal{O}_{a\cdot
s.}\left(\lambda_{n}^{\prime}+\lambda_{n}^{2}\right),$

where $\beta$ is given in (A4)(ii).

######  Remark 3.3.

The almost sure uniform convergence rates of the estimators $m_{n,1}(x)$ and
$U_{n,1}(x)$ remain the same as those obtained in Theorems 3.1 and 3.2,
respectively.

####  3.4.2. Asymptotic distribution

######  Theorem 3.6.

Suppose that conditions of Theorem 3.3 are satisfied. Then

$\sqrt{n\phi_{2}\left(h_{2}\right)}\left\\{U_{n,1}(x)-U(x)\right\\}\overset{\mathcal{D}}{\longrightarrow}\mathcal{N}\left(0,\sigma^{{}^{\prime}2}(x)\right),$

where
$\sigma^{{}^{\prime}2}(x)=\frac{M_{2,W,2}U^{2}(x)\omega(x)\pi(x)}{M_{1,W,2}^{2}f_{1}(x)}=\sigma_{0}^{2}(x)+\frac{M_{2,W\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0},2}U^{2}(x)(\pi^{2}(x)-1)\omega(x)}{\pi(x)M_{1,W,2}^{2}f_{1}(x)}.$

Note that when $\pi(x)\rightarrow 1$ (low MAR rate) $\sigma^{{}^{\prime}2}(x)$
and $\sigma_{0}^{2}(x)$ will converge to the asymptotic conditional variance
obtained in Chaouch (2019) in the case of complete data. On the other side,
for high MAR rate ($\pi(x)\rightarrow 0$) one observes that
$\sigma_{0}^{2}(x)\rightarrow\infty$ while
$\sigma^{{}^{\prime}2}(x)\rightarrow 0.$ In other words nonparametric imputed
estimator is more efficient that the simplified one when the missing rate is
high.

####  3.4.3. Asymptotic confidence intervals

Similar to the simplified estimator, we use the following corollary to build
asymptotic confidence intervals for $U(x).$

######  Corollary 3.2.

Under the conditions of Theorem 3.6, we have

(3.12)
$\left\\{\frac{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widehat{M}_{1,W,2}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sqrt{\widehat{M}_{2,W,2}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}}\sqrt{\frac{n\widehat{F}_{x,2}(h)}{\omega_{n}(x)\pi_{n}(x)U^{2}_{n,1}(x)}}\right\\}\left\\{U_{n,1}(x)-U(x)\right\\}\overset{\mathcal{D}}{\longrightarrow}\mathcal{N}\left(0,1\right),\,\
\hbox{as}\,\ n\rightarrow\infty.$

Then, an asymptotic $100(1-\nu)\%$ confidence interval for $U(x)$ is

(3.13) $\mathrm{CI}^{NI}_{\nu}=U_{n,1}(x)\left\\{1\pm
q_{\nu/2}\frac{\sqrt{\widehat{M}_{2,W,2}}}{\widehat{M}_{1,W,2}}\sqrt{\frac{\omega_{n}(x)\pi_{n}(x)}{n\widehat{F}_{x,2}(h)}}\right\\},$

where $q_{\nu/2}$ is the upper $\nu/2$ quantile of the of the standard normal
distribution.

A comparison between the simplified-based and the nonparametric imputation-
based confidence intervals given in equations (3.11) and (3.13), we can
observe that higher is the missing rate ($\pi(x)\rightarrow 0$), larger will
be $\mathrm{CI}_{\nu}^{S}$ compared to $\mathrm{CI}_{\nu}^{NI}.$ In other
words, missing data imputation allows to obtain more accurate interval
estimation of $U(x)$.

##  4\. Finite sample properties

In this section we carry out a simulation study to assess the quality of the
proposed estimation methods. Let us consider
$(X_{t},Y_{t},\delta_{t})_{t=1,\cdots,n}$ to be a strict stationary process
such that, for any $t=1,\dots,n,$ the functional covariate
$\\{X_{t}(\lambda):\lambda\in[-1,1]\\}$ is sampled at $100$ equally spaced
points in $[-1,1]$, and generated as follows:

$X_{t}(\lambda)=A(2-\cos(\pi\lambda\omega))+(1-A)\cos(\pi\lambda\omega),$

where $\omega\sim\mathcal{N}(0,1)$,
$A\sim\text{Bernoulli}\bigg{(}\dfrac{1}{2}\bigg{)}.$ A sample of $n=100$
simulated curves is displayed in Figure 2.

Figure 2. A sample of simulated curves $X_{t}(\lambda)$.

The response variable is generated according to the following heteroscedastic
functional regression model:

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

where the regression and variance operators are defined, at any fixed point
$x$, as

(4.1) $m(x)=\int_{-1}^{1}\lambda x(\lambda)d\lambda,\quad
U(x)=\int_{-1}^{1}|\lambda|x^{2}(\lambda)d\lambda.$

Regarding the errors $\varepsilon_{t}$ we suppose that they are generated
according to one of the following models:

Model 1: The $\varepsilon_{t}$’s are i.i.d, distributed according to
$\mathcal{N}(0,1)$.
Model 2: $\varepsilon_{t}=\dfrac{1}{2}\varepsilon_{t-1}+\xi_{t}$, where
$\xi_{t}\sim\mathcal{N}(0,1)$.
Model 3: $\varepsilon_{t}=-\dfrac{1}{2}\varepsilon_{t-1}+\xi_{t}$, where
$\xi_{t}\sim\mathcal{N}(0,1)$.
Model 4: $\varepsilon_{t}=\dfrac{1}{2}\varepsilon_{t-1}+\xi_{t}$, where
$\xi_{t}\sim\texttt{Bernoulli}\bigg{(}\dfrac{1}{2}\bigg{)}$.

We consider here four different models corresponding to three different
dependence structures. Indeed, Model 1 corresponds to the case where data are
independent and identically distributed. Models 1 and 2 cover the case where
the process is $\alpha$-mixing. Finally, Model 4 is an example of ergodic but
not mixing process. Figure 3 shows the generated response variable for each of
the four models above.

(a)

(b)

(c)

(d)

Figure 3. The generated process $Y_{t}$ for Model 1 (a), Model 2 (b), Model 3
(c) and Model 4 (d).

We suppose that missing at random observations in the response variable $Y$
are generated according to the following conditional probability distribution:

$\pi(x)=\mathbb{P}(\delta=1|X=x)=\text{expit}\bigg{(}2\eta\int_{-1}^{1}x^{2}(\lambda)d\lambda\bigg{)},$

where $\text{expit}(u)=\dfrac{e^{u}}{1+e^{u}}$ and $\eta\in\\{0.2,0.8\\}$.

Observe that, according to the value of $\eta,$ one gets a different MAR rate.
Higher is the value of $\eta$, higher will be $\pi(x)$. Therefore, the smaller
will be the missing data rate. Indeed, when $\eta=0.8$ the MAR rate will be
$20\%$ and $60\%$ for $\eta=0.2$. Figure 4 shows an example of the process
$Y_{t}$ that is affected by MAR mechanism for $\eta=0.2$ and $0.8$,
respectively.

(a)

(b)

Figure 4. Missing at random in the generated process $Y_{t}$ when $\eta=0.2$
(a) and $0.8$ (b).

Regarding the tuning parameters used to calculate the simplified estimator we
consider the quadratic kernel $K(u)=\frac{3}{2}(1-u^{2})1\kern-1.99997pt{\rm
l}_{[0,1]}(u).$ The choice of the bandwidth is based on the cross-validation
criterion based on the $\kappa$-nearest neighbors as detailed in Ferraty and
Vieu (2006). Because of the smoothness of the curves $X_{t}(\lambda)$, we
consider a semi-metric, for the regression and the conditional variance
functions, as the usual $L_{2}$-norm of the first derivatives of the curves,
which is defined as follows:

$d(X_{t},X_{s})=\bigg{[}\int_{-1}^{1}\bigg{\\{}X^{(1)}_{t}(\lambda)-X^{(1)}_{s}(\lambda)\bigg{\\}}^{2}d\lambda\bigg{]}^{1/2},\quad\forall
t\neq s.$

Our purpose is to estimate the conditional variance at a fixed curve

$x_{0}(\lambda)=\cos(\pi\lambda/4)\qquad\text{for}\quad\lambda\in[-1,1].$

To assess the consistency of the estimator, we consider $B=500$ replications
and for each replication we estimate the conditional variance and evaluate the
square error. That is, at each replication $b\in\\{1,\dots B\\}$, we calculate

$\text{SE}_{b}=\left(\mathcal{U}_{n,b}(x_{0})-U(x_{0})\right)^{2},$

where $\mathcal{U}_{n,b}(x_{0})$ denotes either the complete, simplified, or
nonparametric imputation conditional variance estimator obtained at
replication $b.$

Tables LABEL:tab:MSE20 and LABEL:tab:MSE60 display some summary statistics of
the square errors obtained for each model with missing at random rate of
$20\%$ and $60\%$, respectively. We can see that the estimator obtained after
missing data imputation provides better results. Moreover, one can observe
that the higher the MAR rate, the lower the quality of estimation. Finally,
one can notice that the dependence structure in the data plays an important
role in determining the quality of estimation. Indeed, small errors are
obtained in model 1 (corresponding to the i.i.d case), then the higher is the
dependence structure in the data the higher will be the estimation errors.
Absolute errors obtained for model 2 and 3 (corresponding to $\alpha$-mixing
processes) are higher than the i.i.d. case. However, the highest errors are
obtained for the non-mixing but ergodic process given in model 4.

Table 1. Quartiles of the $(\text{SE}_{b})_{b=1,\dots,B}$ obtained for each model when MAR = 20% Model |  Comp($\times 10^{-2}$) |  Simp.($\times 10^{-2}$) |  NPI.($\times 10^{-2}$)
---|---|---|---
| $Q_{25\%}$ | $Q_{50\%}$ | $Q_{75\%}$ | $Q_{25\%}$ | $Q_{50\%}$ | $Q_{75\%}$ | $Q_{25\%}$ | $Q_{50\%}$ | $Q_{75\%}$
Model1 | 0.4 | 3.57 | 11.48 | 0.58 | 5.06 | 14.06 | 0.51 | 4.54 | 11.02
Model2 | 0.85 | 3.88 | 19.79 | 0.97 | 5.88 | 23.77 | 0.94 | 4.46 | 19.42
Model3 | 0.91 | 4.00 | 22.32 | 0.97 | 8.24 | 26.87 | 0.99 | 5.56 | 20.92
Model4 | 0.36 | 6.79 | 19.22 | 0.38 | 7.03 | 21.10 | 0.34 | 6.95 | 19.67
Table 2. Quartiles of the $(\text{SE}_{b})_{b=1,\dots,B}$ obtained for each model when MAR = 60%. Model |  Comp($\times 10^{-2}$) |  Simp.($\times 10^{-2}$) |  NPI.($\times 10^{-2}$)
---|---|---|---
| $Q_{25\%}$ | $Q_{50\%}$ | $Q_{75\%}$ | $Q_{25\%}$ | $Q_{50\%}$ | $Q_{75\%}$ | $Q_{25\%}$ | $Q_{50\%}$ | $Q_{75\%}$
Model1 | 1.25 | 4.03 | 21.97 | 1.41 | 4.62 | 24.73 | 1.25 | 4.29 | 21.67
Model2 | 1.78 | 7.50 | 35.46 | 2.27 | 8.42 | 39.84 | 2.04 | 7.42 | 35.9
Model3 | 2.10 | 5.07 | 39.10 | 2.63 | 6.93 | 44.81 | 2.12 | 6.08 | 39.56
Model4 | 2.21 | 7.38 | 46.04 | 2.28 | 7.85 | 46.59 | 2.24 | 7.55 | 46.38

##  5\. Application to high-frequency financial data

In this section we are interested in estimating and forecasting the volatility
of the daily log return of Brent crude oil closing price given the intraday
(1-minute frequency) curve of natural gas closing price.

The relationship between Brent crude oil and natural gas prices has been a
topic of interest to researchers and market practitioners for many years. In
particular, the volatility of oil prices has been a major concern for
investors and market participants. A number of studies have investigated the
impact of natural gas prices on the volatility of oil prices, with the aim of
developing models that can better capture this relationship. Liu et al. (2013)
used Vector AutoRegressive (VAR) model to examine the relationship between
crude oil and natural gas prices in North America. Another study by Chen et
al. (2019) used a BEKK-GARCH model to investigate the volatility spillovers
between crude oil and natural gas prices in the United States. Moreover, Aloui
et al. (2013) used a copula-GARCH model to examine the relationship between
Brent oil and natural gas prices. The authors found that the volatility of
Brent oil prices was indeed affected by natural gas prices, and that their
model outperformed other traditional GARCH models in terms of forecasting
accuracy. The authors found that the relationship between the two prices
varies depending on the market regime, and suggested that this could be
important for risk management purposes.

###  5.1. Data preliminary analysis

The data cover trading days from February 14, 2020 to February 14, 2023.
Figure 5 displays a 1-day frequency time series of Brent crude oil and natural
gas closing prices. One can see that overall there is a correlation between
the two prices. Figure 6 shows that the most likely prices for the Brent oil
and the natural gas do not in general exceed $80 and $5, respectively. While
Brent crude oil closing price, say $P_{t}^{o},$ is observed at a daily
frequency from February 14, 2020 to February 14, 2023, the natural gas closing
price, say $P_{m}^{g},$ is observed every minute over the same period. Returns
of Brent crude oil are calculated as follows:
$r_{t}^{o}:=\log(P_{t}^{o}/P_{t-1}^{o})$, where $P_{t}^{o}$ is the daily
closing price at day $t$ of the Brent crude oil. Similarly, the 1-minute
frequency returns of natural gas are obtained according to the following
formula: $r_{m}^{g}:=\log(P_{m}^{g}/P_{m-1}^{g})$, where $P_{m}^{g}$ is the
price of natural gas observed at a minute $m.$

Figure 5. Closing Price for Brent Crude Oil and Natural Gas. Figure 6. Joint
density estimation of daily Brent oil and Natural Gas prices.

###  5.2. Random sample construction

Our sample here is denoted as follows: $(X_{t},Y_{t})_{t=1,\cdots,1096}$,
where the sample size $n=1096$ is the total number of trading days from
February 14, 2020 to February 14, 2023. Note that $Y_{t}=r_{t}^{o}$ and the
functional-valued process is build as follows:

$X_{t}(m)=r_{t}^{g}\left(m+(t-1)\times
1439\right),\quad\quad\text{for}\;t=1,\dots,n,\;\text{and}\;\forall
m\in[1,1439].$

Figure 7(a) displays a sample of three 1-minute frequency curve of the natural
gas. Figure 7(b) shows all intraday (1-minute frequency) curves from February
14, 2020 to February 14, 2023. One can observe that the price of natural gas
reached its highest values since the beginning of the Russo-Ukrainian war in
February 2022.

(a)

(b)

Figure 7. (a) Sample of three intraday (1-minute frequency) Natural Gas price
curves. (b) All historical intraday (1-minute frequency) Natural Gas price
curves.

Observe that the data are initially completely observed. Therefore, we
artificially generate missing observations in order to validate our
methodology. We assume that the missing at random mechanism is generated
according to the following probability distribution:

$\pi(x)=\mathbb{P}(\delta=1\mid
X=x)=\operatorname{expit}\left(\int_{1}^{1439}x^{2}(m)dm\right),$

This choice of probability distribution leads to 12% of missing observations
in the oil price return process.

###  5.3. Daily Brent oil return volatility estimation and forecasting

Our purpose is to estimate and forecast the daily volatility of Brent Oil
price log-return using as predictor the intraday (1-minute) frequency log
return of natural gas. We split the original sample into training (In-Sample)
and testing (Out-Of-Sample) subsamples. The training subsample is selected
from February 14, 2020 to June 30, 2022. However, the remaining period from
July 1, 2022 to February 13, 2023 will be used to evaluate forecasts of the
daily volatility of Brent Oil price return. For the tuning parameters, we
considered here the quadratic kernel, the bandwidth is chosen using cross-
validation technique. For the semi-metric, because the curves of natural gas
return are not smooth we use the PCA-semi-metric, say
$d_{4}^{\mathrm{PCA}}(\cdot,\cdot)$, based on the projection on the four
eigenfunctions, $v_{1}(\cdot),\dots,v_{4}(\cdot)$, associated to the four
largest eigenvalues of the empirical covariance operator of the functional
predictor $X$:

(5.1)
$d_{4}^{\mathrm{PCA}}\left(X_{t},X_{s}\right)=\sqrt{\sum_{k=1}^{4}\left(\int_{1}^{1439}\left(X_{t}(m)-X_{s}(m)\right)v_{k}(m)dm\right)^{2}}.$

Note that the term “Volatility” refers to a latent variable, which cannot be
observed directly but can be approximated from other variables that can be
observed. To evaluate the estimation and forecast of the volatility, one
considers the so-called realized volatility (see Merton (1980)) computed based
on the 1-hour frequency of Brent oil returns over the same period. Thus, the
realized volatility is considered as an approximation of the true value of
volatility that we can refer to in order to assess the performance of our
estimators. Given 1-hour frequency one calculates the realized volatility on a
specific day $t$ as follows:

$RV_{t}=\sum_{h=1}^{24}r_{t,h}^{2},$

where $r_{t,h}$ is the value of the log return of the Brent observed at hour
$h$ on the day $t.$

Figure 8 shows the In-Sample and Out-of-Sample (the green-shaded area) sets
for the realized and estimated volatility of the Brent returns. Figure 8(a)
shows that the most volatile period for the Brent is the year 2020. This is
due to an important event named “the 2020 Russia-Saudi Arabia conflict,
COVID-19”, which is a combination of two factors that led to a significant
drop in oil prices, including a supply-demand oil conflict between Russia and
Saudi Arabia, and another supply-demand disruption from the COVID-19 pandemic,
which has clearly impacted oil supply-demand because of the lockdowns around
the world. The confinements and shutdowns of economic activity lowered demand,
resulting in the collapse of oil prices. Besides, Vienna, a major oil
producer, were unable to reach an agreement to reduce oil production in
response to the COVID-19 pandemic, and immediately after that, Saudi Arabia
and Russia began a pricing war that significantly lowered the price of oil.
However, the period of lowest volatility is between 2021 and 2022, displayed
in Figure 8(a). That is because in January 2021 oil prices started to rise due
to demand outside Europe and supply oil reductions made OPEC countries. It can
also be noted that the volatility increased steeply during the starting year
2022. Reaching a new peak after 2020, indicating another major high-risk event
which is the war between Russia and Ukraine and its impact on the global
economy after the COVID-19 epidemic. However, the Out-of-Sample set (green-
shaded area) shows that the forecast of the realized volatility of the Brent
return is getting stable, indicating that the volatility levels are low
starting from July 1, 2022 to February 13, 2023. Figure 8(b) seems to indicate
that the estimated volatility of the Brent return fits well with the true
values of volatility. Figure 8(c) and (d) present the estimated volatility of
the Brent return at a $12\%$ MAR rate. It is clear that the four graphs retain
the same pattern. As a criterion for measuring estimators’ accuracy in
estimating (resp. forecasting) the In-Sample (resp. Out-of-Sample) volatility,
we calculate the daily absolute error defined as

$AE_{t}:=|\mathcal{U}_{t}(X_{t})-RV_{t}|,$

where $\mathcal{U}_{t}(X_{t})$ denotes the volatility estimation/forecast
obtained either with complete data, missing data or imputed data.

Table 3 provides the distribution (through the calculation of the first,
second, third quartiles as well as the mean) of the absolute error for each
estimator at a $12\%$ MAR rate. Results show that for the In-Sample (resp.
Out-of-Sample) the estimation (resp. forecast) of the volatility of brent
returns is accurately estimated (resp. forecasted) with complete data. On the
other side, one observes that volatiltiy estimation (resp. forecast), after
missing data imputation, is more accurate that the one obtained with
simplified estimator.

(a)

(b)

(c)

(d)

Figure 8.  (a) Realized daily Volatility Brent log-returns. (b) Daily Volatility based on complete data. (c) Daily Volatility based on Simplified estimator. (d) Daily volatility based on nonparametric Imputed estimator. Green-shaded area represents the out-of-sample period. Table 3. Summary Statistics of the AE obtained for each estimator when MAR=$12\%$. Estimators |  In-Sample (IS)$\times 10^{-4}$ |  Out-of-Sample (OoS)$\times 10^{-4}$
---|---|---
| $\small Q_{25\%}$ | $\small Q_{50\%}$ | $\small Q_{75\%}$ | Mean | $\small Q_{25\%}$ | $\small Q_{50\%}$ | $\small Q_{75\%}$ | Mean
Complete | $0.0496$ | $0.4954$ | $7.3966$ | $7.5302$ | $1.41353$ | $6.12136$ | $7.75438$ | $6.29141$
Simplified | $0.0401$ | $0.8101$ | $7.9516$ | $7.5359$ | $1.59297$ | $6.24067$ | $7.87776$ | $6.36422$
NP Imp. | $0.0416$ | $0.7807$ | $7.6034$ | $7.5697$ | $1.27913$ | $6.13609$ | $7.68484$ | $6.30530$

##  6\. Discussion

In this paper we introduced nonparametric estimation of regression and
variance operators when the data generating process is assumed to be a
nonlinear heteroscedastic functional regression model and observations are
affected by a missing at random mechanism. The simplified estimator
(calculated from observed data only) is then used to impute the missing data
in the original process. Finally, a nonparametric imputed estimator is
calculated making use of both observed and imputed observations. We
investigated several asymptotic properties of both simplified and
nonparametric imputed estimators. That includes pointwise and uniform almost
sure consistency rate, identification of the asymptotic distribution and
estimation of asymptotic confidence intervals. To assess and compare the
performance of volatility estimators, we conducted a numerical analysis.
First, we assessed the quality of estimators using simulated data. Then, an
application to high-frequency financial data was discussed. The daily
volatility of the Brent Oil price log-return has been estimated and forecasted
using the intraday (one-minute) frequency log-return of natural gas as a
predictor. The results showed that the nonparametric imputed estimator
demonstrates superior performance compared to the simplified estimator.
This work can be extended from different perspectives. First, one can think of
reducing the dimensionality of the predictor by using a single functional
index model (SFIM) to estimate the volatility. SFIM has shown its efficiency
in improving the consistency of the regression operator estimator. However, to
the best of our knowledge, there is no work for the conditional variance.
Moreover, in several situations, in finance or economics, the volatility does
not only depend on one functional predictor. Other real-valued covariates that
may describe some economic or geopolitical circumstances may be useful to
explain high volatile periods. In such framework, it would be worthwhile to
extend the obtained results to the semi-functional partial linear regression
model ( see Pérez and Vieu (2006)) to the case where errors are not
homoscedatic.

## Proof of main results

In order to prove the main results of this paper, we introduce additional
notations and necessary Lemmas. Let $j\in\\{1,2\\}.$ Let

(6.1) $\displaystyle m^{[j]}_{n,0}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{n\mathbb{E}(K_{1}(x))}\sum_{t=1}^{n}Y^{j-1}_{t}\delta_{t}K_{t}(x),$
$\displaystyle\overline{m}^{[j]}_{n,0}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{n\mathbb{E}(K_{1}(x))}\sum_{t=1}^{n}\mathbb{E}[Y^{j-1}_{t}\delta_{t}K_{t}(x)|\mathcal{F}_{t-1}],$

where $K_{t}(x)=K\left(\frac{d_{1}(x,X_{t})}{h_{1}}\right)$,
and

(6.2)
$V_{n,0}(x)=\frac{1}{n\mathbb{E}\left[K_{1}(x)\right]}\sum_{t=1}^{n}\delta_{t}\sqrt{U(X_{t})}\varepsilon_{t}K_{t}(x)\text{.}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$

######  Lemma 6.1 (Laïb and Louani (2016)).

Let $(M_{n})_{n\geq 1}$ be a sequence of martingale differences with respect
to the sequence of $\sigma$-fields
$\mathcal{F}_{n}=\\{\sigma(M_{1},\ldots,M_{n}):n\geq 1\\}$, where
$\sigma(M_{1},\ldots,M_{n})$ is the $\sigma$-field generated by the random
variables $M_{1},\ldots,M_{n}$. Set $S_{n}=\displaystyle\sum_{t=1}^{n}M_{t}$.
Assume that there exist for any $t\in\mathbb{N}$ some nonnegative constants
$C$ and $d_{t}$ such that $|M_{t}|\leq C$ almost surely, and
$\mathbb{E}(M_{t}^{2}|\mathcal{F}_{t-1})\leq d_{t}^{2}$ almost surely.
Then, for any $\varepsilon>0$,

$\mathbb{P}(|S_{n}|>\varepsilon)\leq
2\exp\left\\{\frac{-\varepsilon^{2}}{(4D_{n}+2C\varepsilon)}\right\\},$

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

######  Lemma 6.2 (Laïb and Louani (2010)).

Suppose that assumptions (A1)(i), (A2)(i)-(ii) and (A2)(iv) hold true. For
$k\in\\{1,2,3\\}$, let
$\mathcal{K}_{t,k}(x)=\mathcal{K}\left(\frac{d_{k}(x,X_{t})}{h_{n,k}}\right)$
and
$M_{j,\mathcal{K},k}=\mathcal{K}^{j}(1)-\displaystyle\int_{0}^{1}(\mathcal{K}^{j})^{\prime}\tau_{0,k}(u)du$,
where $1\leq j\leq 2+\kappa$, with $\kappa>0$, we have

$(i)$:

$(\phi_{k}(h_{n,k}))^{-1}\mathbb{E}(\mathcal{K}_{t,k}^{j}(x)|\mathcal{F}_{t-1})=M_{j,\mathcal{K},k}f_{t,1}(x)+\mathcal{O}_{a.s.}\left(\psi_{t,x}(h_{n,k})/\phi_{k}(h_{n,k})\right)$;

$(ii)$:

$(\phi_{k}(h_{n,k}))^{-1}\mathbb{E}(\mathcal{K}_{1,k}^{j}(x))=M_{j,\mathcal{K},k}f_{1}(x)+o(1)$.

######  Lemma 6.3.

Suppose that assumptions (A1),(A2)(i)-(vi),(A3)(v), and condition (3.6) is
satisfied. Then,

(6.3)
$\underset{n\rightarrow\infty}{\lim}\sup_{x\in\mathcal{C}}\left|m_{n,0}^{[1]}(x)-\pi(x)\right|=0\quad
a.s.$

###### Proof.

Note that

(6.4)
$\sup_{x\in\mathcal{C}}|m^{[1]}_{n,0}(x)|\leq\sup_{x\in\mathcal{C}}|m^{[1]}_{n,0}(x)-\overline{m}^{[1]}_{n,0}(x)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}|+\sup_{x\in\mathcal{C}}|\overline{m}^{[1]}_{n,0}(x)|.$

Then, the first term in right-hand side of (6.4) could be decomposed as
follows. For $\eta>0$ and
$B(c_{k},\eta):=\\{x\in\mathcal{C}:d_{\mathcal{C}}(x,c_{k})<\eta\\}$, we have

(6.5)
$\displaystyle\displaystyle{\sup_{x\in\mathcal{C}}}\left|m_{n,0}^{[1]}(x)-\overline{m}_{n,0}^{[1]}(x)\right|$
$\displaystyle\leq$ $\displaystyle\displaystyle{\max_{1\leq k\leq
N(\eta,\mathcal{C},d_{\mathcal{C}})}\sup_{x\in
B(c_{k},\eta)}}\left|m_{n,0}^{[1]}(x)-\overline{m}_{n,0}^{[1]}(x)\right|$
$\displaystyle\leq$ $\displaystyle\displaystyle{\max_{1\leq k\leq
N(\eta,\mathcal{C},d_{\mathcal{C}})}\sup_{x\in
B(c_{k},\eta)}}\left|m_{n,0}^{[1]}(x)-m_{n,0}^{[1]}(c_{k})\right|+\displaystyle{\max_{1\leq
k\leq
N(\eta,\mathcal{C},d_{\mathcal{C}})}}\left|m_{n,0}^{[1]}(c_{k})-\overline{m}_{n,0}^{[1]}(c_{k})\right|$
$\displaystyle\quad+\displaystyle{\max_{1\leq k\leq
N(\eta,\mathcal{C},d_{\mathcal{C}})}\sup_{x\in
B(c_{k},\eta)}}\left|\overline{m}_{n,0}^{[1]}(x)-\overline{m}_{n,0}^{[1]}(c_{k})\right|$
$\displaystyle=:$
$\displaystyle\mathcal{H}_{1}+\mathcal{H}_{2}+\mathcal{H}_{3}.$

We start with studying the term $\mathcal{H}_{1}.$ Observe that, for any $x\in
B(c_{k}$, $\eta)$ and making use of the definition of $m_{n,0}^{[1]}(x)$ in
(6.1), one gets

$\displaystyle m_{n,0}^{[1]}(x)-m_{n,0}^{[1]}(c_{k})$
$\displaystyle=\frac{1}{n\mathbb{E}(K_{1}(x))\mathbb{E}(K_{1}(c_{k}))}{\sum_{t=1}^{n}}\delta_{t}\left[K_{t}(x)\mathbb{E}(K_{1}(c_{k}))-K_{t}(c_{k})\mathbb{E}(K_{1}(x)\right]$
$\displaystyle=\frac{1}{n\mathbb{E}(K_{1}(x))}{\sum_{t=1}^{n}}\delta_{t}\left[K_{t}(x)-K_{t}(c_{k})\right]$
$\displaystyle\quad+\frac{1}{n\mathbb{E}(K_{1}(x))\mathbb{E}(K_{1}(c_{k}))}{\sum_{t=1}^{n}}\delta_{t}K_{t}(c_{k})\left[\mathbb{E}(K_{1}(c_{k}))-\mathbb{E}(K_{1}(x))\right]$
$\displaystyle=\mathcal{H}_{1,1}+\mathcal{H}_{1,2}.$

Condition (A1)(ii) and the fact that $|\delta_{t}|<1$ almost surely, one can
bound the term $\mathcal{H}_{1,1}$ from above as follows:
$\begin{aligned}
\left|\mathcal{H}_{1,1}\right|&\leq\frac{1}{n\left|\mathbb{E}(K_{1}(x))\right|}\displaystyle{\sum_{t=1}^{n}}\left|\delta_{t}\right|\left|K_{t}(x)-K_{t}(c_{k})\right|\\\
&\leq\frac{1\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}{n\left|\mathbb{E}(K_{1}(x))\right|}\displaystyle{\sum_{t=1}^{n}}|K_{t}(x)-K_{t}(c_{k})|\\\
&\leq\frac{1\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}{n\left|\mathbb{E}(K_{1}(x))\right|}\left(a_{0}\displaystyle{\sum_{t=1}^{n}}\left|\frac{d_{1}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(x,X_{t})-d_{1}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(c_{k},X_{t})}{h_{1}}\right|^{\gamma}\right)\\\
&\leq\frac{1\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}{n\left|\mathbb{E}(K_{1}(x))\right|}\left(a_{0}n\left[\frac{\eta\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}{h_{1}}\right]^{\gamma}\right)\\\
&\leq\frac{\eta^{\gamma}a_{0}}{h_{1}^{\gamma}|\mathbb{E}(K_{1}(x))|}.\end{aligned}$
On the other hand, by condition (A1)(iii) and Lemma 6.2, we have
$\frac{K_{t}(x)}{|\mathbb{E}(K_{1}(x))|}\leq\frac{a_{2}}{a_{1}}=:a_{3},\;\forall
x\in\mathcal{E}.$ Hence

$\left|\mathcal{H}_{1,2}\right|\leq\frac{1}{n\mathbb{E}(K_{1}(x))}\displaystyle{\sum_{t=1}^{n}}\left|\delta_{t}\right|\left|\frac{K_{t}(c_{k})}{\mathbb{E}(K_{1}(c_{k}))}\right|\left|\mathbb{E}(K_{1}(c_{k}))-\mathbb{E}(K_{1}(x))\right|.$

Using condition (A1)(ii) and the almost sure boundedness of $\delta$, we get
$\left|\mathcal{H}_{1,2}\right|\leq\frac{a_{3}\eta\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}^{\gamma}a_{0}}{h_{1}^{\gamma}\left|\mathbb{E}(K_{1}(x))\right|}.$
Therefore, for any $1\leq k\leq N(\eta,\mathcal{C},d_{\mathcal{C}})$, we have
$|m_{n,0}^{[1]}(x)-m_{n,0}^{[1]}(c_{k})|\leq\frac{(1+a_{3})a_{0}\eta^{\gamma}}{h_{1}^{\gamma}a_{1}}.$
Then, by taking $\eta=\eta_{n}=o(h_{1})$, one gets

(6.6) $\mathcal{H}_{1}=\displaystyle{\max_{1\leq k\leq
N(\eta,\mathcal{C},d_{\mathcal{C}})}\sup_{x\in
B(c_{k},\eta)}}\left|m_{n,0}^{[1]}(x)-m_{n,0}^{[1]}(c_{k})\right|=o(1).$

Similar to $\mathcal{H}_{1}$, one can show that when (A3)(v) holds true, the
term $\mathcal{H}_{3}$ in (6.5) is negligible. That is

(6.7) $\mathcal{H}_{3}=\displaystyle{\max_{1\leq k\leq
N(\eta,\mathcal{C},d_{\mathcal{C}})}\sup_{x\in
B(c_{k},\eta\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})}}\left|\overline{m}_{n,0}^{[1]}(x)-\overline{m}_{n,0}^{[1]}(c_{k})\right|=o(1).$

Now we turn our attention to the study of the term $\mathcal{H}_{2}.$ Note
that

$m_{n,0}^{[1]}(c_{k})-\overline{m}_{n,0}^{[1]}(c_{k})=\frac{1}{n\mathbb{E}(K_{1}(c_{k}))}\sum_{t=1}^{n}L_{n,t}(c_{k}),$

where
$L_{n,t}(c_{k})=\delta_{t}K_{t}(c_{k})-\mathbb{E}(\delta_{t}K_{t}(c_{k})|\mathcal{F}_{t-1})$
is a martingale difference with respect to the $\sigma$-field
$\mathcal{F}_{t-1}$, for $t=1,2,\ldots,n.$ Then, we can use Lemma 6.1 to
handle the convergence of
$m_{n,0}^{[1]}(c_{k})-\overline{m}_{n,0}^{[1]}(c_{k})$. To be able to use this
Lemma, we need first to check its conditions.
First, since $|\delta_{t}|<1$ almost surely and by using assumptions
(A1)(iii), it follows that

$\displaystyle|L_{n,t}(c_{k})|$ $\displaystyle\leq
2\left|\delta_{t}K_{t}(c_{k})\right|$ $\displaystyle\leq 2a_{2}=:C.$

In addition, making use of $C_{r}$-inequality, assumptions (A1)(i), (A2)(iv),
(A3)(v), Lemma 6.2(i) and the fact that $f_{t,1}$ is bounded by a
deterministic quantity $b_{t}(x)$, one gets
$\begin{aligned} \mathbb{E}(L^{2}_{n,t}(c_{k})|\mathcal{F}_{t-1})&\leq
2\mathbb{E}(\delta_{t}K^{2}_{t}(c_{k})|\mathcal{F}_{t-1})\\\ &\leq
2(\pi(x)+o(1))\mathbb{E}(K^{2}_{t}(c_{k})|\mathcal{F}_{t-1})\\\
&\leq(2\pi(x)+o(1))\phi_{1}(h_{1})(M_{2,K,1}b_{t}(x)+1)=:d^{2}_{t}.\end{aligned}$
Then, by assumption (A2)(v), we can write
$n^{-1}D_{n}=(2\pi(x)+o(1))\phi_{1}(h_{1})[M_{2,K,1}D(x)+o_{a.s.}(1)]$, which
means that $D_{n}=\mathcal{O}(n\phi_{1}(h_{1}))$. Thus, by applying Lemma 6.1,
we have

$\displaystyle\mathbb{P}\left({\max_{1\leq k\leq
N(\eta,\mathcal{C},d_{\mathcal{C}})}}\left|m_{n,0}^{[1]}(c_{k})-\overline{m}_{n,0}^{[1]}(c_{k})\right|>\overline{\lambda}_{n}\right)$
$\displaystyle\leq{\sum_{k=1}^{N(\eta,\mathcal{C},d_{\mathcal{C}})}}\mathbb{P}\left(\left|m_{n,0}^{[1]}(c_{k})-\overline{m}_{n,0}^{[1]}(c_{k})\right|>\overline{\lambda}_{n}\right)$
$\displaystyle\leq{\sum_{k=1}^{N(\eta,\mathcal{C},d_{\mathcal{C}})}}\mathbb{P}\left(\left|\sum_{t=1}^{n}L_{n,t}(c_{k})\right|>\overline{\lambda}_{n}n\mathbb{E}[K_{1}(c_{k})]\right)$
$\displaystyle\leq
2N(\eta,\mathcal{C},d_{\mathcal{C}})\exp\left(-\frac{\left(\overline{\lambda}_{n}n\mathbb{E}[K_{1}(c_{k})]\right)^{2}}{4D_{n}+2C\overline{\lambda}_{n}n\mathbb{E}[K_{1}(c_{k})]}\right)$
$\displaystyle\leq
2N(\eta,\mathcal{C},d_{\mathcal{C}})\exp\left(-\frac{\mathcal{O}(n\phi_{1}(h_{1}))^{2}\overline{\lambda}_{n}^{2}}{\mathcal{O}(n\phi_{1}(h_{1}))+\overline{\lambda}_{n}\mathcal{O}(n\phi_{1}(h_{1}))}\right)$
$\displaystyle\leq
2N(\eta,\mathcal{C},d_{\mathcal{C}})\exp\left(-\frac{\mathcal{O}(n\phi_{1}(h_{1}))\overline{\lambda}_{n}^{2}}{1+\overline{\lambda}_{n}}\right)$
$\displaystyle\leq
2\exp\left(-\mathcal{O}(n\phi_{1}(h_{1}))\overline{\lambda}_{n}^{2}\left[1-\frac{\log
N(\eta,\mathcal{C},d_{\mathcal{C}})}{\mathcal{O}(n\phi_{1}(h_{1}))\overline{\lambda}_{n}^{2}}\right]\right),$

where $\overline{\lambda}_{n}=\lambda_{n}\ell_{n}^{-1}$, $\lambda_{n}$ and
$\ell_{n}$ are defined in Theorem 3.1. Hence, using condition (3.6), one gets

$\displaystyle{\sum_{n\geq 1}}\mathbb{P}\left(\displaystyle{\max_{1\leq k\leq
N(\eta,\mathcal{C},d_{\mathcal{C}})}}\left|m_{n,0}^{[1]}(c_{k})-\overline{m}_{n,0}^{[1]}(c_{k})\right|>\overline{\lambda}_{n}\right)<\infty.$

Then, using Borel-Cantelli Lemma we have

(6.8) $\displaystyle{\max_{1\leq k\leq
N(\eta,\mathcal{C},d_{\mathcal{C}})}}\left|m_{n,0}^{[1]}(c_{k})-\overline{m}_{n,0}^{[1]}(c_{k})\right|=\mathcal{O}_{a.s.}(\overline{\lambda}_{n})=o_{a.s.}(1).$

Combining (6.5) with (6.6), (6.7) and (6.8), one deduces that the first term
in the right-hand side of inequality (6.4) converges to zero as $n$ goes to
infinity. Regarding the second term
$\displaystyle\sup_{x\in\mathcal{C}}|\overline{m}^{[1]}_{n,0}(x)|$, note that
assumption(A3)(v), along with a double conditioning with respect to
$\mathcal{G}_{t-1}$, allows to obtain

$\displaystyle\overline{m}^{[1]}_{n,0}(x)-\pi(x)$
$\displaystyle=\frac{1}{n\mathbb{E}(K_{1}(x))}\sum_{t=1}^{n}\mathbb{E}\left\\{\mathbb{E}\left[\delta_{t}K_{t}(x)\mid\mathcal{G}_{t-1}\right]\mid\mathcal{F}_{t\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-1}\right\\}-\pi(x)$
$\displaystyle=\frac{1}{n\mathbb{E}(K_{1}(x))}\sum_{t=1}^{n}\mathbb{E}[(\pi(x)+o(1))K_{t}(x)\mid\mathcal{F}_{t-1}]-\pi(x)$
$\displaystyle=(\pi(x)+o(1))\frac{1}{n\mathbb{E}(K_{1}(x))}\sum_{t=1}^{n}\mathbb{E}\left(K_{t}(x)\mid\mathcal{F}_{t-1}\right)-\pi(x)$
$\displaystyle=\pi(x)\left[\frac{1}{n\mathbb{E}(K_{1}(x))}\sum_{t=1}^{n}\mathbb{E}\left(K_{t}(x)\mid\mathcal{F}_{t-1}\right)-1\right]+o(1),$

where $o(1)$ is uniformly in $x$. Finally, Lemma 7 in Laïb and Louani (2011)
allows to conclude the proof of this Lemma. ∎

######  Lemma 6.4.

Suppose that assumptions (A1), (A2)(i)-(ii),(iv)-(vi), (A3)(i),(v), and
conditions (3.5) and (3.6) are satisfied. Then, we have

$\displaystyle{\sup_{x\in\mathcal{C}}}\left|V_{n,0}(x)\right|=\mathcal{O}_{a.s.}\left(\lambda_{n}\right)\,\
\hbox{as}\quad n\rightarrow\infty\text{.}$

###### Proof.

Let

(6.9)
$V_{n,0}(x)=\frac{1}{n\mathbb{E}\left[K_{1}(x)\right]}\sum_{t=1}^{n}\mathcal{L}_{t}K_{t}(x)=\left(V_{n,0}(x)-V^{\top}_{n,0}(x)\right)+\widetilde{V}_{n,0}(x)+V^{-}_{n,0}(x),$

where
$V^{\top}_{n,0}(x)=\displaystyle{\frac{1}{n\mathbb{E}\left[K_{1}(x)\right]}\sum_{t=1}^{n}\mathcal{L}_{t}1\kern-1.99997pt{\rm
l}_{\left(|\mathcal{L}_{t}|\leq\ell_{n}\right)}K_{t}(x)},\quad\widetilde{V}_{n,0}(x)=\displaystyle{\frac{1}{n\mathbb{E}\left[K_{1}(x)\right]}\sum_{t=1}^{n}\mathbb{E}(\mathcal{L}_{t}1\kern-1.99997pt{\rm
l}_{\left(|\mathcal{L}_{t}|\leq\ell_{n}\right)}\mid\mathcal{F}_{t-1})K_{t}(x)},$
$V^{-}_{n,0}(x)=\displaystyle{\frac{1}{n\mathbb{E}\left[K_{1}(x)\right]}\sum_{t=1}^{n}\left(\mathcal{L}_{t}1\kern-1.99997pt{\rm
l}_{\left(|\mathcal{L}_{t}|\leq\ell_{n}\right)}-\mathbb{E}(\mathcal{L}_{t}1\kern-1.99997pt{\rm
l}_{\left(|\mathcal{L}_{t}|\leq\ell_{n}\right)}\mid\mathcal{F}_{t-1})\right)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}K_{t}(x)},$
such that $\mathcal{L}_{t}=\delta_{t}\sqrt{U(X_{t})}\varepsilon_{t}$ and
$\ell_{n}$ is as defined in Theorem 3.1.
Lemma 6.8 allows to conclude that the first term in (6.9) equals zero almost
surely as $n\rightarrow\infty.$ Moreover, making use of Lemma 6.5 and Lemma
6.6 below, the second term is
$\mathcal{O}\left(\\{\ell^{\rho-1}_{n}\phi_{1}(h_{1})^{\rho-1/\rho}\\}^{-1}\right)$
and the third term is $\mathcal{O}_{a.s.}\left(\lambda_{n}\right),$
respectively. Finally, since
$(\lambda_{n}\ell^{\rho-1}_{n}\phi_{1}(h_{1})^{\rho-1/\rho})^{-1}=o(1)$ for n
large enough, the proof of this lemma is achieved. ∎

######  Lemma 6.5.

Suppose that assumptions (A1)(i), (A2)(i)-(ii),(iv),(vi), (A3)(i),(v) and
condition (3.5) are satisfied. Then

$\displaystyle{\sup_{x\in\mathcal{C}}}|\widetilde{V}_{n,0}(x)|=\mathcal{O}\left(\\{\ell^{\rho-1}_{n}\phi_{1}(h_{1})^{\rho-1/\rho}\\}^{-1}\right).$

The proof of this Lemma is detailed in the Appendix.

######  Lemma 6.6.

Under assumptions (A1), (A2)(i)-(ii),(iv)-(vi), (A3)(i),(v) and condition
(3.6), we have, as $n\rightarrow\infty,$

$\displaystyle{\sup_{x\in\mathcal{C}}}|V^{-}_{n,0}(x)|=\mathcal{O}_{a.s.}(\lambda_{n}).$

###### Proof.

The proof is similar to the proof of Lemma C in Chaouch (2019). ∎

### Proof of Theorem 3.1

By Model (2.1), we have

$m_{n,0}(x)-m(x)=\frac{1}{m_{n,0}^{[1]}(x)}\left[\frac{1}{n\mathbb{E}(K_{1}(x))}\displaystyle{\sum_{t=1}^{n}}\delta_{t}\\{m(X_{t})-m(x)+\sqrt{U(X_{t})}\varepsilon_{t}\\}K_{t}(x)\right].$

Then,

(6.10)
$\displaystyle{\sup_{x\in\mathcal{C}}}|m_{n,0}(x)-m(x)|\leq\displaystyle{\sup_{u\in
B(x,h_{1})}}|m(u)-m(x)|+\\{\displaystyle{\inf_{x\in\mathcal{C}}}m_{n,0}^{[1]}(x)\\}^{-1}\times\displaystyle{\sup_{x\in\mathcal{C}}}|V_{n,0}(x)|.$

Further, observe that

$\displaystyle{\inf_{x\in\mathcal{C}}}\left|m_{n,0}^{[1]}(x)\right|>\displaystyle{\inf_{x\in\mathcal{C}}}\left|\pi(x)\right|-\displaystyle{\sup_{x\in\mathcal{C}}}\left|m_{n,0}^{[1]}(x)-\pi(x)\right|.$

Thus, making use of Lemma 6.3 and assumption (A2)(vii), we get

(6.11)
$\displaystyle{\inf_{x\in\mathcal{C}}}\left|m_{n,0}^{[1]}(x)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\right|>\theta_{1}\,\
a.s.,$

where $\theta_{1}$ is defined in assumption (A2)(vii).
Finally, using Lemma 6.4, combined with assumption (A4)(i), equation (6.10)
and equation (6.11), one concludes the proof of this theorem.

### Proof of Theorem 3.2

Let

$U_{n,0}(x)-U(x)=\left(\vartheta_{n,0}^{[1]}(x)+\vartheta_{n,0}^{[2]}(x)+\vartheta_{n,0}^{[3]}(x)+\vartheta_{n,0}^{[4]}(x)\right)/U_{n,0}^{[1]}(x),$

where

(6.12) $\displaystyle\vartheta_{n,0}^{[1]}(x)$ $\displaystyle=$
$\displaystyle\displaystyle\frac{1}{n\mathbb{E}\left[W_{1}(x)\right]}\sum_{t=1}^{n}\delta_{t}\left(m\left(X_{t}\right)-m_{n,0}\left(X_{t}\right)\right)^{2}W_{t}(x),$
(6.13) $\displaystyle\vartheta_{n,0}^{[2]}(x)$ $\displaystyle=$
$\displaystyle\displaystyle\frac{2}{n\mathbb{E}\left[W_{1}(x)\right]}\sum_{t=1}^{n}\delta_{t}\left(m\left(X_{t}\right)-m_{n,0}\left(X_{t}\right)\right)\sqrt{U\left(X_{t}\right)}\varepsilon_{t}W_{t}(x),$
(6.14) $\displaystyle\vartheta_{n,0}^{[3]}(x)$ $\displaystyle=$
$\displaystyle\displaystyle\frac{1}{n\mathbb{E}\left[W_{1}(x)\right]}\sum_{t=1}^{n}\delta_{t}U\left(X_{t}\right)W_{t}(x)\left(\varepsilon_{t}^{2}-1\right),$
$\displaystyle\vartheta_{n,0}^{[4]}(x)$ $\displaystyle=$
$\displaystyle\displaystyle\frac{1}{n\mathbb{E}\left[W_{1}(x)\right]}\sum_{t=1}^{n}\delta_{t}\left(U\left(X_{t}\right)-U(x)\right)W_{t}(x),$
$\displaystyle U_{n,0}^{[1]}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{n\mathbb{E}\left[W_{1}(x)\right]}\sum_{t=1}^{n}\delta_{t}W_{t}(x),$

with $W_{t}(x)=W\left(\frac{d_{2}(x,X)}{h_{2}}\right).$ Our main objective is
to establish the uniform consistency rate, with respect to $x,$ of
$U_{n,0}(x)$. For this purpose, let us consider

(6.15)
$\displaystyle{\sup_{x\in\mathcal{C}}}\left|U_{n,0}(x)-U(x)\right|\leq\big{\\{}\displaystyle{\sup_{x\in\mathcal{C}}}|\vartheta_{n,0}^{[1]}(x)|+\displaystyle{\sup_{x\in\mathcal{C}}}|\vartheta_{n,0}^{[2]}(x)|+\displaystyle{\sup_{x\in\mathcal{C}}}|\vartheta_{n,0}^{[3]}(x)|+\displaystyle{\sup_{x\in\mathcal{C}}}|\vartheta_{n,0}^{[4]}(x)|\big{\\}}\times\big{\\{}\displaystyle{\inf_{x\in\mathcal{C}}}|U_{n,0}^{[1]}(x)|\big{\\}}^{-1}.$

Note that $U_{n,0}^{[1]}(x)$ has similar form as $m_{n,0}^{[1]}(x)$ when $K$
is replaced by $W$. Thus, similar to the proof of (6.3) and (6.11), one can
show that, under assumptions (A1), (A2), (A3)(v) and condition (3.8), one gets

(6.16)
$\underset{n\rightarrow\infty}{\lim}\sup_{x\in\mathcal{C}}\left|U_{n,0}^{[1]}(x)-\pi(x)\right|=0\quad
a.s.,\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$

and

(6.17)
$\displaystyle{\inf_{x\in\mathcal{C}}}\left|U_{n,0}^{[1]}(x)\right|>\theta_{1}\quad
a.s.\,\ $

Study of the term $\vartheta_{n,0}^{[1]}(x)$. We have

$\left|\vartheta_{n,0}^{[1]}(x)\right|\leq\frac{1}{n\mathbb{E}\left[W_{1}(x)\right]}\sum_{t=1}^{n}\delta_{t}\left|m\left(X_{t}\right)-m_{n,0}\left(X_{t}\right)\right|^{2}W_{t}(x)\leq\sup_{x\in\mathcal{C}}\left|m_{n,0}(x)-m(x)\right|^{2}\times\frac{1}{n\mathbb{E}\left[W_{1}(x)\right]}\sum_{t=1}^{n}\delta_{t}W_{t}(x)\text{.}$

Then, in view of Theorem 3.1 and equation (6.16), we get

(6.18)
$\displaystyle{\sup_{x\in\mathcal{C}}}\left|\vartheta_{n,0}^{[1]}(x)\right|=\mathcal{O}_{a.s.}(h_{1}^{2\alpha})+\mathcal{O}_{a.s.}\left(\lambda_{n}^{2}\right)\text{.}$

Study of the term $\vartheta_{n,0}^{[2]}(x)$. Observe that

$\vartheta_{n,0}^{[2]}(x)\leq
2\sup_{x\in\mathcal{C}}\left|m_{n,0}(x)-m(x)\right|\times\overline{V}_{n,0}(x),\,\
$

where

$\overline{V}_{n,0}(x)=\frac{1}{n\mathbb{E}\left[W_{1}(x)\right]}\sum_{t=1}^{n}\delta_{t}\sqrt{U(X_{t})}\varepsilon_{t}W_{t}(x)\text{.}$

In addition, note that $\overline{V}_{n,0}(x)$ has similar form as
$V_{n,0}(x)$ when $K$ is replaced by $W$. Therefore, by Lemma 6.4, it follows
that

(6.19)
$\displaystyle{\sup_{x\in\mathcal{C}}}\left|\overline{V}_{n,0}(x)\right|=\mathcal{O}_{a.s.}\left(\lambda_{n}^{\prime}\right),\,\
$

where $\lambda_{n}^{\prime}$ is defined in Theorem 3.2.
Making use of Theorem 3.1 and equation (6.19), we find

(6.20)
$\displaystyle{\sup_{x\in\mathcal{C}}}\left|\vartheta_{n,0}^{[2]}(x)\right|=\mathcal{O}_{a.s.}\left\\{\left(h_{1}^{\alpha}+\lambda_{n}\right)\lambda_{n}^{\prime}\right\\}\text{.}$

Study of the term $\vartheta_{n,0}^{[3]}(x).$ Observe that
$\vartheta_{n,0}^{[3]}(x)$ has the same form as $\overline{V}_{n,0}(x)$ when
$\varepsilon$ and $\sqrt{U}$ are replaced by $\varepsilon^{2}-1$ and $U,$
respectively. Similar to the proof of Lemma 6.4, under assumptions (A1),
(A2)(i)-(ii), (iv)-(vi), (A3)(iv)-(v) and condition (3.8), we obtain

(6.21)
$\displaystyle{\sup_{x\in\mathcal{C}}}\left|\vartheta_{n,0}^{[3]}(x)\right|=\mathcal{O}_{a.s.}\left(\lambda_{n}^{\prime}\right)\text{.}$

Study of the term $\vartheta_{n,0}^{[4]}(x).$ One can easily show, using
assumption (A4)(ii) and equation (6.16), that

(6.22)
$\displaystyle{\sup_{x\in\mathcal{C}}}\left|\vartheta_{n,0}^{[4]}(x)\right|=\mathcal{O}_{a.s.}\left(h_{2}^{\beta}\right)\text{.}$

Finally, using equation (6.15) combining with the results obtained in
(6.17)-(6.18) and (6.20)-(6.22), we conclude the proof of the theorem .The
following Lemma gives the asymptotic normality of $\vartheta_{n,0}^{[3]}(x)$
which is needed to prove Theorem 3.3.

######  Lemma 6.7.

Suppose that assumptions (A1)(i), (A2)(i)-(iv), (vi), (A3)(ii)-(iii), (v),
(A4)(ii)-(iii) and condition (3.7) are satisfied. Then, as $n\to\infty$, we
have,

$\sqrt{n\phi_{2}(h_{2})}\vartheta_{n,0}^{[3]}(x)\overset{\mathcal{D}}{\longrightarrow}\mathcal{N}\left(0,\sigma^{{}^{\prime}2}(x)\right).$

where $\sigma^{{}^{\prime}2}(x)$ is defined in Theorem 3.6.

###### Proof.

Define
$\xi_{n,t}=\\{\sqrt{\phi_{2}\left(h_{2}\right)/n}\\}\left[\delta_{t}U\left(X_{t}\right)W_{t}(x)\left(\varepsilon_{t}^{2}-1\right)/\mathbb{E}\left\\{W_{1}(x)\right\\}\right],$
$t=1,...,n.$ Further, observe that

(6.23)
$\sqrt{n\phi_{2}\left(h_{2}\right)}\vartheta_{n,0}^{[3]}(x)=\sum_{i=1}^{n}\xi_{n,t},$

where, for any $x\in\mathcal{E},$ the summands in $(\ref{36})$ form a
triangular array of stationary martingale differences with respect to the
$\sigma$-field $\mathcal{F}_{t-1}$. Then, similar to the proof of Lemma 4 in
Laïb and Louani (2010), we apply the Central Limit Theorem for discrete-time
arrays of real-valued martingales to provide the asymptotic normality of
$\vartheta_{n,0}^{[3]}(x)$ (see Hall and Heyde (1980)). For that, we have to
establish the following statements:

  * (i)

$\underset{n\rightarrow\infty}{\lim}\sum_{t=1}^{n}\mathbb{E}\left(\xi_{n,t}^{2}\mid\mathcal{F}_{t-1}\right)\stackrel{{\scriptstyle\mathbb{P}}}{{=}}\sigma^{{}^{\prime}2}(x),$

  * (ii)

$n\mathbb{E}\left\\{\xi_{n,t}^{2}1\kern-1.99997pt{\rm
l}_{\left(\left|\xi_{n,t}\right|>\zeta\right)}\right\\}=o(1)$ holds for any
$\zeta>0$.

Proof of part (i). Notice that
$\begin{aligned}
\sum_{t=1}^{n}\mathbb{E}\left(\xi_{n,t}^{2}\mid\mathcal{F}_{t-1}\right)=&\frac{\phi_{2}\left(h_{2}\right)}{n\left[\mathbb{E}\left(W_{1}(x)\right)\right]^{2}}\sum_{t=1}^{n}\mathbb{E}\left[\delta_{t}\left\\{U\left(X_{t}\right)-U(x)\right\\}^{2}W_{t}^{2}(x)\left(\varepsilon_{t}^{2}-1\right)^{2}\mid\mathcal{F}_{t-1}\right]\\\
&+\frac{\phi_{2}\left(h_{2}\right)}{n\left[\mathbb{E}\left(W_{1}(x)\right)\right]^{2}}\sum_{t=1}^{n}\mathbb{E}\left[\delta_{t}U^{2}(x)W_{t}^{2}(x)\left(\varepsilon_{t}^{2}-1\right)^{2}\mid\mathcal{F}_{t-1}\right]\\\
&+\frac{2\phi_{2}\left(h_{2}\right)}{n\left[\mathbb{E}\left(W_{1}(x)\right)\right]^{2}}\sum_{t=1}^{n}\mathbb{E}\left[\delta_{t}\left\\{U\left(X_{t}\right)-U(x)\right\\}U(x)W_{t}^{2}(x)\left(\varepsilon_{t}^{2}-1\right)^{2}\mid\mathcal{F}_{t-1}\right]\\\
&\equiv\Upsilon_{n,1}+\Upsilon_{n,2}+\Upsilon_{n,3}.\end{aligned}$
Regarding the term $\Upsilon_{n,1}.$ Due to (2.6), $\delta_{t}$ and
$\varepsilon_{t}$ are independent. Then, by using Lemma 6.2 and assumptions
(A1)(i), (A2)(iii)-(iv), (vi), (A3)(ii), (v), and (A4)(ii), we can show that:

$\begin{aligned}
|\Upsilon_{n,1}|&=\mathcal{O}_{a.s.}(h_{2}^{2\beta})\times\left\\{\frac{\phi_{2}\left(h_{2}\right)}{n\left[\mathbb{E}\left\\{W_{1}(x)\right\\}\right]^{2}}\sum_{t=1}^{n}\mathbb{E}\left[\mathbb{E}\left\\{\delta_{t}W_{t}^{2}(x)\left(\varepsilon_{t}^{2}-1\right)^{2}\mid\mathcal{G}_{t-1}\right\\}\mid\mathcal{F}_{t-1}\right]\right\\}\\\
&\leq\mathcal{O}_{a.s.}(h_{2}^{2\beta})\times\left\\{\pi(x)+\sup_{u\in
B(x,h)}|\pi(u)-\pi(x)|\right\\}\left\\{\frac{M_{2,W,2}}{M_{1,W,2}^{2}}\frac{1}{f_{1}(x)}+o_{a.s.}(1)\right\\}\\\
&\times\left\\{\omega(x)+\sup_{u\in
B(x,h)}|\omega(u)-\omega(x)|\right\\}\longrightarrow 0\,\ \hbox{as}\,\
n\to\infty.\\\ \end{aligned}$
Regarding the term $\Upsilon_{n,2}.$ By considering the independence of
$\delta$ and $\varepsilon_{t},$ using Lemma 6.2 and assumptions (A1)(i),
(A2)(iii)-(iv),(vi), (A3)(ii),(v), we obtain

$\begin{aligned}
|\Upsilon_{n,2}|&=U^{2}(x)\times\left\\{\frac{\phi_{2}\left(h_{2}\right)}{n\left[\mathbb{E}\left\\{W_{1}(x)\right\\}\right]^{2}}\sum_{t=1}^{n}\mathbb{E}\left[\mathbb{E}\left\\{\delta_{t}W_{t}^{2}(x)\left(\varepsilon_{t}^{2}-1\right)^{2}\mid\mathcal{G}_{t-1}\right\\}\mid\mathcal{F}_{t-1}\right]\right\\}\\\
&\leq U^{2}(x)\times\left\\{\pi(x)+\sup_{u\in
B(x,h)}|\pi(u)-\pi(x)|\right\\}\left\\{\frac{M_{2,W,2}}{M_{1,W,2}^{2}}\frac{1}{f_{1}(x)}+o_{a.s.}(1)\right\\}\left\\{\omega(x)+\sup_{u\in
B(x,h)}|\omega(u)-\omega(x)|\right\\}\\\ &\longrightarrow
U^{2}(x)\pi(x)\left\\{\frac{M_{2,W,2}}{M_{1,W,2}^{2}}\frac{1}{f_{1}(x)}\right\\}\omega(x)\,\
\hbox{as}\,\ n\to\infty.\end{aligned}$
Regarding the term $\Upsilon_{n,3}$. Similarly as for $\Upsilon_{n,1},$ using
assumptions (A1)(i), (A2)(iii)-(iv),(vi), (A3)(ii),(v), and (A4)(ii) together
with Lemma 6.2, we get

$\begin{aligned} |\Upsilon_{n,3}|&=\mathcal{O}_{a.s.}(h_{2}^{\beta})\times
U(x)\left\\{\frac{\phi_{2}\left(h_{2}\right)}{n\left[\mathbb{E}\left\\{W_{1}(x)\right\\}\right]^{2}}\sum_{t=1}^{n}\mathbb{E}\left[\mathbb{E}\left\\{\delta_{t}W_{t}^{2}(x)\left(\varepsilon_{t}^{2}-1\right)^{2}\mid\mathcal{G}_{t-1}\right\\}\mid\mathcal{F}_{t-1}\right]\right\\}\\\
&\leq\mathcal{O}_{a.s.}(h_{2}^{\beta})\times U(x)\left\\{\pi(x)+\sup_{u\in
B(x,h)}|\pi(u)-\pi(x)|\right\\}\left\\{\frac{M_{2,W,2}}{M_{1,W,2}^{2}}\frac{1}{f_{1}(x)}+o_{a.s.}(1)\right\\}\\\
&\times\left\\{\omega(x)+\sup_{u\in
B(x,h)}|\omega(u)-\omega(x)|\right\\}\longrightarrow 0\,\ \hbox{as}\,\
n\to\infty.\end{aligned}$

Finally, we get

$\displaystyle\lim_{n\longrightarrow\infty}\sum_{t=1}^{n}\mathbb{E}\left(\xi_{n,t}^{2}\mid\mathcal{F}_{t-1}\right)=\lim_{n\longrightarrow\infty}\left(\Upsilon_{n,1}+\Upsilon_{n,2}+\Upsilon_{n,3}\right)=\frac{M_{2,W,2}U^{2}(x)\pi(x)\omega(x)}{M_{1,W,2}^{2}f_{1}(x)}=\sigma^{{}^{\prime}2}(x)\,\
\hbox{a.s.,}$

whenever $f_{1}(x)>0.$

Proof of part (ii). Consider $a>1$ and $b>1$ such that
$\displaystyle{\frac{1}{a}+\frac{1}{b}=1}$, by using Hölder’s and Markov’s
inequalities one can have, for all $\zeta>0$,

$\mathbb{E}\left\\{\xi_{n,t}^{2}1\kern-1.99997pt{\rm
l}_{\left(\left|\xi_{n,t}\right|>\zeta\right)}\right\\}\leq\frac{\mathbb{E}|\xi_{n,t}|^{2a}}{\zeta^{2a/b}}\text{.}$

Taking $2a=\kappa+2$ (where $\kappa$ is given in assumption (A3)(iii)) and let
$C_{0}$ be a positive constant. Under assumptions (A1)(i),
(A2)(iii)-(iv),(vi), (A3)(iii), (v), (A4)(iii) and by using Lemma 6.2, we
obtain

$\displaystyle n\mathbb{E}\left\\{\xi_{n,t}^{2}1\kern-1.99997pt{\rm
l}_{\left(\left|\xi_{n,t}\right|>\zeta\right)}\right\\}$ $\displaystyle\leq
C_{0}\left\\{\phi_{2}\left(h_{2}\right)/n\right\\}^{(2+\kappa)/2}\times\frac{n}{\left[\mathbb{E}\left(W_{1}(x)\right\\}\right]^{2+\kappa}}\mathbb{E}\left\\{\delta_{t}U^{2+\kappa}\left(X_{t}\right)W_{t}^{2+\kappa}(x)\left(\varepsilon_{t}^{2}-1\right)^{2+\kappa}\right\\}$
$\displaystyle\leq
C_{0}\left\\{n\phi_{2}\left(h_{2}\right)\right\\}^{-\kappa/2}\frac{M_{2+\kappa,W,2}f_{1}(x)+o(1)}{M_{1,W,2}^{2+\kappa}f_{1}^{2+\kappa}(x)+o(1)}\left\\{U^{2+\kappa}(x)+o(1)\right\\}\left\\{\pi(x)+o(1)\right\\}$
$\displaystyle=\mathcal{O}\left[\left\\{n\phi_{2}\left(h_{2}\right)\right\\}^{-\kappa/2}\right]\text{.}$

Finally, since $n\phi_{2}(h_{2})\longrightarrow\infty$ as
$n\longrightarrow\infty$, we get

$n\mathbb{E}\left\\{\xi_{n,t}^{2}1\kern-1.99997pt{\rm
l}_{\left(\left|\xi_{n,t}\right|>\zeta\right)}\right\\}=o(1).$

Therefore, Lemma 6.7 holds. ∎

### Proof of Theorem 3.3.

Note that statements in (6.18), (6.20) and (6.22), allow to say that
$\vartheta_{n,0}^{[1]}(x)$, $\vartheta_{n,0}^{[2]}(x)$,
$\vartheta_{n,0}^{[4]}(x)$ are negligible as $n$ goes to $\infty$. Moreover,
by using result given in (6.16), we can conclude that $U^{[1]}_{n,0}(x)$
converges almost surely to $\pi(x)$ as $n\rightarrow\infty$. Hence, the
asymptotic normality of $U_{n,0}(x)$ is achieved by the application of Lemma
6.7 along with Slutsky’s theorem.

### Proof of Corollary 3.1

Let us consider

$\dfrac{\widehat{M}_{1,W,2}}{\sqrt{\widehat{M}_{2,W,2}}}\sqrt{\dfrac{n\widehat{F}_{x,2}(h)\pi_{n}(x)}{\omega_{n}(x)U^{2}_{n,0}(x)}}(U_{n,0}(x)-U(x))\\\
=\dfrac{\widehat{M}_{1,W,2}\sqrt{M_{2,W,2}}}{\sqrt{\widehat{M}_{2,W,2}}M_{1,W,2}}\sqrt{\dfrac{n\widehat{F}_{x,2}(h)\pi_{n}(x)U^{2}(x)\omega(x)}{\omega_{n}(x)\pi(x)U^{2}_{n,0}(x)n\phi_{2}(h_{2})f_{1}(x)}}\dfrac{M_{1,W,2}}{\sqrt{M_{2,W,2}}}\sqrt{\dfrac{n\phi_{2}(h_{2})f_{1}(x)\pi(x)}{U^{2}(x)\omega(x)}}(U_{n,0}(x)-U(x)).$

By Theorem 3.3, we have

$\dfrac{M_{1,W,2}}{\sqrt{M_{2,W,2}}}\sqrt{\dfrac{n\phi_{2}(h_{2})f_{1}(x)\pi(x)}{U^{2}(x)\omega(x)}}(U_{n,0}(x)-U(x))\overset{\mathcal{D}}{\longrightarrow}\mathcal{N}\left(0,1\right).$

Then, by (A1)(i), (A2)(i),(iv), and following the same steps as the proof of
Corollary 1 in Laïb and Louani (2010), we get

(6.24)
$\widehat{M}_{1,W,2}\stackrel{{\scriptstyle\mathbb{P}}}{{\rightarrow}}M_{1,W,2},\quad\widehat{M}_{2,W,2}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\stackrel{{\scriptstyle\mathbb{P}}}{{\rightarrow}}M_{2,W,2},\quad\frac{\widehat{F}_{x,2}(h)}{\phi_{2}(h_{2})f_{1}(x)}\stackrel{{\scriptstyle\mathbb{P}}}{{\rightarrow}}1\quad\hbox{as}\quad
n\rightarrow\infty.$

In addition, from Theorem 3.2, we have
$U_{n,0}(x)\stackrel{{\scriptstyle\mathbb{P}}}{{\rightarrow}}U(x)$ and by
equation (5.25) in Ling et al. (2015)(p. 86), we have

(6.25)
$\quad\pi_{n}(x)\stackrel{{\scriptstyle\mathbb{P}}}{{\rightarrow}}\pi(x)\quad\hbox{as}\quad
n\rightarrow\infty.$

Then, it remains to prove that

(6.26)
$\quad\omega_{n}(x)\stackrel{{\scriptstyle\mathbb{P}}}{{\rightarrow}}\omega(x)\quad\hbox{as}\quad
n\rightarrow\infty.$

Observe that

$\displaystyle\omega_{n}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{G_{n}(x)}\frac{1}{n\mathbb{E}(H_{1}(x)}\sum_{t=1}^{n}\left(\frac{(Y_{t}-m_{n,0}(X_{t}))^{2}-U_{n,0}(X_{t})}{U_{n,0}(X_{t})}\right)^{2}H_{t}(x),$

where
$G_{n}(x)=\left(n\mathbb{E}(H_{1}(x))\right)^{-1}\displaystyle\sum_{t=1}^{n}H_{t}(x).$

Making use of Lemma 3 in Laïb and Louani (2011), we get

(6.27)
$G_{n}(x)\stackrel{{\scriptstyle\mathbb{P}}}{{\rightarrow}}1\quad\hbox{as}\quad
n\rightarrow\infty.$

Further, let $\mathfrak{{L}}_{t}$ be defined as

$\mathfrak{{L}}_{t}=\left(\frac{(Y_{t}-m_{n,0}(X_{t}))^{2}-U_{n,0}(X_{t})}{U_{n,0}(X_{t})}\right)^{2}H_{t}(x),\quad\hbox{for}\quad
t\in\\{1,\ldots,n\\}.$

Then

$\frac{1}{n\mathbb{E}(H_{1}(x)}\sum_{t=1}^{n}\mathfrak{{L}}_{t}=\mathfrak{T}_{1,n}(x)+\mathfrak{T}_{2,n}(x),$

where

$\mathfrak{T}_{1,n}(x)=\frac{1}{n\mathbb{E}(H_{1}(x)}\sum_{t=1}^{n}\left\\{\mathfrak{{L}}_{t}-\mathbb{E}(\mathfrak{{L}}_{t}|\mathcal{F}_{t-1})\right\\},\quad\mathfrak{T}_{2,n}(x)=\frac{1}{n\mathbb{E}(H_{1}(x)}\sum_{i=1}^{n}\mathbb{E}(\mathfrak{{L}}_{t}|\mathcal{F}_{t-1}).$

Let us turn our attention to the study of the second term
$\mathfrak{T}_{2,n}(x)$. By employing a double conditioning with respect to
the $\sigma$-field $\mathcal{G}_{t-1}$ and the $C_{r}$-inequality, we derive

(6.28)
$\mathfrak{T}_{2,n}(x)=\frac{1}{n\mathbb{E}(H_{1}(x)}\sum_{t=1}^{n}\mathbb{E}\left(H_{t}(x)\,\mathbb{E}\left(\left.\left\\{\frac{(Y_{t}-m_{n,0}(X_{t}))^{2}-U_{n,0}(X_{t})}{U_{n,0}(X_{t})}\right\\}^{2}\bigg{|}\mathcal{G}_{t-1}\right)\right|\mathcal{F}_{t-1}\right).$

Now, it is worth noting that Equation (2.1) provides:

$\displaystyle\mathbb{E}\left[\left\\{\frac{(Y_{t}-m_{n,0}(X_{t}))^{2}-U_{n,0}(X_{t})}{U_{n,0}(X_{t})}\right\\}^{2}\bigg{|}\mathcal{G}_{t-1}\right]$
$\displaystyle=\mathbb{E}\left[\left(\mathcal{A}+\mathcal{B}\right)^{2}|\mathcal{G}_{t-1}\right],$

where
$\mathcal{A}=\frac{\left\\{m(X_{t})-m_{n,0}(X_{t})\right\\}^{2}+\left\\{U(X_{t})-U_{n,0}(X_{t})\right\\}\varepsilon_{t}^{2}+2\left\\{m(X_{t})-m_{n,0}(X_{t})\right\\}\sqrt{U(X_{t})}\varepsilon_{t}}{U_{n,0}(X_{t})},$
and $\mathcal{B}=\varepsilon_{t}^{2}-1$
Using assumption (A3)(i), the first part of condition (2.2), and Theorems 3.1
and 3.2, we have

$\mathbb{E}\left(\mathcal{A}^{2}|\mathcal{G}_{t-1}\right)=o_{\mathbb{P}}(1).$

Similarly, condition (2.2), assumption (A3)(i), and Theorems 3.1 and 3.2 imply

$\mathbb{E}(\mathcal{A}\mathcal{B}\big{|}\mathcal{G}_{t-1})=o_{\mathbb{P}}(1).$

Finally, assumption (A3)(ii) ensures that, as $n\to\infty$, we get

(6.29)
$\mathbb{E}\left[\left\\{\frac{(Y_{t}-m_{n,0}(X_{t}))^{2}-U_{n,0}(X_{t})}{U_{n,0}(X_{t})}\right\\}^{2}\bigg{|}\mathcal{G}_{t-1}\right]=\omega(x)+o(1).$

From (6.28) with (6.29), we obtain that, as $n\to\infty$,

$\mathfrak{T}_{2,n}(x)=(\omega(x)+o(1))\frac{1}{n\mathbb{E}(H_{1}(x))}\sum_{t=1}^{n}\mathbb{E}\left(H_{t}(x)|\mathcal{F}_{t-1}\right).$

Then, by using Lemma 7 in Laïb and Louani (2011), under assumptions (A1)(i)
and (A2)(i)-(vi), it follows that

$\frac{1}{n\mathbb{E}(H_{1}(x))}\sum_{t=1}^{n}\mathbb{E}\left(H_{t}(x)|\mathcal{F}_{t-1}\right){\longrightarrow}1\,\
\hbox{a.s.}\,\ \text{ as }\,\ n\to\infty.$

Hence, we obtain

$\mathfrak{T}_{2,n}(x)\stackrel{{\scriptstyle\mathbb{P}}}{{\longrightarrow}}\omega(x)\,\
\text{ as }\,\ n\to\infty.$

Now, we need to show that $\mathfrak{T}_{1,n}(x)$ goes to zero in probability
as $n\to\infty$. Using Markov’s, Burkholder’s, and Jensen’s inequalities, one
obtains, for any $\eta>0$,

$\mathbb{P}\left\\{|\mathfrak{T}_{1,n}(x)|>\eta\right\\}\leq\frac{C\mathbb{E}(\mathfrak{{L}}_{1}^{2})}{{n\eta^{2}\left(\mathbb{E}(H_{1}(x))\right)^{2}}},$

where $C$ is a generic positive constant. Using the $C_{r}$-inequality, we
have

$\mathbb{E}(\mathfrak{{L}}_{t}^{2}|X_{t}))\leq
8\left(\frac{1}{U^{4}_{n,0}(X_{t})}\mathbb{E}\left\\{\mathbb{E}\left[\left(m(X_{t})-m_{n}(X_{t})+\sqrt{U(X^{t})}\varepsilon^{t}\right)^{8}\mid
X_{t}\right]\right\\}+\mathbb{E}\left((-1)^{4}\mid
X_{t}\right)\right).\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$

By using Binomial Theorem, we have

$\displaystyle\left((m(X_{t})-m_{n,0}(X_{t}))+\sqrt{U(X_{t})}\varepsilon_{t}\right)^{8}$
$\displaystyle=\sum_{k=0}^{8}\binom{8}{k}(m(X_{t})-m_{n,0}(X_{t}))^{k}(\sqrt{U(X_{t})}\varepsilon_{t})^{8-k}$
$\displaystyle=(m(X_{t})-m_{n,0}(X_{t}))^{8}+U^{4}(X_{t})\varepsilon_{t}^{8}$
$\displaystyle+8\left((m(X_{t})-m_{n,0}(X_{t}))^{7}(\sqrt{U(X_{t})}\varepsilon_{t})\right)$
$\displaystyle+28\left((m(X_{t})-m_{n,0}(X_{t}))^{6}U(X_{t})\varepsilon_{t}^{2}\right)$
$\displaystyle+56\left((m(X_{t})-m_{n,0}(X_{t}))^{5}\sqrt{U(X_{t})}U(X_{t})\varepsilon_{t}^{3}\right)$
$\displaystyle+70\left((m(X_{t})-m_{n,0}(X_{t}))^{4}U^{2}(X_{t})\varepsilon_{t}^{4}\right]$
$\displaystyle+56\left((m(X_{t})-m_{n,0}(X_{t}))^{3}\sqrt{U(X_{t})}U^{2}(X_{t})\varepsilon_{t}^{5}\right)$
$\displaystyle+28\left((m(X_{t})-m_{n,0}(X_{t}))^{2}U^{3}(X_{t})\varepsilon_{t}^{6}\right)$
$\displaystyle+8\left((m(X_{t})-m_{n,0}(X_{t}))\sqrt{U(X_{t})}U^{3}(X_{t})\varepsilon_{t}^{7}\right).$

Making use of Theorems 3.1 and 3.2, and assumptions (A3)(i),(iv), (A4)(ii), we
can prove that

$\mathbb{E}\left[\left(m(X_{t})-m_{n}(X_{t})+\sqrt{U(X_{t})}\varepsilon_{t}\right)^{8}\mid
X_{t}\right]<\infty.$

Therefore, By using Theorem 3.2 and assumption (A4)(ii), one gets

$\mathbb{E}(\mathbb{E}(\mathfrak{{L}}_{1}^{2}|X_{1}))<\infty.$

Thus, by Lemma 6.2, we have

$P\left\\{|\mathfrak{T}_{1,n}(x)|>\eta\right\\}\frac{C\mathbb{E}(H_{1}^{2}(x))}{{n\eta^{2}E^{2}(H_{1}(x))}}\leq\frac{C(M_{2,H,3}f_{1}(x)+o(1))}{n\eta^{2}(M_{1,H,3}^{2}f_{1}^{2}(x)+o(1))},$

which goes to zero as $n\to\infty$. The proof of equation (6.26) is complete.

Finally, we obtain

$\frac{\widehat{M}_{1,W,2}}{\sqrt{\widehat{M}_{2,W,2}}}\frac{\sqrt{M_{2,W,2}}}{M_{1,W,2}}\sqrt{\frac{n\widehat{F}_{x,2}(h)\pi_{n}(x)U^{2}(x)\omega(x)}{\omega_{n}(x)U^{2}_{n,0}(x)\pi(x)n\phi_{2}(h_{2})f_{1}(x)}}\stackrel{{\scriptstyle\mathbb{P}}}{{\rightarrow}}1\quad\hbox{as}\quad
n\rightarrow\infty.$

Hence, the proof of Corollary 3.1 is achieved.

### Proof of Theorem 3.4.

By using Model (2.1), we have

$\displaystyle m_{n,1}(x)-m(x)$
$\displaystyle=\frac{1}{G^{\prime}_{n}(x)}\left[\frac{1}{n\mathbb{E}(K_{1}(x))}{\sum_{t=1}^{n}}[\delta_{t}\\{m(X_{t})+\sqrt{U(X_{t})}\varepsilon_{t}\\}+(1-\delta_{t})m_{n,0}(X_{t})-m(x)]K_{t}(x)\right]$
$\displaystyle=\frac{1}{G^{\prime}_{n}(x)}\left[\mathcal{I}_{n,1}(x)+\mathcal{I}_{n,2}(x)+V_{n,0}(x)\right],$

where $V_{n,0}(x)$ is defined in (6.2) and

$\displaystyle\mathcal{I}_{n,1}(x)$
$\displaystyle=\frac{1}{n\mathbb{E}\left[K_{1}(x)\right]}\sum_{t=1}^{n}(1-\delta_{t})\left[m_{n,0}\left(X_{t}\right)-m\left(X_{t}\right)\right]K_{t}(x),$
$\displaystyle\mathcal{I}_{n,2}(x)$
$\displaystyle=\frac{1}{n\mathbb{E}\left[K_{1}(x)\right]}\sum_{t=1}^{n}\left(m\left(X_{t}\right)-m(x)\right)K_{t}(x),$
$G^{\prime}_{n}(x)=\frac{1}{n\mathbb{E}\left[K_{1}(x)\right]}\sum_{t=1}^{n}K_{t}(x).$

Then, we find

(6.29)
$\displaystyle{\sup_{x\in\mathcal{C}}}\left|m_{n,1}(x)-m(x)\right|\leq\frac{\displaystyle{\sup_{x\in\mathcal{C}}}\left|\mathcal{I}_{n,1}(x)\right|+\displaystyle{\sup_{x\in\mathcal{C}}}\left|\mathcal{I}_{n,2}(x)\right|+\displaystyle{\sup_{x\in\mathcal{C}}}\left|V_{n,0}(x)\right|}{\displaystyle{\inf_{x\in\mathcal{C}}}\left|G^{\prime}_{n}(x)\right|}.$

Laïb and Louani (2011) (in p. 371) have proved that, as $n\rightarrow\infty$,

(6.30)
$\displaystyle{\inf_{x\in\mathcal{C}}}\left|G^{\prime}_{n}(x)\right|>1\quad\text{a.s.}$

Next, by using assumption (A4)(i) and the almost sure convergence of
$G^{\prime}_{n}(x)$ to 1 uniformly in $x$ (see Lemma 7 in Laïb and Louani
(2011)), it follows that

(6.31)
$\sup_{x\in\mathcal{C}}|\mathcal{I}_{n,2}(x)|=\mathcal{O}_{a.s.}(h_{1}^{\alpha}).$

Furthermore, by the use of Theorem 3.1, the almost sure boundedness of
$\delta$ by 1 and the almost sure uniform convergence of $G^{\prime}_{n}(x)$
to 1, we get

(6.32)
$\displaystyle{\sup_{x\in\mathcal{C}}}\left|\mathcal{I}_{n,1}(x)\right|=\mathcal{O}_{a.s.}(h_{1}^{\alpha})+\mathcal{O}_{a.s.}\left(\lambda_{n}\right)\text{.}$

Finally, using equation (6.29) combining with Lemma 6.4 and results (6.30),
(6.31), (6.32), we obtain

$\displaystyle{\sup_{x\in\mathcal{C}}}\left|m_{n,1}(x)-m(x)\right|=\mathcal{O}_{a.s.}(h_{1}^{\alpha})+\mathcal{O}_{a.s.}\left(\lambda_{n}\right)\text{.}$

### Proof of Theorem 3.5.

Let

(6.33) $\displaystyle U_{n,1}(x)-U(x)$
$\displaystyle=\frac{1}{\displaystyle\sum_{t=1}^{n}W_{t}(x)}{\displaystyle\sum_{t=1}^{n}[\delta_{t}(Y_{t}-m_{n,0}(X_{t}))^{2}+(1-\delta_{t})U_{n,0}(X_{t})]W_{t}(x)}-U(x)$
$\displaystyle=\frac{1}{G^{\prime\prime}_{n}(x)}\\{\vartheta_{n,0}^{[1]}(x)+\vartheta_{n,0}^{[2]}(x)+\vartheta_{n,0}^{[3]}(x)+\Theta_{n,1}(x)+\Theta_{n,2}(x)\\},$

where
$\vartheta_{n,0}^{[1]}(x),\vartheta_{n,0}^{[2]}(x),\vartheta_{n,0}^{[3]}(x)$
are defined in (6.12), (6.13), (6.14), respectively and

$\begin{array}[]{lll}\Theta_{n,1}(x)&=&\displaystyle\frac{1}{{n\mathbb{E}(W_{1}(x))}}\displaystyle\displaystyle\sum_{t=1}^{n}(1-\delta_{t})[U_{n,0}(X_{t})-U(X_{t})]W_{t}(x),\\\
\Theta_{n,2}(x)&=&\displaystyle\frac{1}{{n\mathbb{E}(W_{1}(x))}}\displaystyle\displaystyle\sum_{t=1}^{n}[U(X_{t})-U(x)]W_{t}(x),\\\
G^{\prime\prime}_{n}(x)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&=&\displaystyle\frac{1}{n\mathbb{E}\left[W_{1}(x)\right]}\sum_{t=1}^{n}W_{t}(x).\end{array}$
Then,

(6.34)
$\displaystyle{\sup_{x\in\mathcal{C}}}\left|U_{n,1}(x)-U(x)\right|\leq\\{\displaystyle{\sup_{x\in\mathcal{C}}}\left|\vartheta_{n,0}^{[1]}(x)\right|+\displaystyle{\sup_{x\in\mathcal{C}}}\left|\vartheta_{n,0}^{[2]}(x)\right|+\displaystyle{\sup_{x\in\mathcal{C}}}\left|\vartheta_{n,0}^{[3]}(x)\right|+\displaystyle{\sup_{x\in\mathcal{C}}}\left|\Theta_{n,1}(x)\right|+\displaystyle{\sup_{x\in\mathcal{C}}}\left|\Theta_{n,2}(x)\right|\\}\times\\{\displaystyle{\inf_{x\in\mathcal{C}}}\left|G^{\prime\prime}_{n}(x)\right|\\}$

According to Lemma 7 in Laïb and Louani (2011), we can deduce that:

(6.35)
$\lim_{n\rightarrow\infty}\sup_{x\in\mathcal{C}}|G^{\prime\prime}_{n}(x)-1|=0\quad\text{a.s.}$

Similar as for (6.30), one has

(6.36)
$\displaystyle{\inf_{x\in\mathcal{C}}}\left|G^{\prime\prime}_{n}(x)\right|>1\quad\text{a.s.}\
\quad\text{as}\quad n\rightarrow\infty$

Making use of Theorem 3.2, equation (6.35) and the almost sure boundedness of
$\delta$, one gets

(6.37)
$\displaystyle{\sup_{x\in\mathcal{C}}}\left|\Theta_{n,1}(x)\right|=\mathcal{O}_{a.s.}(h_{1}^{2\alpha}+h_{2}^{\beta})+\mathcal{O}_{a\cdot
s.}(\lambda_{n}^{\prime}+\lambda_{n}^{2}).$

In addition, by using assumption (A4)(ii) and equation (6.35), we obtain

(6.38)
$\displaystyle{\sup_{x\in\mathcal{C}}}|\Theta_{n,2}(x)|=\mathcal{O}_{a.s.}(h_{2}^{\beta}).$

Finally, decomposition (6.34), combined with equations (6.18), (6.20), (6.21),
(6.36),(6.37), (6.38) allows to conclude that

$\displaystyle{\sup_{x\in\mathcal{C}}}|U_{n,1}(x)-U(x)|=\mathcal{O}_{a.s.}(h_{1}^{2\alpha}+h_{2}^{\beta})+\mathcal{O}_{a\cdot
s.}(\lambda_{n}^{\prime}+\lambda_{n}^{2}).$

### Proof of Theorem 3.6.

Using decomposition in (6.33), equations (6.18), (6.20) and (6.37), (6.38)
allow to conclude that $\vartheta_{n,0}^{[1]}(x)$, $\vartheta_{n,0}^{[2]}(x)$,
$\Theta_{n,1}(x)$ and $\Theta_{n,2}(x)$ are negligible as $n\to\infty.$
Furthermore, according to equation (6.35), $G^{\prime\prime}_{n}(x)$ converges
almost surely to 1. Finally, we can conclude that the asymptotic distribution
of the nonparametric imputed conditional variance is determined by the
asymptotic variance of the term $\vartheta_{n,0}^{[3]}(x)$ which is specified
in Lemma 6.7.

### Proof of Corollary 3.2

First, we note that
$\dfrac{\widehat{M}_{1,W,2}}{\sqrt{\widehat{M}_{2,W,2}}}\sqrt{\dfrac{n\widehat{F}_{x,2}(h)}{\omega_{n}(x)\pi_{n}(x)U^{2}_{n,1}(x)}}(U_{n,1}(x)-U(x))\\\
=\dfrac{\widehat{M}_{1,W,2}\sqrt{M_{2,W,2}}}{\sqrt{\widehat{M}_{2,W,2}}M_{1,W,2}}\sqrt{\dfrac{n\widehat{F}_{x,2}(h)\omega(x)\pi(x)U^{2}(x)}{\omega_{n}(x)\pi_{n}(x)U^{2}_{n,1}(x)n\phi_{2}(h_{2})f_{1}(x)}}\dfrac{M_{1,W,2}}{\sqrt{M_{2,W,2}}}\sqrt{\dfrac{n\phi_{2}(h_{2})f_{1}(x)}{\omega(x)\pi(x)U^{2}(x)}}(U_{n,1}(x)-U(x)).$

By Theorem 3.6, we find

$\frac{M_{1,W,2}}{\sqrt{M_{2,W,2}}}\sqrt{\frac{n\phi_{2}(h_{2})f_{1}(x)}{\omega(x)\pi(x)U^{2}(x)}}(U_{n,1}(x)-U(x))\overset{\mathcal{D}}{\longrightarrow}\mathcal{N}\left(0,1\right).$

Then, in view of Theorem 3.5 and equations (6.24), (6.25) and (6.26), we
obtain

$\frac{\widehat{M}_{1,W,2}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}{\sqrt{\widehat{M}_{2,W,2}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}\frac{\sqrt{M_{2,W,2}}}{M_{1,W,2}}\sqrt{\frac{n\widehat{F}_{x,2}(h)\omega(x)\pi(x)U^{2}(x)}{\omega_{n}(x)\pi_{n}(x)U^{2}_{n,1}(x)n\phi_{2}(h_{2})f_{1}(x)}}\stackrel{{\scriptstyle\mathbb{P}}}{{\rightarrow}}1\quad\hbox{as}\quad
n\rightarrow\infty.$

Therefore, the proof of Corollary 3.2 is completed.

## Appendix

######  Lemma 6.8.

Assume that $\left(X_{t},\varepsilon_{t}\right)$ is a strictly stationary
ergodic process and suppose that assumptions (A3)(i),(v) hold. Then, for each
$\omega$ outside a null set $D$, there exists a positive integer
$n_{0}(\omega)$ such that $V_{n,0}(x)=V_{n,0}^{\top}(x)$ for $n\geq
n_{0}(\omega)$ and $x\in\mathcal{E}$.

###### Proof.

Recall that $\mathcal{L}_{t}=\delta_{t}\sqrt{U(X_{t})}\varepsilon_{t}.$ Then,
for every $\eta>0$ and by using Markov’s inequality, one has

$\displaystyle\mathbb{P}\left({\left|\frac{\mathcal{L}_{t}}{\ell_{n}}\right|}>\eta\right)=\mathbb{P}\left(|\mathcal{L}_{t}|>\eta\ell_{n}\right)$
$\displaystyle\leq{\frac{\mathbb{E}(|\mathcal{L}_{t}|)}{\eta\ell_{n}}}$
$\displaystyle\leq{\frac{\mathbb{E}(\delta_{t}|\sqrt{U(X_{t})}\varepsilon_{t}|)}{\eta\ell_{n}}}.$

Let $a>1$ and $b>1$ be real numbers such that $1/a+1/b=1$. By using the
Hölder’s inequality, it follows that

$\begin{aligned}
\mathbb{P}\left(\displaystyle{\left|\frac{\mathcal{L}_{t}}{\ell_{n}}\right|}>\eta\right)&\leq\frac{1}{\eta\ell_{n}}\mathbb{E}^{1/a}\left(\delta_{t}\left|\sqrt{U(X_{t})}\right|^{a}\right)\times\mathbb{E}^{1/b}(|\varepsilon_{t}|^{b})\end{aligned}.$

By a double conditioning with respect to the $\sigma$-field
$\mathcal{G}_{t-1}$ and by taking $a=\rho/(\rho-1)$ and $b=\rho,$ we get

$\displaystyle\mathbb{P}\left({\left|\frac{\mathcal{L}_{t}}{\ell_{n}}\right|}>\eta\right)$
$\displaystyle\leq\frac{1}{\eta\ell_{n}}\mathbb{E}^{(\rho-1)/\rho}\left(\delta_{t}\left|\sqrt{U(X_{t})}\right|^{\rho/(\rho-1)}\right)\times\mathbb{E}^{1/\rho}[\mathbb{E}(|\varepsilon_{t}|^{\rho}|\mathcal{G}_{t-1})].$

Making use the second part of assumption (A3)(i), one finds

$\displaystyle\mathbb{P}\left({\left|\frac{\mathcal{L}_{t}}{\ell_{n}}\right|}>\eta\right)$
$\displaystyle\leq\frac{C}{\eta\ell_{n}}\mathbb{E}^{(\rho-1)/\rho}\left(\delta_{t}\left|\sqrt{U(X_{t})}\right|^{\rho/(\rho-1)}\right).$

Note that another use of the Hölder inequality allows us to bound the quantity
$\mathbb{E}^{(\rho-1)/\rho}\left(\delta_{t}\left|\sqrt{U(X_{t})}\right|^{\rho/(\rho-1)}\right)$
as follows:

$\displaystyle\left|\mathbb{E}^{(\rho-1)/\rho}\left(\delta_{t}\left|\sqrt{U(X_{t})}\right|^{\rho/(\rho-1)}\right)\right|$
$\displaystyle\leq\mathbb{E}^{(\rho-1)/b\rho}\left(\left|\sqrt{U(X_{t})}\right|^{b\rho/(\rho-1)}\right)\times\mathbb{E}^{(\rho-1)/a\rho}\left(|\delta_{t}|^{a}\right).$

By a double conditioning with respect to the $\sigma$-field
$\mathcal{G}_{t-1}$ and taking the same values of a and b with the use of
assumption (A3)(v), it follows that

$\displaystyle\left|\mathbb{E}^{(\rho-1)/\rho}\left(\delta_{t}\left|\sqrt{U(X_{t})}\right|^{\rho/(\rho-1)}\right)\right|$
$\displaystyle\leq\mathbb{E}^{(\rho-1)/b\rho}\left(\left|\sqrt{U(X_{t})}\right|^{b\rho/(\rho-1)}\right)\times\mathbb{E}^{(\rho-1)/a\rho}[\mathbb{E}\left(\delta_{t}|\mathcal{G}_{t-1}\right)]$
$\displaystyle\leq\mathbb{E}^{(\rho-1)/b\rho}\left(\left|\sqrt{U(X_{t})}\right|^{b\rho/(\rho-1)}\right)\times(\pi(x)+o(1))^{(\rho-1)/a\rho}.$

By putting $M=(\pi(x)+o(1))^{(\rho-1)/a\rho}$, one has

$\displaystyle\left|\mathbb{E}^{(\rho-1)/\rho}\left(\delta_{t}\left|\sqrt{U(X_{t})}\right|^{\rho/(\rho-1)}\right)\right|$
$\displaystyle\leq
M\mathbb{E}^{(\rho-1)/\rho^{2}}\left(\left|\sqrt{U(X_{t})}\right|^{\rho^{2}/(\rho-1)}\right).$

which is finite by the first part of assumption (A3)(i).
Thus,

$\displaystyle\mathbb{P}\left({\left|\frac{\mathcal{L}_{t}}{\ell_{n}}\right|}>\eta\right)$
$\displaystyle\leq\frac{\hat{C}}{\eta\ell_{n}}.$

Since $0<\ell_{n}\uparrow\infty$ and $\eta>0$, we get

$\mathcal{L}_{t}/\ell_{n}\longrightarrow 0\quad\text{a.s.}$

Therefore, for $\omega\in D^{c}$ with $\mathbb{P}(D^{c})=1$ and some positive
integer $n_{0}(\omega)$, it follows that

$|\mathcal{L}_{t}(\omega)|<\ell_{n},\quad\quad t=1\ldots n,\quad n\geq
n_{0}(\omega).$

Finally, $V_{n,0}(x)=V^{\top}_{n,0}(x).$

∎

Proof of Lemma 6.5

Let $a>1$ and $b>1$ be real numbers such that $1/a+1/b=1$. By using assumption
A3(v) and the Hölder’s and Markov’s inequalities, we can get

$\displaystyle\left|\mathbb{E}\left\\{\mathcal{L}_{t}1\kern-1.99997pt{\rm
l}_{\left(\left|\mathcal{L}_{t}\right|\leq\ell_{n}\right)}\mid\mathcal{F}_{t-1}\right\\}\right|$
$\displaystyle\leq\mathbb{E}^{1/a}\left(\left|\mathcal{L}_{t}\right|^{a}\mid\mathcal{F}_{t-1}\right)\times\mathbb{E}^{1/b}\left\\{1\kern-1.99997pt{\rm
l}_{\left(\left|\mathcal{L}_{t}\right|\leq\ell_{n}\right)}\mid\mathcal{F}_{t-1}\right\\}$
$\displaystyle\leq\mathbb{E}^{1/a}\left(\left|\mathcal{L}_{t}\right|^{a}\mid\mathcal{F}_{t-1}\right)\times\frac{\mathbb{E}^{1/b}\left(\left|\mathcal{L}_{t}\right|\mid\mathcal{F}_{t-1}\right)}{\ell_{n}^{1/b}}$
$\displaystyle\leq\frac{\mathbb{E}\left(\left|\mathcal{L}_{t}\right|^{a}\mid\mathcal{F}_{t-1}\right)}{\ell_{n}^{a/b}}=\frac{\mathbb{E}\left(|\delta_{t}|\left|\sqrt{U\left(X_{t}\right)}\right|^{a}\left|\varepsilon_{t}\right|^{a}\mid\mathcal{F}_{t-1}\right)}{\ell_{n}^{a/b}}$
$\displaystyle\leq(\pi(x)+o(1))\frac{\mathbb{E}\left(\left|\sqrt{U\left(X_{t}\right)}\right|^{a}\left|\varepsilon_{t}\right|^{a}\mid\mathcal{F}_{t-1}\right)}{\ell_{n}^{a/b}}.$

Then, by using Lemma 6.2, assumptions (A1), (A2)(i), (iv), (vi) and (A3)(i),
along with the first part of condition (3.5) and by following similar steps as
the proof of Lemma B in Chaouch (2019) with specific values for a and b
($a=\rho$ and $b=\rho/(\rho-1)$), we obtain

$\sup_{x\in\mathcal{C}}\left|\widetilde{V}_{n,0}(x)\right|\leq\frac{C}{\ell_{n}^{\rho-1}\phi_{1}\left(h_{1}\right)^{(\rho-1)/\rho}}\,\
\hbox{almost surely as}\,\ n\rightarrow\infty.$

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