# Modified wavelet variation for the Hermite processes

L. Loosveldt*** Université de Liège, Département de Mathématique – zone
Polytech 1, 12 allée de la Découverte, Bât. B37, B-4000 Liège.
<EMAIL_ADDRESS>C.A. Tudor†††Université de Lille, CNRS, UMR 8524 -
Laboratoire Paul Painlevé, F-59000 Lille<EMAIL_ADDRESS>
###### Abstract

We define an asymptotically normal wavelet-based strongly consistent estimator
for the Hurst parameter of any Hermite processes. This estimator is obtained
by considering a modified wavelet variation in which coefficients are wisely
chosen to be, up to negligeable remainders, independent. We use Stein-
Malliavin calculus to prove that this wavelet variation satisfies a
multidimensional Central Limit Theorem, with an explicit bound for the
Wasserstein distance.

Keywords: Hermite process, multiple Wiener-Itô integrals, wavelet analysis,
Stein-Malliavin calculus, asymptotic normality, central limit theorem, self-
similarity, Hurst parameter estimation, strong consistency.

2020 MSC: 60G18, 60H05, 60H07, 62F12, 60F05

##  1 Introduction

A stochastic process $\\{X_{t}\\}_{t\geq 0}$ is said to be $H$-self-similar
if, for all $a>0$, the processes $\\{X_{at}\\}_{t\geq 0}$ and
$\\{a^{H}X_{t}\\}_{t\geq 0}$ have the same finite dimensional distributions.
This property is predominant in real life applications, for instance, in
astronomy [34, 47], biology [1, 24], climatology [14, 30], hydrology [39, 52],
image processing [31], internet traffic modelling [58, 59], mathematical
finance [28, 37, 46, 50] and physics [40, 48]. We also refer to the three
monographs [11, 23, 25] for an insight on numerous results, applications and
methodologies concerning self-similar processes.

Without any doubt, fractional Brownian motion is the most popular self-similar
stochastic process. It was first introduced by Kolmogorov, in the paper [33]
from 1940, to define “Gaussian spirals” in Hilbert spaces. The name
“fractional Brownian motion” is used since the article [38] from Mandelbrot
and Van Ness, where the first systematic study of this process was carried
out. Indeed, it is itself a generalization of the Brownian motion, defined by
the botanist Robert Brown to describe the movements of pollen grains of the
plant Clarkia Pulchella suspended in the water [13]. For any $H\in(0,1)$, the
fractional Brownian motion of Hurst parameter $H$ on a probability space
$(\Omega,\mathcal{F},\mathbf{P})$ is the unique centred Gaussian process
$\\{B^{H}_{t}\\}_{t\geq 0}$ with covariance function

$\mathbf{E}[B^{H}_{t}B^{H}_{S}]=\frac{1}{2}(|t|^{2H}+|s|^{2H}-|t-s|^{2H}),\mbox{
for every }s,t\geq 0.$

The case $H=1/2$ corresponds to the usual Brownian motion. In other words, the
fractional Brownian motion $\\{B^{H}_{t}\\}_{t\geq 0}$ is the only Gaussian
process with stationary increments which is $H$-self-similar. This property
justifies the fact that the fractional Brownian is greatly used to model
natural phenomena [24, 30, 34, 40, 47].

In some applications, it happens that the self-similarity and the
stationnarity of increments are desirable properties for a stochastic process,
while Gaussianity is not a reasonable assumption [28, 50, 52, 58, 60]. In
these contexts, good candidates for simulation are given by Hermite processes.
In the sequel, for all integer number $q\geq 1$, $I_{q}$ denotes the multiple
stochastic integral of order $q$ with respect to the two-sided Brownian motion
$\\{B_{y}\\}_{y\in\mathbb{R}}$, see the Appendix for a precise definition. Let
$H\in\left(\frac{1}{2},1\right)$, the Hermite process of order $q$ and Hurst
parameter $H$ is the stochastic process $\\{Z^{(q,H)}_{t}\\}_{t\geq 0}$
defined, for every $t\geq 0$, by

$Z^{(q,H)}_{t}=I_{q}(L^{(q,H)}_{t}),$ (1)

where the kernel function $L^{(q,H)}_{t}$ is given, for all
$y_{1},...,y_{q}\in\mathbb{R}$, by

$L^{(q,H)}_{t}(y_{1},...,y_{q})=c_{q,H}\int_{\mathbb{R}}(u-y_{1})_{+}^{-\left(\frac{1}{2}+\frac{1-H}{q}\right)}\ldots(u-y_{q})_{+}^{-\left(\frac{1}{2}+\frac{1-H}{q}\right)}du,$
(2)

with $c_{q,H}$ a strictly positive constant which is chosen such that, for all
$t\geq 0$,
$\mathbf{E}[(Z^{(q,H)}_{t})^{2}]=q!\|L^{(q,H)}_{t}\|^{2}_{L^{2}({R}^{q})}=t^{2H}$.
It is a $H$-self-similar stochastic process with stationary increments. Its
exhibits long-range dependence and its sample paths are Hölder continuous of
order $\delta$ for every $\delta\in(0,H)$. Hermite processes first appeared as
limit of partial sums of correlated random variables, in the so-called Non-
Central Limit Theorem [22, 51, 53]. The class of Hermite processes contains
the fractional Brownian motion which is obtained for $q=1$ and it is the only
Gaussian process in this class. It also contained the Rosenblatt process
(obtained for $q=2$). We refer to the recent monograph [56] for a concise
presentation of these processes and their stochastic analysis. The results
present on this paper do not depend on the exact value of $q$ and $H$. For
this reason, to ease the notation, we omit the indices and we write
$\\{Z_{t}\\}_{t\geq 0}$ for a Hermite process as well as $L_{t}$ for its
associated kernel.

Generally speaking, while working with self-similar processes, a question of
great interest is to estimate the parameter $H$ here over. It is essential
because this parameter, governs the main properties of the analysed process,
such as the long-range dependence and the Hölder regularity. Therefore, in the
Hermite case, many authors have proposed statistical estimators from various
perspectives, such as variations [15, 18, 55], wavelet analysis [10, 7, 17,
29] or least-squares methods [44].

Unfortunately, all the estimators presented in the publications listed here
over failed to be asymptotically normal as soon as $q\geq 2$. Also, in the
case $q=1$, the asymptotic normality only holds when $H\in(0,\frac{3}{4}]$.
Note that, in this last Gaussian case, the authors in [32] propose an
estimator based on higher order increments of the fractional Brownian motion
and obtain the asymptotic normality of this estimator. But, as remarked in
[16], this strategy is not applicable for $q\geq 2$. Of course, this non-
Gaussian behavior is a serious drawback for statistical estimation of the
Hurst parameter $H$, as the limit distribution is a Rosenblatt one, much less
desirable for practical purposes.

But, very recently, in [6], Ayache and Tudor define a new estimator for $H$,
based on a modified quadratic variation, which is asymptotically normal, even
in the case $q\geq 2$. The construction of this estimator is based on an idea
from [2] where increments of Hermite processes are split in two parts: one
which satisfies some independence properties while the second part is clearly
dominated, in $L^{2}(\Omega)$-norm, by the first one. These observations help
to select “good” increments to compute quadratic variations which are then
collected in a simple modification of the estimators previously introduced in
[15, 55].

Nevertheless, in practice, estimators for the Hurst parameter based on wavelet
analysis provide numerous advantages, compared to the ones obtained with other
strategies. Indeed, wavelet-based estimators are defined using a log-
regression of wavelet coefficients at various scales, see Section 2 for the
definition, and a great literature is available concerning goodness-of-fit
tests for such models [8, 9, 7]. Moreover, such estimators are numerically
efficient thanks to the Mallat’s algorithm for computing wavelet coefficients
[35, 36]. Also, if the wavelet have enough vanishing moment, see equation (3)
below, wavelet-based estimators are not sensible to polynomials trends and are
thus very robust.

In the paper [21], inspired by [2], Daw and Loosveldt adapted the “splitting
method” in the context of wavelet coefficients for the Rosenblatt process (aka
the Hermite process of order $2$). In this context, they manage to express any
wavelet coefficient as the sum of two random variables, one which satisfies
some independence property and one which is negligible.

The aim of this paper is to bring together the ideas from [6] and [21] and to
propose a new strongly consistent estimator for the Hurst parameter of any
Hermite processes. It belongs to the “family” of the estimators introduced in
[10, 7, 17, 29], in the sense that it relies on a wavelet variation. But the
wavelet coefficients used in this “modified” variation are precisely chosen
such that this new estimator is asymptotically normal. In short, this
estimator is particularly well-suited for applications as it is an
asymptotically normal wavelet-based strongly consistent estimator. Let us note
that, up to our knowledge, it is anyway the first time that a wavelet-based
estimator is defined for Hermite processes of any orders. Indeed, the papers
[10, 7, 17, 29] only deal with the fractional Brownian motion and the
Rosenblatt process, thus the Hermite processes of order one and two.

In Section 2, we recall some basic facts about wavelet analysis on which
relies the rest of our paper. Then we extend to any Hermite process the
results from [21] concerning the decomposition of wavelet coefficients.

In Section 3, we introduce the modified wavelet variation which is the key
object to define later our estimator. First, we present the special wavelet
coefficients that are used in the variation. Then, we show that a part of this
variation is clearly negligible compared to the other part. It is a
consequence of the decomposition of wavelet coefficients provided in Section
2. Finally, we prove a multidimensional Central Limit Theorem (CLT) for the
modified wavelet variation. Note that the use of a multidimensional CLT is a
major difference with [6], where only a “classic” CLT is used. It is a
consequence of the fact that our estimator relies on wavelet coefficients and
not directly on increments of the process. Indeed, as the reader can see all
along this paper, the use of a wavelet $\Psi$ make appeared a constant
$C_{\Psi}(H)$ in all our computations, see equation (7) below. As this
constant depends, in a non-trivial way, on the value of the Hurst parameter
$H$, it must be eliminated in the definition of the estimator. Then, the
standard strategy from [10, 7] consists in performing a $\log$-regression of
the wavelet variation onto several scales. Thus, we need to insure the
asymptotic normality for a vector of wavelet variation instead of an unique
variation based on increments.

In order to obtain an estimator which can be numerically computed, in Section
4 we study a Riemann approximation of the wavelet variation defined in Section
3. The main result there states that this approximation converges, in
$L^{1}(\Omega)$, to the wavelet variation and thus also satisfies a
multidimensional CLT.

Finally, in Section 5, we use a $\log$-regression to define an estimator for
the Hurst parameter $H$. We show that this estimator is strongly consistent
and asymptotically normal.

Main facts of stochastic analysis, in particular Malliavin-Stein method, are
recalled, for the reader convenience, in the Appendix.

##  2 The wavelet coefficients and their decomposition

Generally speaking, a wavelet is a smooth function $\Psi$ such that the set

$\\{2^{\frac{j}{2}}\Psi(2^{j}\cdot-k)\,:\,(j,k)\in{Z}^{2}\\}$

is an orthonormal basis of $L^{2}({R})$, see [20, 36, 41] for a very complete
view on this subject. Morally speaking, it is not surprising that such bases,
constructed by scaling modifications of an unique mother wavelet $\Psi$ are
particularly well-adapted to study self-similar stochastic processes, as this
definition somehow means that the process is invariant by scaling changes. In
particular, in the context of Hermite processes and their generalizations,
wavelets are used, for instance, for expansion and approximation [3, 4, 5, 42,
49], estimation [10, 7] and precise study of the pointwise regularity [21, 26,
27].

All along this paper, we work with a continuously differentiable mother
wavelet $\Psi:\mathbb{R}\to\mathbb{R}$ with support included in the unit
interval $[0,1]$ such as the one constructed in [19, 20]. We say that such a
wavelet $\Psi$ has $Q\geq 1$ vanishing moments if, for all $0\leq p<Q$,

$\int_{\mathbb{R}}x^{p}\Psi(x)dx=0.$ (3)

In fact, in this paper, we only need that $\Psi$ has one vanishing moment.
But, in practice, authors generally work with higher $Q$, for numerical
purposes. Let us fix $a>0$. We define, for any integer number $k\geq 0$, the
wavelet coefficient associated to the Hermite process $\\{Z_{t}\\}_{t\geq 0}$
by

$c(a,k)=\sqrt{a}\int_{\mathbb{R}}\Psi(x)Z_{a(x+k)}dx.$ (4)

Due to the scaling property, the stationarity of the increments of the Hermite
process and the vanishing moment of $\Psi$, we notice

$\displaystyle
c(a,k)=\sqrt{q}\int_{\mathbb{R}}\Psi(x)(Z_{a(x+k)}-Z_{ak})dx=^{(d)}\sqrt{a}\int_{\mathbb{R}}\Psi(x)Z_{ax}dx=^{(d)}a^{H+\frac{1}{2}}c(1,0),$
(5)

where we denoted by “$=^{(d)}$” the equality in distribution. In particular,
we have

$\mathbf{E}[c(a,k)^{2}]=a^{2H+1}\mathbf{E}[c(1,0)^{2}]=a^{2H+1}C_{\Psi}(H),$
(6)

with

$C_{\Psi}(H)=-\frac{1}{2}\int_{\mathbb{R}}\int_{\mathbb{R}}dxdy\Psi(x)\Psi(y)|x-y|^{2H}.$
(7)

By using (1) and the assumption (3), we can express the wavelet coefficient as
a multiple stochastic integral. Indeed, we write, via a Fubini argument,

$\displaystyle c(a,k)$ $\displaystyle=$
$\displaystyle\sqrt{a}\int_{\mathbb{R}}\Psi(x)(Z_{a(x+k)}-Z_{ak})dx=\sqrt{a}\int_{\mathbb{R}}\Psi(x)I_{q}\left(L_{a(x+k)}-L_{ak}\right)dx$
$\displaystyle=$ $\displaystyle
I_{q}\left(\sqrt{a}\int_{\mathbb{R}}\Psi(x)\left(L_{a(x+k)}-L_{ak}\right)dx\right)$
$\displaystyle=$ $\displaystyle
I_{q}\left(\sqrt{a}\int_{\mathbb{R}}dz\Psi(x)\int_{ak}^{a(x+k)}f_{u}du\right),$

where $L$ is the kernel of the Hermite process given by (2) and

$f_{u}(y_{1},...,y_{q})=c_{q,H}(u-y_{1})_{+}^{-\left(\frac{1}{2}+\frac{1-H}{q}\right)}\ldots(u-y_{q})_{+}^{-\left(\frac{1}{2}+\frac{1-H}{q}\right)},$

for every $y_{1},...,y_{q}\in\mathbb{R}$, with $c_{q,H}$ the constant in (2).
Since $\int_{ak}^{a(x+k)}f_{u}(y_{1},...,y_{q})du$ vanishes if there exists
$i=1,..,q$ such that $y_{i}\geq a(k+1)$ (recall that the support of $\Psi$ is
included in $[0,1]$), we can also write

$c(a,k)=I_{q}\left(\sqrt{a}\int_{\mathbb{R}}dx\Psi(x)\mathbbm{1}_{(-\infty,a(k+1))}^{\otimes
q}\int_{ak}^{a(x+k)}f_{u}du\right).$ (8)

Let $M>0$. We decompose the coefficient $c(a,k)$ as follows

$c(a,k)=\widetilde{c(a,k,M)}+\widecheck{c(a,k,M)}$

where

$\widetilde{c(a,k,M)}=I_{q}\left(\sqrt{a}\int_{\mathbb{R}}dx\Psi(x)\mathbbm{1}_{(a(k-M),a(k+1))}^{\otimes
q}\int_{ak}^{a(x+k)}f_{u}du\right)$ (9)

and

$\widecheck{c(a,k,M)}=I_{q}\left(\sqrt{a}\int_{\mathbb{R}}dx\Psi(x)\left(\mathbbm{1}_{(-\infty,a(k+1))^{q}\setminus(a(k-M),a(k+1))^{q}}\right)\int_{ak}^{a(x+k)}f_{u}du\right).$
(10)

This decomposition will play a crucial role for the development of our
arguments. The key fact is that, for suitable choices of the parameters
$a,M>0$, the random variables $\widetilde{c(a,k,M)}$ and
$\widetilde{c(a,j,M)}$ will become independent when $k\not=j$ (this happens
once the intervals $(a(k-M),a(k+1))$ and $(a(j-M),a(j+1))$ become disjoints).
On the other hand, the terms $\widecheck{c(a,k,M)}$ will be negligible, for
the $L^{2}(\Omega)$-norm, with respect to their tilde counterparts.

Let us first evaluate the squared mean of $\widecheck{c(a,k,M)}$. We denoted
by $C(q,H)$ a strictly positive constant depending only on $q,H$. A similar
notation will be used for constants depending on other parameters.

######  Lemma 1.

Let $a,M>0$ and let $k$ be a positive integer. We have

$\mathbf{E}\Big{[}\widecheck{c(a,k,M)}^{2}\Big{]}\leq
C(q,H)a^{2H+1}(M+1)^{\frac{2H-2}{q}}.$

Proof: We first notice that, for every $a,k,M$, we have, by the same arguments
as in the proofs of Lemma 2 below,

$\widecheck{c(a,k,M)}=^{(d)}a^{H+\frac{1}{2}}I_{q}\left(\mathbbm{1}_{(-\infty,1)^{q}\setminus(-M,1)^{q}}\int_{\mathbb{R}}dx\Psi(x)\int_{0}^{x}f_{u}du\right).$

Thus, by the isometry property (72) of the multiple stochastic integral
$I_{q}$, we get

$\displaystyle\mathbf{E}\Big{[}\widecheck{c(a,k,M)}^{2}\Big{]}$
$\displaystyle=$ $\displaystyle
a^{2H+1}q!c_{q,H}^{2}\int_{\mathbbm{1}_{(-\infty,1)^{q}\setminus(-M,1)^{q}}}dy_{1}...dy_{q}\int_{\mathbb{R}}\int_{\mathbb{R}}dxdy\Psi(x)\Psi(y)$
(11)
$\displaystyle\int_{0}^{x}dy\int_{0}^{y}dvf_{u}(y_{1},...,y_{q})f_{v}(y_{1},...,y_{q})$
$\displaystyle\leq$ $\displaystyle
C(q,H)a^{2H+1}\int_{0}^{1}\int_{0}^{1}dxdy|\Psi(x)|\cdot|\Psi(y)|\int_{0}^{x}du\int_{0}^{y}dv$
$\displaystyle\times\left(\int_{\mathbb{R}}dy(u-y)_{+}^{-\left(\frac{1}{2}+\frac{1-H}{q}\right)}(v-y)_{+}^{-\left(\frac{1}{2}+\frac{1-H}{q}\right)}\right)^{q-1}$
$\displaystyle\left(\int_{-M}^{1}dy(u-y)_{+}^{-\left(\frac{1}{2}+\frac{1-H}{q}\right)}(v-y)_{+}^{-\left(\frac{1}{2}+\frac{1-H}{q}\right)}\right)$
$\displaystyle=$ $\displaystyle
C(q,H)a^{2H+1}\int_{0}^{1}\int_{0}^{1}dxdy|\Psi(x)|\cdot|\Psi(y)|\int_{0}^{x}du\int_{0}^{y}dv|u-v|^{(2H-2)\frac{q-1}{q}}$
$\displaystyle\left(\int_{-M}^{1}dy(u-y)_{+}^{-\left(\frac{1}{2}+\frac{1-H}{q}\right)}(v-y)_{+}^{-\left(\frac{1}{2}+\frac{1-H}{q}\right)}\right).$

We have the following formula, obtained via the change of variables
$z=\frac{u\wedge v-y}{u\vee v-y}$,

$\displaystyle\int_{-M}^{1}dy(u-y)_{+}^{-\left(\frac{1}{2}+\frac{1-H}{q}\right)}(v-y)_{+}^{-\left(\frac{1}{2}+\frac{1-H}{q}\right)}=\int_{-M}^{u\wedge
v}(u-y)^{-\left(\frac{1}{2}+\frac{1-H}{q}\right)}(v-y)^{-\left(\frac{1}{2}+\frac{1-H}{q}\right)}dy$
$\displaystyle=$ $\displaystyle|u-v|^{\frac{2H-2}{q}}\int_{0}^{\frac{u\wedge
v+M}{u\vee
v+M}}z^{\frac{H-1}{q}-\frac{1}{2}}(1-z)^{\frac{2-2H}{q}-1}dz=\beta\left(\frac{u\wedge
v+M}{u\vee v+M},\frac{H-1}{q}+\frac{1}{2},\frac{2-2H}{q}\right),$

where $\beta(x,a,b)$ is the incomplete beta function given, for $a,b>0$ and
$x\in(0,1]$, by $\beta(x,a,b)=\int_{0}^{x}z^{a-1}(1-z)^{b-1}dz$. In
particular, $\beta(1,a,b)=\beta(a,b)$ is the usual beta function. Let us also
introduce the notation

$\bar{\beta}(x,a,b)=\beta(a,b)-\beta(x,a,b)=\int_{x}^{1}z^{a-1}(1-z)^{b-1}dz$

for $a,b>0$ and $x\in(0,1).$ We have the following estimate

$\displaystyle\bar{\beta}\left(\frac{u\wedge v+M}{u\vee
v+M},\frac{H-1}{q}+\frac{1}{2},\frac{2-2H}{q}\right)=\int_{\frac{u\wedge
v+M}{u\vee v+M}}^{1}z^{\frac{H-1}{q}-\frac{1}{2}}(1-z)^{\frac{2-2H}{q}-1}dz$
(12) $\displaystyle\leq$
$\displaystyle\int_{\frac{M}{M+1}}^{1}z^{\frac{H-1}{q}-\frac{1}{2}}(1-z)^{\frac{2-2H}{q}-1}dz\leq\int_{\frac{M}{M+1}}^{1}(1-z)^{\frac{2-2H}{q}-1}dz$
$\displaystyle=$ $\displaystyle\frac{2-2H}{q}(M+1)^{\frac{2H-2}{q}}.$

Therefore, by plugging (12) into (11), we get

$\displaystyle\mathbf{E}\Big{[}\widecheck{c(a,k,M)}^{2}\Big{]}$
$\displaystyle\leq$ $\displaystyle
C(q,H)a^{2H+1}(M+1)^{\frac{2H-2}{q}}\int_{0}^{1}\int_{0}^{1}dxdy|\Psi(x)|\cdot|\Psi(y)|\int_{0}^{x}du\int_{0}^{y}dv|u-v|^{(2H-2)\frac{q-1}{q}}$
$\displaystyle\leq$ $\displaystyle
C(q,H)a^{2H+1}(M+1)^{\frac{2H-2}{q}}\int_{0}^{1}\int_{0}^{1}dxdy|\Psi(x)|\cdot|\Psi(y)|\int_{0}^{x}du\int_{0}^{y}dv|u-v|^{2H-2}$
$\displaystyle=$ $\displaystyle
C(q,H)a^{2H+1}(M+1)^{\frac{2H-2}{q}}\int_{0}^{1}\int_{0}^{1}dxdy|\Psi(x)|\cdot|\Psi(y)|(x^{2H}+y^{2H}-|x-y|^{2H})$
$\displaystyle\leq$ $\displaystyle C(q,H)a^{2H+1}(M+1)^{\frac{2H-2}{q}}.$

##  3 The modified wavelet variation

We will define and analyse the wavelet variation of the Hermite process based
on some particular wavelet coefficients. We first introduce these special
coefficients and we also study their main properties.

###  3.1 The special wavelet coefficients

We start with some notations. Consider two real numbers

$0<\gamma<\beta<1$

and for every $N\geq 1$, let us set

$\mathcal{L}_{N}=\mathbb{N}\cap\left[1,\frac{2^{N}}{2^{[N^{\beta}]}}\right]=\left[1,2^{N-[N^{\beta}]}\right]$

and

$\mathcal{L}_{N,\gamma}=\mathcal{L}_{N}\cap\left[1,2^{[N^{\gamma}]}\right].$

We have, for every $N\geq 1$,

$\mathcal{L}_{N,\gamma}\subset\mathcal{L}_{N}\mbox{ and
}|\mathcal{L}_{N,\gamma}|\leq 2^{[N^{\gamma}]}$ (13)

and for $N$ large enough,

$|\mathcal{L}_{N,\gamma}|\geq 2^{[N^{\gamma}]}-1.$ (14)

For $\ell\in\mathcal{L}_{N,\gamma}$ and $N\geq 1$, we set

$e_{\ell,N,\beta}=\frac{\ell 2^{[N^{\beta}]}}{2^{N}}.$ (15)

Now, we describe the wavelet coefficients on which we will focus in the rest
of this work. Let $d\geq 1$. For $\ell\in\mathcal{L}_{N,\gamma}$ and
$M=1,...,d$, we define

$A_{M}(\ell,N)=c\left(\frac{1}{M2^{N}},\ell M2^{[N^{\beta}]}\right).$ (16)

From (8), we express

$\displaystyle A_{M}(\ell,N)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{M2^{N}}}I_{q}\left(1^{\otimes
q}_{(-\infty,\frac{\ell
M2^{[N^{\beta}]}+M}{M2^{N}})}\int_{\mathbb{R}}dx\Psi(x)\int_{\frac{\ell
M2^{[N^{\beta}]}}{M^{2^{N}}}}^{{\frac{\ell
M2^{[N^{\beta}]}+x}{M^{2^{N}}}}}f_{u}du\right)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{M2^{N}}}I_{q}\left(1^{\otimes
q}_{(-\infty,e_{\ell,N,\beta}+2^{-N})}\int_{\mathbb{R}}dx\Psi(x)\int_{e_{\ell,N,\beta}}^{e_{\ell,N,\beta}+xM^{2^{-N}}}f_{u}du\right),$

where we used the notation (15). We have, by (5),

$\mathbf{E}\big{[}A_{M}(\ell,N)^{2}\big{]}=2^{-N(2H+1)}M^{-(2H+1)}C_{\Psi}(H),$
(17)

with $C_{\Psi}(H)$ from (7). We will decompose this coefficient as

$A_{M}(\ell,N)=\widetilde{A_{M}(\ell,N)}+\widecheck{A_{M}(\ell,N)},$ (18)

where, with the notations (9) and (10),

$\displaystyle\widetilde{A_{M}(\ell,N)}$ $\displaystyle=$
$\displaystyle\widetilde{c\left(\frac{1}{M2^{N}},M\ell
2^{[N^{\beta}]},M(2^{[N^{\beta}]}-1)\right)}$ (19) $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{M2^{N}}}I_{q}\left(1^{\otimes
q}_{(e_{\ell-1,N,\beta}+2^{-N},e_{\ell,N,\beta}+2^{-N})}\int_{\mathbb{R}}dx\Psi(x)\int_{e_{\ell,N,\beta}}^{e_{\ell,N,\beta}+xM^{2^{-N}}}f_{u}du\right)$
$\displaystyle=$ $\displaystyle I_{q}(\widetilde{g}_{M}(\ell,N)),$

with

$\displaystyle\widetilde{g}_{M}(\ell,N)(y_{1},...,y_{q})$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{M2^{N}}}\mathbbm{1}^{\otimes
q}_{(e_{\ell-1,N,\beta}+2^{-N},e_{\ell,N,\beta}+2^{-N})}(y_{1},...,y_{q})$
$\displaystyle\int_{\mathbb{R}}dx\Psi(x)\int_{e_{\ell,N,\beta}}^{e_{\ell,N,\beta}+xM^{2^{-N}}}f_{u}(y_{1},...,y_{q})du$

and

$\displaystyle\widecheck{A_{M}(\ell,N)}$ $\displaystyle=$
$\displaystyle\widecheck{c\left(\frac{1}{M2^{N}},M\ell
2^{[N^{\beta}]},M(2^{[N^{\beta}]}-1)\right)}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{M2^{N}}}I_{q}\left(\mathbbm{1}_{(-\infty,e_{\ell,N,\beta}+2^{-N})^{q}\setminus(e_{\ell-1,N,\beta}+2^{-N},e_{\ell,N,\beta}+2^{-N})^{q}}\int_{\mathbb{R}}dx\Psi(x)\int_{e_{\ell,N,\beta}}^{e_{\ell,N,\beta}+xM^{2^{-N}}}f_{u}du\right)$

Before going further, let us emphasize the roles for statistical inference of
the various parameters introduced in this section. The estimator
$\widehat{H}_{N}$ for the Hurst parameter defined in equation (65) below
relies on the wavelet variations (25) in which the corresponding wavelet
coefficients are precisely chosen. First, we need to consider $d$ scales, that
is the parameter $a$ in (4), to perform a $\log$-regression. To define
$\widehat{H}_{N}$, we take the scales $(M2^{N})^{-1}$ for $M=1,\dots,d$. Then,
at such a scale $(M2^{N})^{-1}$ we only work with wavelet coefficients at
position, that is the parameter $k$ in (4), which are multiple of
$M2^{[N^{\beta}]}$. Thus, we exactly get the definition of the coefficients
$A_{M}(\ell,N)$ in (16). This choice of wavelet coefficients has the double
advantage to offer, as already stated, some independence properties for the
“tilde part”, see also Lemma 2 below, while the “check counterpart” are
negligible, for the $L^{2}(\Omega)$-norm. Nevertheless, if all coefficients
$A_{M}(\ell,N)$, with $\ell\in\mathcal{L}_{N}$, were chosen in (25), then the
proofs of the Central Limit Theorems (Theorem 1 and 2 below) would
unfortunately not hold. To overcome this situation, we just have to select a
proportion of the coefficients $A_{M}(\ell,N)$, that is the one for which
$\ell\in\mathcal{L}_{N,\gamma}$. The simple fact that $\gamma<\beta$ allows to
deduce precious convergence to $0$ in $L^{1}(\Omega)$, see for instance
Proposition 1 below. Note that increasing $\beta$ allows to also increase
$\gamma$ and thus improve the rate of convergence to the normal distribution,
see the bound for the Wasserstein distance in Theorems 1 and 2, but, on the
other side, reduce the number of data used in the statistical inference, as a
consequence of the definition of $\mathcal{L}_{N}$ and
$\mathcal{L}_{N,\gamma}$. For this reason, it is a difficult question to know
what should be the more “practicable” value for the parameter $\beta$.

Let us come back to the main properties of the special wavelet coefficients
introduced here over. First, the following facts will play an important role
for the proofs of our main results.

######  Lemma 2.

  1. 1.

For every $N\geq 1$, the random vectors
$\left((\widetilde{A_{1}(\ell,N)},\ldots,\widetilde{A_{d}(\ell,N)}),\ell\in\mathcal{L}_{N,\gamma}\right)$
are mutually independent.

  2. 2.

For every $N\geq 1$ and $M=1,...,d$, the random variables
$(\widetilde{A_{M}(\ell,N)},\ell\in\mathcal{L}_{N,\gamma})$ are identically
distributed.

  3. 3.

For every $N\geq 1$ and $M=1,...,d$, the random variables
$(\widecheck{A_{M}(\ell,N)},\ell\in\mathcal{L}_{N,\gamma})$ are identically
distributed.

Proof:  To prove the first point, we notice that for every
$\ell,j\in\mathcal{L}_{N,\gamma}$ with $\ell\not=j$ and for every
$M_{1},M_{2}=1,...,d$, we clearly have

$\widetilde{g_{M_{1}}(\ell,N)}\otimes_{1}\widetilde{g_{M_{2}}(j,N)}=0$

almost everywhere on $\mathbb{R}^{2q-2}$ (the notation $\otimes_{1}$ stands
for the contraction of order one, see (74) in the Appendix). This is because
the intervals that appear in the expression of the kernels
$\widetilde{g_{M_{1}}(\ell,N)}$ and $\widetilde{g_{M_{2}}(j,N)}$ are disjoint.
The independence follows from the Ustúnel-Zakai criterion (see e.g. [57,
Proposition 1]).

Concerning the second point, for any integer $h$ such that
$\ell+h\in\mathcal{L}_{N,\gamma}$, we can write

$\displaystyle\widetilde{A_{M}(\ell+h,N)}$ $\displaystyle=$ $\displaystyle
d(H)\sqrt{\frac{1}{M2^{N}}}\int_{\mathbb{R}^{q}}dB(y_{1})...dB(y_{q})$
$\displaystyle\mathbbm{1}_{(e_{\ell+h-1,N,\beta}+2^{-N},e_{\ell+h,N,\beta}+2^{-N})}^{\otimes
q}(y_{1},...,y_{q})$
$\displaystyle\int_{\mathbb{R}}dx\Psi(x)\int_{e_{\ell+h,N,\beta}}^{e_{\ell+h,N,\beta}+xM2^{-N}}(u-y_{1})_{+}^{-\left(\frac{1}{2}+\frac{1-H}{q}\right)}\ldots(u-y_{q})_{+}^{-\left(\frac{1}{2}+\frac{1-H}{q}\right)}du$

and with the successive changes of variables $\tilde{u}=u-e_{h,N,\beta}$ and
$\tilde{y}_{j}=y_{j}-e_{h,N,\beta}$ for $j=1,...,q$, we arrive at

$\displaystyle\widetilde{A_{M}(\ell+h,N)}$ $\displaystyle=$ $\displaystyle
d(H)\sqrt{\frac{1}{M2^{N}}}\int_{\mathbb{R}^{q}}dB(y_{1})...dB(y_{q})$
$\displaystyle\mathbbm{1}_{(e_{\ell+h-1,N,\beta}+2^{-N},e_{\ell+h,N,\beta}+2^{-N})}^{\otimes
q}(y_{1},...,y_{q})$
$\displaystyle\int_{\mathbb{R}}dx\Psi(x)\int_{e_{\ell,N,\beta}}^{e_{\ell,N,\beta}+xM2^{-N}}(u-y_{1}+e_{h,N,\beta})_{+}^{-\left(\frac{1}{2}+\frac{1-H}{q}\right)}\ldots(u-y_{q}+e_{h,N,\beta})_{+}^{-\left(\frac{1}{2}+\frac{1-H}{q}\right)}du$
$\displaystyle=$ $\displaystyle
d(H)\sqrt{\frac{1}{M2^{N}}}\int_{\mathbb{R}^{q}}dB(y_{1}+e_{h,N,\beta})...dB(y_{q}+e_{h,N,\beta})$
$\displaystyle\mathbbm{1}_{(e_{\ell-1,N,\beta}+2^{-N},e_{\ell,N,\beta}+2^{-N})}^{\otimes
q}(y_{1},...,y_{q})$
$\displaystyle\int_{\mathbb{R}}dx\Psi(x)\int_{e_{\ell,N,\beta}}^{e_{\ell,N,\beta}+xM2^{-N}}(u-y_{1})_{+}^{-\left(\frac{1}{2}+\frac{1-H}{q}\right)}\ldots(u-y_{q})_{+}^{-\left(\frac{1}{2}+\frac{1-H}{q}\right)}du.$

Since the Brownian motion has stationary increments, we have

$\displaystyle\widetilde{A_{M}(\ell+h,N)}$ $\displaystyle=^{(d)}$
$\displaystyle
d(H)\sqrt{\frac{1}{M2^{N}}}\int_{\mathbb{R}^{q}}dB(y_{1})...dB(y_{q})$
$\displaystyle\mathbbm{1}_{(e_{\ell-1,N,\beta}+2^{-N},e_{\ell,N,\beta}+2^{-N})}^{\otimes
q}(y_{1},...,y_{q})$
$\displaystyle\int_{\mathbb{R}}dx\Psi(x)\int_{e_{\ell,N,\beta}}^{e_{\ell,N,\beta}+xM2^{-N}}(u-y_{1})_{+}^{-\left(\frac{1}{2}+\frac{1-H}{q}\right)}\ldots(u-y_{q})_{+}^{-\left(\frac{1}{2}+\frac{1-H}{q}\right)}du$
$\displaystyle=$ $\displaystyle\widetilde{A_{M}(\ell,N)}.$

The third point follows immediately, using the same arguments as above.

The next result is then a direct consequence of Lemma 1.

######  Lemma 3.

For every $N\geq 1,M=1,...,d$ and for every $\ell\in\mathcal{L}_{N,\gamma}$,
we have

$\mathbf{E}\Big{[}\widecheck{A_{M}(\ell,N)}^{2}\Big{]}\leq
C(q,H,M)2^{-N(2H+1)}2^{[N^{\beta}]\frac{2H-2}{q}}.$

We will need some auxiliary results concerning the behavior of the second and
fourth moments of the random variable (19).

######  Lemma 4.

For $N$ sufficiently large and for all
$\ell\in\mathcal{L}_{N,\gamma},M=1,...,d$, we have

$\left|2^{N(2H+1)}\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}^{2}\Big{]}-M^{-(2H+1)}C_{\Psi}(H)\right|\leq
C2^{[N^{\beta}]\frac{2H-2}{q}}.$ (21)

In particular,

$2^{N(2H+1)}\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}^{2}\Big{]}\underset{{N\to\infty}}{\longrightarrow}M^{-(2H+1)}C_{\Psi}(H).$

Also, for $N$ sufficiently large and for
$\ell\in\mathcal{L}_{N,\gamma},M=1,...,d$, we have

$\left|2^{2N(2H+1)}\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}^{4}\Big{]}-M^{-(4H+2)}\mathbf{E}\big{[}c(1,0)^{4}\big{]}\right|\leq
C2^{[N^{\beta}]\frac{2H-2}{q}},$ (22)

which implies

$2^{2N(2H+1)}\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}^{4}\Big{]}{{N\to\infty}}M^{-(4H+2)}\mathbf{E}\big{[}c(1,0)^{4}\big{]}.$

Proof:  Since $\widetilde{A_{M}(\ell,N)}$ and $\widecheck{A_{M}(\ell,N)}$ are
independent random variables, we have from (18),

$\mathbf{E}\Big{[}A_{M}(\ell,N)^{2}\Big{]}=\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}^{2}\Big{]}+\mathbf{E}\Big{[}\widecheck{A_{M}(\ell,N)}^{2}\Big{]}.$
(23)

Let us note that, the above inequality combined with equality (17) and Lemme 3
entail

$2^{N(2H+1)}\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}^{2}\Big{]}\leq C.$ (24)

The relation (23) gives

$\displaystyle
2^{N(2H+1)}\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}^{2}\Big{]}$
$\displaystyle=$ $\displaystyle
2^{N(2H+1)}\mathbf{E}\Big{[}A_{M}(\ell,N)^{2}\Big{]}-2^{N(2H+1)}\mathbf{E}\Big{[}\widecheck{A_{M}(\ell,N)}^{2}\Big{]}$
$\displaystyle=$ $\displaystyle
M^{-(2H+1)}C_{\Psi}(H)-2^{N(2H+1)}\mathbf{E}\Big{[}\widecheck{A_{M}(\ell,N)}^{2}\Big{]},$

and the inequality (21) is then a direct consequence of Lemma 3.

To prove (22), we write

$\displaystyle\mathbf{E}\Big{[}A_{M}(\ell,N)^{4}\Big{]}$ $\displaystyle=$
$\displaystyle\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}^{4}\Big{]}+\mathbf{E}\Big{[}\widecheck{A_{M}(\ell,N)}^{4}\Big{]}+6\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}^{2}\widecheck{A_{M}(\ell,N)}^{2}\Big{]}$
$\displaystyle+4\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}^{3}\widecheck{A_{M}(\ell,N)}\Big{]}+4\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}\widecheck{A_{M}(\ell,N)}^{3}\Big{]}$
$\displaystyle=$
$\displaystyle\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}^{4}\Big{]}+\mathbf{E}\Big{[}\widecheck{A_{M}(\ell,N)}^{4}\Big{]}+6\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}^{2}\widecheck{A_{M}(\ell,N)}^{2}\Big{]}+4\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}\widecheck{A_{M}(\ell,N)}^{3}\Big{]},$

because, by [6, Lemma 2], we have
$\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}^{3}\widecheck{A_{M}(\ell,N)}\Big{]}=0$.
Thus, we get, using the equality (5),

$\displaystyle
2^{2N(2H+1)}\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}^{4}\Big{]}$
$\displaystyle=$ $\displaystyle
2^{2N(2H+1)}\mathbf{E}\Big{[}A_{M}(\ell,N)^{4}\Big{]}-2^{2N(2H+1)}\mathbf{E}\Big{[}\widecheck{A_{M}(\ell,N)}^{4}\Big{]}$
$\displaystyle-6\times
2^{2N(2H+1)}\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}^{2}\widecheck{A_{M}(\ell,N)}^{2}\Big{]}-4\times
2^{2N(2H+1)}\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}\widecheck{A_{M}(\ell,N)}^{3}\Big{]}$
$\displaystyle=$ $\displaystyle
M^{-(4H+2)}\mathbf{E}\Big{[}c(1,0)^{4}\Big{]}-2^{2N(2H+1)}\mathbf{E}\Big{[}\widecheck{A_{M}(\ell,N)}^{4}\Big{]}$
$\displaystyle-6\times
2^{2N(2H+1)}\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}^{2}\widecheck{A_{M}(\ell,N)}^{2}\Big{]}-4\times
2^{2N(2H+1)}\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}\widecheck{A_{M}(\ell,N)}^{3}\Big{]}.$

The hypercontractivity property (75), the inequality (24) and Lemma 3 imply

$2^{2N(2H+1)}\mathbf{E}\Big{[}\widecheck{A_{M}(\ell,N)}^{4}\Big{]}\leq
C2^{2N(2H+1)}\left(\mathbf{E}\Big{[}\widecheck{A_{M}(\ell,N)}^{2}\Big{]}\right)^{2}\leq
C2^{[N^{\beta}]\frac{4H-4}{q}}$

while, by, the Cauchy-Schwarz’s inequality, (75), Lemma 3 and (24), we have

$\displaystyle\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}^{2}\widecheck{A_{M}(\ell,N)}^{2}\Big{]}$
$\displaystyle\leq$
$\displaystyle\left(\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}^{4}\Big{]}\right)^{\frac{1}{2}}\left(\mathbf{E}\Big{[}\widecheck{A_{M}(\ell,N)}^{4}\Big{]}\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
C\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}^{2}\Big{]}\mathbf{E}\Big{[}\widecheck{A_{M}(\ell,N)}^{2}\Big{]}$
$\displaystyle\leq$ $\displaystyle
C2^{-2N(2H+1)}2^{[N^{\beta}]\frac{2H-2}{q}}$

and

$\displaystyle\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}\widecheck{A_{M}(\ell,N)}^{3}\Big{]}$
$\displaystyle\leq$ $\displaystyle
C\left(\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}^{2}\Big{]}\right)^{\frac{1}{2}}\left(\mathbf{E}\Big{[}\widecheck{A_{M}(\ell,N)}^{6}\Big{]}\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
C\left(\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}^{2}\Big{]}\right)^{\frac{1}{2}}\left(\mathbf{E}\Big{[}\widecheck{A_{M}(\ell,N)}^{2}\Big{]}\right)^{\frac{3}{2}}$
$\displaystyle\leq$ $\displaystyle
C2^{-2N(2H+1)}2^{[N^{\beta}]\frac{3H-3}{q}}.$

The above three bounds lead to inequality (22).

######  Lemma 5.

Let $M_{1},M_{2}=1,...,d$. Then for $N$ sufficiently large, we have

$\left|2^{2N(2H+1)}\mathbf{E}\Big{[}\widetilde{A_{M_{1}}(\ell,N)}^{2}\widetilde{A_{M_{2}}(\ell,N)}^{2}\Big{]}-\mathbf{E}\Big{[}c(M_{1}^{-1},0)^{2}c(M_{2}^{-1},0)^{2}\Big{]}\right|\leq
C2^{[N^{\beta}]\frac{2H-2}{q}}.$

Proof:  By the scaling property and the stationarity of the increments of the
Hermite process we can show that for every $M_{1},M_{2}=1,...,d$,

$(A_{M_{1}}(\ell,N),A_{M_{2}}(\ell,N))=^{(d)}2^{-N(H+\frac{1}{2})}(c(M_{1}^{-1},0),c(M_{2}^{-1},0)).$

This implies

$2^{2N(2H+1)}\mathbf{E}\Big{[}A_{M_{1}}(\ell,N)^{2}A_{M_{2}}(\ell,N)^{2}\Big{]}=\mathbf{E}\Big{[}c(M_{1}^{-1},0)^{2}c(M_{2}^{-1},0)^{2}\Big{]}.$

By using the decomposition (21) for $A_{M_{1}}(\ell,N)$ and
$A_{M_{2}}(\ell,N)$ and by applying the estimate in Lemma 3 for the negligible
parts $\widecheck{A_{M_{1}}(\ell,N)}$ and $\widecheck{A_{M_{2}}(\ell,N)}$, we
can conclude.

###  3.2 The modified wavelet variation and its negligible part

Let us now introduce the main object of this work, namely the modified wavelet
variation of the Hermite process. This wavelet variation is defined by using,
not all the wavelet coefficients of the Hermite process, but only the special
coefficient given by (16). More exactly, for $N\geq 1,M=1,...,d$, we set

$V_{N,M}=\frac{1}{\sqrt{|\mathcal{L}_{N,\gamma}|}}\sum_{\ell\in\mathcal{L}_{N,\gamma}}\left[\frac{A_{M}(\ell,N)^{2}}{\mathbf{E}\big{[}A_{M}(\ell,N)^{2}\big{]}}-1\right].$
(25)

From (17), we can also write

$V_{N,M}=\frac{1}{\sqrt{|\mathcal{L}_{N,\gamma}|}}\frac{2^{N(2H+1)}M^{2H+1}}{C_{\Psi}(H)}\sum_{\ell\in\mathcal{L}_{N,\gamma}}\left(A_{M}(\ell,N)^{2}-\mathbf{E}\big{[}A_{M}(\ell,N)^{2}\big{]}\right).$
(26)

The purpose is to find the limit behavior in distribution, as $N\to\infty$, of
the sequence $(V_{N,M},N\geq 1)$. To this end, we decompose $V_{N,M}$ as
follows

$V_{N,M}=V_{N,M,1}+V_{N,M,2}+V_{N,M,3},$ (27)

where, for all $N\geq 1$ and $M=1,...,d$, we set

$V_{N,M,1}=\frac{1}{\sqrt{|\mathcal{L}_{N,\gamma}|}}\frac{2^{N(2H+1)}M^{2H+1}}{C_{\Psi}(H)}\sum_{\ell\in\mathcal{L}_{N,\gamma}}\left(\widetilde{A_{M}(\ell,N)}^{2}-\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}^{2}\Big{]}\right),$
(28)

$V_{N,M,2}=\frac{1}{\sqrt{|\mathcal{L}_{N,\gamma}|}}\frac{2^{N(2H+1)}M^{2H+1}}{C_{\Psi}(H)}\sum_{\ell\in\mathcal{L}_{N,\gamma}}\left(\widecheck{A_{M}(\ell,N)}^{2}-\mathbf{E}\Big{[}\widecheck{A_{M}(\ell,N)}^{2}\Big{]}\right),$
(29)

and

$V_{N,M,3}=2\frac{1}{\sqrt{|\mathcal{L}_{N,\gamma}|}}\frac{2^{N(2H+1)}M^{2H+1}}{C_{\Psi}(H)}\sum_{\ell\in\mathcal{L}_{N,\gamma}}\widetilde{A_{M}(\ell,N)}\widecheck{A_{M}(\ell,N)}.$
(30)

In a first step, we will show that the summands denoted by $V_{N,M,2}$ and
$V_{N,M,3}$ above are negligible, and consequently the behavior of the wavelet
variation (25) is given by the sequence $(V_{N,M,1},N\geq 1)$.

######  Proposition 1.

Let $V_{N,M,2},V_{N,M,3}$ be given by (29), (30), respectively. We have

$\mathbf{E}\big{[}|V_{N,M,2}|\big{]}\leq
C(q,H,M)2^{\frac{N^{\gamma}}{2}}2^{[N^{\beta}]\frac{2H-2}{q}}$

and

$\mathbf{E}\big{[}|V_{N,M,3}|\big{]}\leq
C(q,H,M)2^{\frac{N^{\gamma}}{2}}2^{[N^{\beta}]\frac{H-1}{q}}.$

In particular, the sequences $(V_{N,M,2},N\geq 1)$ and $(V_{N,M,3},N\geq 1)$
converge to zero in $L^{1}(\Omega)$, as $N\to\infty$.

Proof:  For $V_{N,M,2}$ we have the following estimates, by (29) and Lemma 3,

$\displaystyle\mathbf{E}\big{[}|V_{N,M,2}|\big{]}$ $\displaystyle\leq$
$\displaystyle
C(q,H,M)\frac{2^{N(2H+1)}}{\sqrt{|\mathcal{L}_{N,\gamma}|}}\sum_{\ell\in\mathcal{L}_{N,\gamma}}\mathbf{E}\Big{[}\widecheck{A_{M}(\ell,N)}^{2}\Big{]}$
$\displaystyle\leq$ $\displaystyle
C(q,H,M)\sqrt{|\mathcal{L}_{N,\gamma}|}2^{-N(2H+1)}2^{[N^{\beta}]\frac{2H-2}{q}}$
$\displaystyle\leq$ $\displaystyle
C(q,H,M)2^{\frac{N^{\gamma}}{2}}2^{[N^{\beta}]\frac{2H-2}{q}}.$

Since $\gamma<\beta$ and $H<1$, we get the converge to zero in
$L^{1}(\Omega)$, as $N\to\infty$, of $V_{N,M,2}$.

Concerning the summand denoted by $V_{N,M,3}$, we have by (30) and Cauchy-
Schwarz’s inequality,

$\displaystyle\mathbf{E}\big{[}|V_{N,M,3}|\big{]}$ $\displaystyle\leq$
$\displaystyle
C(q,H,M)\frac{2^{N(2H+1)}}{\sqrt{|\mathcal{L}_{N,\gamma}|}}\sum_{\ell\in\mathcal{L}_{N,\gamma}}\left(\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}^{2}\Big{]}\right)^{\frac{1}{2}}\left(\mathbf{E}\Big{[}\widecheck{A_{M}(\ell,N)}^{2}\Big{]}\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
C(q,H,M)\frac{2^{N(2H+1)}}{\sqrt{|\mathcal{L}_{N,\gamma}|}}\sum_{\ell\in\mathcal{L}_{N,\gamma}}\sqrt{2^{N(2H+1)}}\sqrt{2^{N(2H+1)}2^{[N^{\beta}]\frac{2H-2}{q}}}$
$\displaystyle\leq$ $\displaystyle
C(q,H,M)2^{\frac{N^{\gamma}}{2}}2^{[N^{\beta}]\frac{H-1}{q}}.$

Again, the assumptions $\gamma<\beta$ and $H<1$ imply that $V_{N,M,3}$
converges to zero in $L^{1}(\Omega)$ as $N\to\infty$.

###  3.3 The Central Limit Theorem for the modified wavelet variation

In this section, we prove that the $d$-dimensional random vector
$(V_{N,M},M=1,...,d)$ satisfies a (multidimensional) Central Limit Theorem. On
this purpose, we first focus on the sequence $(V_{N,M,1},N\geq 1)$ defined by
(28).

Let us start by evaluating the behavior of
$\mathbf{E}\big{[}V_{N,M_{1},1}V_{N,M_{2},1}\big{]}$, with
$M_{1},M_{2}=1,...,d$, as $N\to\infty$.

######  Proposition 2.

Let $V_{N,M,1}$ be given by (28). Then for $N$ large enough, and for every
$M=1,...,d$ , we have

$\left|\mathbf{E}\big{[}V_{N,M,1}^{2}\big{]}-\left(\frac{\mathbf{E}\big{[}c(1,0)^{4}\big{]}}{C_{\Psi}(H)^{2}}-1\right)\right|\leq
C2^{[N^{\beta}]\frac{2H-2}{q}}.$ (31)

If $M_{1},M_{2}=1,...,d$ are such that $M_{1}\not=M_{2}$, then for $N$ large

$\left|\mathbf{E}\big{[}V_{N,M_{1},1}V_{N,M_{2},1}\big{]}-\left(\frac{(M_{1}M_{2})^{2H+1}\mathbf{E}\big{[}c(M_{1}^{-1},0)^{2}c(M_{2}^{-1},0)^{2}\big{]}}{C_{\Psi}(H)^{2}}-1\right)\right|\leq
C2^{[N^{\beta}]\frac{2H-2}{q}}.$ (32)

Proof:  By (28), we have, for $M=1,...,d$ and $N\geq 1$,

$\displaystyle V_{N,M,1}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{|\mathcal{L}_{N,\gamma}|}\frac{2^{2N(2H+1)M^{4H+2}}}{C_{\Psi}(H)^{2}}$
$\displaystyle\sum_{\ell,j\in\mathcal{L}_{N,\gamma}}\mathbf{E}\left[\left(\widetilde{A_{M}(\ell,N)}^{2}-\mathbf{E}\big{[}\widetilde{A_{M}(\ell,N)}^{2}\big{]}\right)\left(\widetilde{A_{M}(j,N)}^{2}-\mathbf{E}\big{[}\widetilde{A_{M}(j,N)}^{2}\big{]}\right)\right]$
$\displaystyle=$
$\displaystyle\frac{1}{|\mathcal{L}_{N,\gamma}|}\frac{2^{2N(2H+1)M^{4H+2}}}{C_{\Psi}(H)^{2}}\sum_{\ell\in\mathcal{L}_{N,\gamma}}\mathbf{E}\left[\left(\widetilde{A_{M}(\ell,N)}^{2}-\mathbf{E}\big{[}\widetilde{A_{M}(\ell,N)}^{2}\big{]}\right)^{2}\right],$

since $\widetilde{A_{M}(\ell,N)}$ and $\widetilde{A_{M}(j,N)}$ are independent
when $\ell\not=j$ (see the first point of Lemma 2). Next, since, by the second
point of Lemma 2, $(\widetilde{A_{M}(\ell,N)},\ell\in\mathcal{L}_{N,\gamma})$
are identically distributed, we can write

$\displaystyle\mathbf{E}\big{[}V_{N,M,1}^{2}\big{]}$ $\displaystyle=$
$\displaystyle\frac{2^{2N(2H+1)M^{4H+2}}}{C_{\Psi}(H)^{2}}\mathbf{E}\left[\left(\widetilde{A_{M}(\ell_{0},N)}^{2}-\mathbf{E}\Big{[}\widetilde{A_{M}(\ell_{0},N)}^{2}\Big{]}\right)^{2}\right]$
$\displaystyle=$
$\displaystyle\frac{2^{2N(2H+1)M^{4H+2}}}{C_{\Psi}(H)^{2}}\left(\mathbf{E}\Big{[}\widetilde{A_{M}(\ell_{0},N)}^{4}\Big{]}-\left(\mathbf{E}\Big{[}\widetilde{A_{M}(\ell_{0},N)}^{2}\Big{]}\right)^{2}\right).$

for some $\ell_{0}\in\mathcal{L}_{N,\gamma}$. Therefore, we obtain

$\displaystyle\mathbf{E}[V_{N,M,1}^{2}]-\left(\frac{\mathbf{E}\big{[}c(1,0)^{4}\big{]}}{C_{\Psi}(H)^{2}}-1\right)$
(33) $\displaystyle=$
$\displaystyle\frac{1}{C_{\Psi}(H)^{2}}M^{4H+2}\left[\left(2^{2N(2H+1)}\mathbf{E}\Big{[}\widetilde{A_{M}(\ell_{0},N)}^{4}\Big{]}-M^{-(4H+2)}\mathbf{E}[c(1,0)^{4}]\right)\right.$
$\displaystyle\left.-\left(2^{2N(2H+1)}\left(\mathbf{E}\Big{[}\widetilde{A_{M}(\ell_{0},N)}^{2}\Big{]}\right)^{2}-M^{-(4H+2)}C_{\Psi}(H)^{2}\right)\right].$

On one hand, by (22), we have

$\left|2^{2N(2H+1)}\mathbf{E}\Big{[}\widetilde{A_{M}(\ell_{0},N)}^{4}\Big{]}-M^{-(4H+2)}\mathbf{E}\big{[}c(1,0)^{4}\big{]}\right|\leq
C2^{[N^{\beta}]\frac{2H-2}{q}}$ (34)

while, on the other hand, by Lemma 4, we get

$\displaystyle\left|2^{2N(2H+1)}\left(\mathbf{E}\Big{[}\widetilde{A_{M}(\ell_{0},N)}^{2}\Big{]}\right)^{2}-M^{-(4H+2)}C_{\Psi}(H)^{2}\right|$
$\displaystyle=\left|2^{N(2H+1)}\mathbf{E}\Big{[}\widetilde{A_{M}(\ell_{0},N)}^{2}\Big{]}-M^{-(2H+1)}C_{\Psi}(H)\right|\left|2^{N(2H+1)}\mathbf{E}\Big{[}\widetilde{A_{M}(\ell_{0},N)}^{2}\Big{]}+M^{-(2H+1)}C_{\Psi}(H)\right|$
$\displaystyle\leq C2^{[N^{\beta}]\frac{2H-2}{q}}.$ (35)

By plugging (34) and (35) into (33), we obtain the inequality (31).

If $M_{1}\not=M_{2}$, by (26) and the independence property in the first point
of Lemma 2, we express

$\displaystyle\mathbf{E}\big{[}V_{N,M_{1},1}V_{N,M_{2},1}\big{]}$
$\displaystyle=$
$\displaystyle\frac{1}{|\mathcal{L}_{N,\gamma}|}\frac{2^{2N(2H+1)M_{1}^{2H+1}M_{2}^{2H+1}}}{C_{\Psi}(H)^{2}}\sum_{\ell\in\mathcal{L}_{N,\gamma}}$
$\displaystyle\mathbf{E}\left[\left(\widetilde{A_{M_{1}}(\ell,N)}^{2}-\mathbf{E}\Big{[}\widetilde{A_{M_{1}}(\ell,N)}^{2}\Big{]}\right)\left(\widetilde{A_{M_{2}}(\ell,N)}^{2}-\mathbf{E}\Big{[}\widetilde{A_{M_{2}}(\ell,N)}^{2}\Big{]}\right)\right]$
$\displaystyle=$
$\displaystyle\frac{1}{|\mathcal{L}_{N,\gamma}|}\frac{2^{2N(2H+1)M_{1}^{2H+1}M_{2}^{2H+1}}}{C_{\Psi}(H)^{2}}\sum_{\ell\in\mathcal{L}_{N,\gamma}}$
$\displaystyle\left(\mathbf{E}\Big{[}\widetilde{A_{M_{1}}(\ell,N)}^{2}\widetilde{A_{M_{2}}(\ell,N)}^{2}\Big{]}-\mathbf{E}\Big{[}\widetilde{A_{M_{1}}(\ell,N)}^{2}\Big{]}\mathbf{E}\Big{[}\widetilde{A_{M_{2}}(\ell,N)}^{2}\Big{]}\right)$
$\displaystyle=$
$\displaystyle\frac{2^{2N(2H+1)M_{1}^{2H+1}M_{2}^{2H+1}}}{C_{\Psi}(H)^{2}}\left(\mathbf{E}\Big{[}\widetilde{A_{M_{1}}(\ell_{0},N)}^{2}\widetilde{A_{M_{2}}(\ell_{0},N)}^{2}\Big{]}-\mathbf{E}\Big{[}\widetilde{A_{M_{1}}(\ell_{0},N)}^{2}\Big{]}\mathbf{E}\Big{[}\widetilde{A_{M_{2}}(\ell_{0},N)}^{2}\Big{]}\right)$

for some $\ell_{0}\in\mathcal{L}_{N,\gamma}$. This time, we write

$\displaystyle\mathbf{E}\big{[}V_{N,M_{1},1}V_{N,M_{2},1}\big{]}-\left(\frac{(M_{1}M_{2})^{2H+1}\mathbf{E}\big{[}c(M_{1}^{-1},0)^{2}c(M_{2}^{-1},0)^{2}\big{]}}{C_{\Psi}(H)^{2}}-1\right)$
$\displaystyle=$
$\displaystyle\frac{M_{1}^{2H+1}M_{2}^{2H+1}}{C_{\Psi}(H)^{2}}\left[2^{2N(2H+1)}\mathbf{E}\Big{[}\widetilde{A_{M_{1}}(\ell_{0},N)}^{2}\widetilde{A_{M_{2}}(\ell_{0},N)}^{2}\Big{]}-\mathbf{E}\big{[}c(M_{1}^{-1},0)^{2}c(M_{2}^{-1},0)^{2}\big{]}\right.$
$\displaystyle-\left.\left(2^{2N(2H+1)}\mathbf{E}\Big{[}\widetilde{A_{M_{1}}(\ell_{0},N)}^{2}\Big{]}\mathbf{E}\Big{[}\widetilde{A_{M_{2}}(\ell_{0},N)}^{2}\Big{]}-M_{1}^{-(2H+1)}M_{2}^{-(2H+1)}C_{\Psi}(H)^{2}\right)\right].$

First, we get, from Lemma 5,

$\displaystyle\left|2^{2N(2H+1)}\mathbf{E}\Big{[}\widetilde{A_{M_{1}}(\ell_{0},N)}^{2}\widetilde{A_{M_{2}}(\ell_{0},N)}^{2}\Big{]}-\mathbf{E}\big{[}c(M_{1}^{-1},0)^{2}c(M_{2}^{-1},0)^{2}\big{]}\right|\leq
C2^{[N^{\beta}]\frac{2H-2}{q}}$ (37)

while we have, using Lemma 4,

$\displaystyle\left|2^{2N(2H+1)}\mathbf{E}\Big{[}\widetilde{A_{M_{1}}(\ell_{0},N)}^{2}\Big{]}\mathbf{E}\Big{[}\widetilde{A_{M_{2}}(\ell_{0},N)}^{2}\Big{]}-M_{1}^{-(2H+1)}M_{2}^{-(2H+1)}C_{\Psi}(H)^{2}\right|$
$\displaystyle\leq\left|2^{N(2H+1)}\mathbf{E}\Big{[}\widetilde{A_{M_{1}}(\ell_{0},N)}^{2}\Big{]}-M_{1}^{-(2H+1)}C_{\Psi}(H)\right|\left|2^{N(2H+1)}\mathbf{E}\Big{[}\widetilde{A_{M_{2}}(\ell_{0},N)}^{2}\Big{]}\right|$
$\displaystyle+\left|M_{1}^{-(2H+1)}C_{\Psi}(H)\right|\left|2^{N(2H+1)}\mathbf{E}\Big{[}\widetilde{A_{M_{2}}(\ell_{0},N)}^{2}\Big{]}-M_{2}^{-(2H+1)}C_{\Psi}(H)\right|$
$\displaystyle\leq C2^{[N^{\beta}]\frac{2H-2}{q}}$ (38)

We conclude the proof by plugging (37) and (38) into (LABEL:4d-7) to obtain
the inequality (32).

For the proof of our main results, we will use the notion of strong
independence. introduced in [54].

######  Definition 1.

Two square integrable random variables $F$ and $G$ admiting the chaos
expansion $F=\sum_{n\geq 0}I_{n}(f_{n}),G=\sum_{n\geq 0}I_{n}(g_{n})$ where
$f_{n},g_{n}\in L^{2}({R}^{n})$ are symmetric for every $n\geq 0$, are
strongly independent if for every $m,n\geq 0$, the random variables
$I_{n}(f_{n})$ and $I_{m}(g_{m})$ are independent.

We need two technical lemmas concerning the strong independent random
variables in Wiener chaos. The first lemma concerns the strong independence of
squares of independent chaotic random variables.

######  Lemma 6.

[12, Lemma 2] Consider two integers $p,q\geq 1$. Let $F=I_{p}(f)$ and
$G=I_{q}(g)$ with $f\in L^{2}({R}^{p}),g\in L^{2}({R}^{q})$. Assume that $F$
and $G$ are independent random variables. Then $F^{2}$ and $G^{2}$ are
strongly independent.

The second lemma deals with scalar product of Malliavin derivative of strongly
independent random variables. Basic notions of Malliavin calculus, together
with the notation used in the following lemma are recalled in the Appendix.

######  Lemma 7.

[54, Lemma 1] and [12, Lemma 4] Assume that $F$ and $G$ are two strongly
independent random variables in $\mathbb{D}^{1,2}$. Then

  1. 1.

$\langle DF,D(-L)^{-1}G\rangle=\langle D(-L)^{-1}F,DG\rangle=0.$

  2. 2.

$\langle DF,D(-L)^{-1}F\rangle$ and $\langle DG,D(-L)^{-1}G\rangle$ are
independent random variables.

Let us now prove the CLT for the sequence (28) when $M=1,...,d$ is fixed. In
the sequel, $d_{W}$ stands for the Wasserstein distance. If $F$ and $G$ are
two random variables, it is defined as

$d_{W}(F,G)=\sup_{h\in\mathcal{A}}|\mathbf{E}[h(F)]-\mathbf{E}[h(G)]|,$

where $\mathcal{A}$ is the set of Lipschitz function $h:{R}\to{R}$ such that

$\sup_{x,y\in{R},x\neq y}\frac{|h(x)-h(y)|}{|x-y|}\leq 1.$

We refer to [43, Appendix C1] for a comprehensive view on distances between
probability distributions.

######  Proposition 3.

Fix $M=1,...,d$ and consider the sequence $(V_{N,M,1},N\geq 1)$ defined by
(28). Let us set

$K_{\Psi,H}=\frac{\mathbf{E}\big{[}c(1,0)^{4}\big{]}}{C_{\Psi}(H)^{2}}-1=\frac{\mathbf{E}\big{[}c(1,0)^{4}\big{]}}{(\mathbf{E}\big{[}c(1,0)^{2}\big{]})^{2}}-1.$
(39)

We have

$V_{N,M,1}\underset{N\to\infty}{\longrightarrow^{(d)}}N(0,K_{\Psi,H}),$

where $\to^{(d)}$ stands for the convergence in distribution. Moreover, for
$N$ large enough,

$d_{W}\left(V_{N,M,1};N(0,K_{\Psi,H})\right)\leq
C2^{-\frac{[N^{\gamma}]}{2}}.$

Proof:  From Theorem 4 and equation (79) in the Appendix, we know that

$\displaystyle d_{W}\left(V_{N,M,1};N(0,K_{\Psi,H})\right)$
$\displaystyle\leq$ $\displaystyle
C\left(\left|\mathbf{E}\big{[}V_{N,M,1}^{2}\big{]}-K_{\Psi,H}\right|+\sqrt{\text{Var}\left[\langle
DV_{N,M,1},D(-L)^{-1}V_{N,M,1}\rangle\right]}\right)$ (40) $\displaystyle\leq$
$\displaystyle
C\left(2^{[N^{\beta}]\frac{2H-2}{q}}+\sqrt{\text{Var}\left(\langle
DV_{N,M,1},D(-L)^{-1}V_{N,M,1}\rangle\right)}\right),$

where we used the estimate in Proposition 2. We start by computing the
quantity $\langle DV_{N,M,1},D(-L)^{-1}V_{N,M,1}\rangle$. We have by (28),

$DV_{N,M,1}=\frac{2^{N(2H+1)}}{C_{\Psi}(H)\sqrt{|\mathcal{L}_{N,\gamma}|}}\sum_{\ell\in\mathcal{L}_{N,\gamma}}D\left(\widetilde{A_{M}(\ell,N)}^{2}-\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}^{2}\Big{]}\right)$

and

$D(-L)^{-1}V_{N,M,1}=\frac{2^{N(2H+1)}}{C_{\Psi}(H)\sqrt{|\mathcal{L}_{N,\gamma}|}}\sum_{\ell\in\mathcal{L}_{N,\gamma}}D(-L)^{-1}\left(\widetilde{A_{M}(\ell,N)}^{2}-\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}^{2}\Big{]}\right).$

Thus, we express

$\displaystyle\langle DV_{N,M,1},D(-L)^{-1}V_{N,M,1}\rangle$ $\displaystyle=$
$\displaystyle\frac{2^{2N(2H+1)}}{C_{\Psi}(H)^{2}|\mathcal{L}_{N,\gamma}|}$
$\displaystyle\sum_{\ell,j\in\mathcal{L}_{N,\gamma}}\langle
D\widetilde{A_{M}(\ell,N)}^{2},D(-L)^{-1}\left(\widetilde{A_{M}(j,N)}^{2}-\mathbf{E}\Big{[}\widetilde{A_{M}(j,N)}^{2}\Big{]}\right)\rangle.$

If $\ell\not=j$, then $\widetilde{A_{M}(\ell,N)}$ and $\widehat{A_{M}(j,N)}$
are independent, by Lemma 2, point 1. By Lemma 6, the random variables
$\widetilde{A_{M}(\ell,N)}^{2}$ and $\widehat{A_{M}(j,N)}^{2}$ are strongly
independent and the Lemma 7 implies that

$\langle
D\widehat{A_{M}(\ell,N)}^{2},D(-L)^{-1}\left(\widehat{A_{M}(j,N)}^{2}-\mathbf{E}\Big{[}\widehat{A_{M}(j,N)}^{2}\Big{]}\right)\rangle=0.$

Therefore, we get

$\displaystyle\langle DV_{N,M,1},D(-L)^{-1}V_{N,M,1}\rangle$
$\displaystyle=\frac{2^{2N(2H+1)}}{C_{\Psi}(H)^{2}|\mathcal{L}_{N,\gamma}|}\sum_{\ell,\in\mathcal{L}_{N,\gamma}}\left\langle
D\widehat{A_{M}(\ell,N)}^{2},D(-L)^{-1}\left(\widehat{A_{M}(\ell,N)}^{2}-\mathbf{E}\Big{[}\widehat{A_{M}(\ell,N)}^{2}\Big{]}\right)\right\rangle$
$\displaystyle=\frac{2^{2N(2H+1)}}{C_{\Psi}(H)^{2}|\mathcal{L}_{N,\gamma}|}\sum_{\ell,\in\mathcal{L}_{N,\gamma}}H_{\ell,N},$

where we used the notation

$H_{\ell,N}=\left\langle
D\widetilde{A_{M}(\ell,N)}^{2},D(-L)^{-1}\left(\widehat{A_{M}(\ell,N)}^{2}-\mathbf{E}\Big{[}\widehat{A_{M}(\ell,N)}^{2}\Big{]}\right)\right\rangle.$

Next, we estimate the variance of $\langle
DV_{N,M,1},D(-L)^{-1}V_{N,M,1}\rangle$. For any $N\geq 1$, we can write

$\displaystyle\text{Var}\left(\langle
DV_{N,M,1},D(-L)^{-1}V_{N,M,1}\rangle\right)$ $\displaystyle=$
$\displaystyle\mathbf{E}\left[\left(\langle
DV_{N,M,1},D(-L)^{-1}V_{N,M,1}\rangle-\mathbf{E}\Big{[}\langle
DV_{N,M,1},D(-L)^{-1}V_{N,M,1}\rangle\Big{]}\right)^{2}\right]$
$\displaystyle=$
$\displaystyle\frac{2^{4N(2H+1)}}{C_{\Psi}(H)^{4}|\mathcal{L}_{N,\gamma}|^{2}}\mathbf{E}\left[\sum_{\ell,j\in\mathcal{L}_{N,\gamma}}\left(H_{\ell,N}-\mathbf{E}\big{[}H_{\ell,N}\big{]}\right)\left(H_{j,N}-\mathbf{E}\big{[}H_{j,N}\big{]}\right)\right].$

By the second part of Lemma 7, for every $\ell\not=j$ we have that
$H_{\ell,N}$ and $H_{j,N}$ are independent and so

$\mathbf{E}\left[\left(H_{\ell,N}-\mathbf{E}\big{[}H_{\ell,N}\big{]}\right)\left(H_{j,N}-\mathbf{E}\big{[}H_{j,N}\big{]}\right)\right]=0.$

Consequently,

$\text{Var}\left[\langle
DV_{N,M,1},D(-L)^{-1}V_{N,M,1}\rangle\right]=\frac{2^{4N(2H+1)}}{C_{\Psi}(H)^{4}|\mathcal{L}_{N,\gamma}|^{2}}\sum_{\ell\in\mathcal{L}_{N,\gamma}}\mathbf{E}\left[\left(H_{\ell,N}-\mathbf{E}\big{[}H_{\ell,N}\big{]}\right)^{2}\right].$
(41)

We claim that for every $N\geq 1$ and $\ell\in\mathcal{L}_{N,\gamma}$ we have

$\mathbf{E}\left[\left(H_{\ell,N}-\mathbf{E}\big{[}H_{\ell,N}\big{]}\right)^{2}\right]\leq
C(q,H)2^{-4N(2H+1)}<\infty,$ (42)

with $C(q,H)>0$ not depending on $N,\ell$. To prove the inequality (42), we
will use the chaos expansion of $H_{\ell,N}$. Recall that
$\widetilde{A_{M}(\ell,N)}=I_{q}(\widehat{g}_{M}(\ell,N)$ with
$\widehat{g}_{M}(\ell,N)$ from (3.1). This implies, via the product formula
(73),

$\widetilde{A_{M}(\ell,N)}^{2}-\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}^{2}\Big{]}=\sum_{r=0}^{q-1}r!\binom{q}{r}^{2}I_{2q-2r}\left(\widehat{g}_{M}(\ell,N)\otimes_{r}\widehat{g}_{M}(\ell,N)\right),$

and

$\displaystyle
D_{\ast}\widetilde{A_{M}(\ell,N)}^{2}=\sum_{r=0}^{q-1}r!\binom{q}{r}^{2}(2q-2r)I_{2q-2r-1}\left((\widehat{g}_{M}(\ell,N)\otimes_{r}\widehat{g}_{M}(\ell,N)(\cdot,\ast)\right),$
$\displaystyle
D_{\ast}(-L)^{-1}\left(\widetilde{A_{M}(\ell,N)}^{2}-\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}^{2}\Big{]}\right)=\sum_{r=0}^{q-1}r!\binom{q}{r}^{2}I_{2q-2r-1}\left((\widehat{g}_{M}(\ell,N)\otimes_{r}\widehat{g}_{M}(\ell,N)(\cdot,\ast)\right).$

As a consequence of the above two relations, we write

$\displaystyle H_{\ell,N}$ $\displaystyle=$
$\displaystyle\sum_{r_{1},r_{2}=0}^{q-1}r_{1}!r_{2}!\binom{q}{r_{1}}^{2}\binom{q}{r_{2}}^{2}(2q-2r_{1})$
(43)
$\displaystyle\int_{\mathbb{R}}dxI_{2q-2r-1}\left((\widehat{g}_{M}(\ell,N)\otimes_{r_{1}}\widehat{g}_{M}(\ell,N)(\cdot,x)\right)I_{2q-2r-1}\left((\widehat{g}_{M}(\ell,N)\otimes_{r_{2}}\widehat{g}_{M}(\ell,N)(\cdot,x)\right)dx$
$\displaystyle=$
$\displaystyle\sum_{r_{1},r_{2}=0}^{q-1}r_{1}!r_{2}!\binom{q}{r_{1}}^{2}\binom{q}{r_{2}}^{2}(2q-2r_{1})$
$\displaystyle\sum_{a=0}^{(2q-2r_{1}-1)\wedge(2q-2r_{2}-1)}a!\binom{2q-2r_{1}-1}{a}\binom{2q-2r_{2}-1}{a}$
$\displaystyle
I_{4q-2r_{1}-2r_{2}-2-2a}\left((\widehat{g}_{M}(\ell,N)\widetilde{\otimes}_{r_{1}}\widehat{g}_{M}(\ell,N)\otimes_{a+1}(\widehat{g}_{M}(\ell,N)\widetilde{\otimes}_{r_{2}}\widehat{g}_{M}(\ell,N)\right).$

We remark, thanks to (24), that

$q!\|\widehat{g}_{M}(\ell,N)\|^{2}_{L^{2}({R}^{q})}=\mathbf{E}\Big{[}\widetilde{A_{M}(\ell,N)}^{2}\big{]}\leq
C2^{-N(2H+1)},$

and, as a consequence, for every $r_{1},r_{2}=0,...,q-1$ and for every
$a=0,...,(2q-2r_{1}-1)\wedge(2q-2r_{2}-1)$, we have

$\displaystyle\mathbf{E}\left[I_{4q-2r_{1}-2r_{2}-2-2a}\left((\widehat{g}_{M}(\ell,N)\widetilde{\otimes}_{r_{1}}\widehat{g}_{M}(\ell,N))\otimes_{a+1}(\widehat{g}_{M}(\ell,N)\widetilde{\otimes}_{r_{2}}\widehat{g}_{M}(\ell,N))\right)^{2}\right]$
(44) $\displaystyle=$ $\displaystyle
C(q,r_{1},r_{2},a)\|(\widehat{g}_{M}(\ell,N)\widetilde{\otimes}_{r_{1}}\widehat{g}_{M}(\ell,N)\otimes_{a+1}(\widehat{g}_{M}(\ell,N)\widetilde{\otimes}_{r_{2}}\widehat{g}_{M}(\ell,N))\|^{2}_{M^{2}(\mathbb{R}^{4q-2r_{1}-2r_{2}-2a-2})}$
$\displaystyle\leq$ $\displaystyle
C(q,r_{1},r_{2},a)\|\widehat{g}_{M}(\ell,N)\widetilde{\otimes}_{r_{1}}\widehat{g}_{M}(\ell,N)\|^{2}_{L^{2}({R}^{2q-2r_{1}})}\|\widehat{g}_{M}(\ell,N)\widetilde{\otimes}_{r_{2}}\widehat{g}_{M}(\ell,N)\|^{2}_{L^{2}({R}^{2q-2r_{2}})}$
$\displaystyle\leq$ $\displaystyle
C(q,r_{1},r_{2},a)\|\widehat{g}_{M}(\ell,N)\|^{8}_{L^{2}({R}^{q})}$
$\displaystyle\leq$ $\displaystyle C(q,r_{1},r_{2},a)2^{-4N(2H+1)}.$

By combining the above estimates (44) and (43), we obtain the claim (42).
Coming back to (41), we get, using (13),

$\displaystyle\text{Var}\left[\langle
DV_{N,M,1},D(-L)^{-1}V_{N,M,1}\rangle\right]$ $\displaystyle\leq$
$\displaystyle
C(q,H,\Psi)\frac{2^{4N(2H+1)}}{|\mathcal{L}_{N,\gamma}|^{2}}\sum_{\ell\in\mathcal{L}_{N,\gamma}}2^{-4N(2H+1)}$
(45) $\displaystyle\leq$ $\displaystyle C(q,H,\Psi)2^{[N^{\gamma}]}.$

The conclusion is obtained via (40) and (45).

We immediately deduce the asymptotic normality of the modified wavelet
variation and its rate of convergence under the Wasserstein distance.

######  Theorem 1.

Consider the sequence $V_{N,M}$ defined by (25). Then

$V_{N,M}\underset{N\to\infty}{\longrightarrow^{(d)}}N(0,K_{\Psi,H}),$

and for $N$ large enough,

$d_{W}\left(V_{N,M};N(0,K_{\Psi,H})\right)\leq C2^{-\frac{[N^{\gamma}]}{2}}.$

Proof:  The proof is a consequence of the results proven in Propositions 1,
and 3. Indeed, by the triangle’s inequality,

$\displaystyle d_{W}\left(V_{N,M};N(0,K_{\Psi,H})\right)$ $\displaystyle\leq$
$\displaystyle
d_{W}\left(V_{N,M,1};N(0,K_{\Psi,H})\right)+d_{W}(V_{N,M,1},V_{N,M})$
$\displaystyle\leq$ $\displaystyle
d_{W}\left(V_{N,M,1};N(0,K_{\Psi,H})\right)+\mathbf{E}\big{[}|V_{N,M,1}-V_{N,M}|\big{]}$
$\displaystyle\leq$ $\displaystyle
d_{W}\left(V_{N,M,1};N(0,K_{\Psi,H})\right)+\mathbf{E}\big{[}|V_{N,M,2}|\big{]}+\mathbf{E}\big{[}|V_{N,M,3}|\big{]}.$

It suffices to use Proposition 3 to bound the first summand in the right-hand
side above and Proposition 1 to bound the second and third summands.

Let us now state and prove a multidimensional CLT for the modified wavelet
variation.

######  Theorem 2.

Let $V_{N,M}$ be given by (25). Then the $d$-dimensional random vector
$(V_{N,M},M=1,,,,,d)$ converges in distribution, as $N\to\ \infty$, to the
$d$-dimensional Gaussian vector $N(0,K)$, where
$K=(K_{M_{1},M_{2}})_{M_{1},M_{2}=1,...,d}$ is such that

$K_{M,M}=K_{\Psi,H}\mbox{ for every }M=1,...,d$ (46)

and

$K_{M_{1},M_{2}}=\frac{(M_{1}M_{2})^{2H+1}\mathbf{E}\big{[}c(M_{1}^{-1},0)^{2}c(M_{2}^{-1},0)^{2}\big{]}}{C_{\Psi}(H)^{2}}-1\mbox{
for }M_{1},M_{2}=1,...,d,M_{1}\not=M_{2}.$ (47)

Proof:  The triangle’s inequality gives

$\displaystyle d_{W}\left((V_{N,M},M=1,...,d),N(0,K)\right)$
$\displaystyle\leq$ $\displaystyle
d_{W}\left((V_{N,M,1},M=1,...,d),N(0,K)\right)$ (48)
$\displaystyle+\sum_{M=1}^{d}\left(\mathbf{E}\big{[}|V_{N,M,2}|\big{]}+\mathbf{E}\big{[}|V_{N,M,3}|\big{]}\right).$

By Theorem 4 and equation (79) in the Appendix, we have

$\displaystyle d_{W}\left((V_{N,M,1},M=1,...,d),N(0,K)\right)$
$\displaystyle\leq$ $\displaystyle
C\left(\sqrt{\sum_{M_{1},M_{2}=1}^{d}\left(\mathbf{E}\big{[}V_{N,M_{1},1}V_{N,M_{2},1}\big{]}-K_{M_{1},M_{2}}\right)^{2}}\right.$
$\displaystyle\left.+\sqrt{\sum_{M_{1},M_{2}=1}^{d}\text{Var}\left[\langle
DV_{N,M_{1},1},D(-L)^{-1}V_{N,M_{2},1}\rangle\right]}\right)$

We need to compute the variance of the quantity $\langle
DV_{N,M_{1},1},D(-L)^{-1}V_{N,M_{2},1}\rangle$. The situation when
$M_{1}=M_{2}$ has been treated in the proof of Proposition 3 and the case
$M_{1}\not=M_{2}$ follows in a similar way. Actually, we express

$\displaystyle\langle DV_{N,M_{1},1},D(-L)^{-1}V_{N,M_{2},1}\rangle$
$\displaystyle=$
$\displaystyle\frac{2^{2N(2H+1)}}{C_{\Psi}(H)^{2}|\mathcal{L}_{N,\gamma}|}$
$\displaystyle=$
$\displaystyle\frac{2^{2N(2H+1)}}{C_{\Psi}(H)^{2}|\mathcal{L}_{N,\gamma}|}\sum_{\ell,\in\mathcal{L}_{N,\gamma}}H_{\ell,N}(M_{1},M_{2})$

where we used the notation

$H_{\ell,N}(M_{1},M_{2})=\left\langle
D\widehat{A_{M_{1}}(\ell,N)}^{2},D(-L)^{-1}\left(\widehat{A_{M_{2}}(\ell,N)}^{2}-\mathbf{E}\Big{[}\widehat{A_{M_{2}}(\ell,N)}^{2}\Big{]}\right)\right\rangle.$

This gives, as in (41),

$\displaystyle\text{Var}\left[\langle
DV_{N,M_{1},1},D(-L)^{-1}V_{N,M_{2},1}\rangle\right]$ (50)
$\displaystyle=\frac{2^{4N(2H+1)}}{C_{\Psi}(H)^{4}|\mathcal{L}_{N,\gamma}|^{2}}\sum_{\ell\in\mathcal{L}_{N,\gamma}}\mathbf{E}\left[\left(H_{\ell,N}(M_{1},M_{2})-\mathbf{E}\big{[}H_{\ell,N}(M_{1},M_{2})\big{]}\right)^{2}\right]$

and we can show that

$\mathbf{E}\left[\left(H_{\ell,N}(M_{1},M_{2})-\mathbf{E}\big{[}H_{\ell,N}(M_{1},M_{2})\big{]}\right)^{2}\right]\leq
C(q,H)2^{-4N(2H+1)}.$

We obtain, from (13),

$\text{Var}\left[\langle
DV_{N,M_{1},1},D(-L)^{-1}V_{N,M_{2},1}\rangle\right]\leq C2^{-[N^{\gamma}]}.$
(51)

By combining (32), (51) and (LABEL:8d-3), we conclude

$d_{W}\left((V_{N,M,1},M=1,...,d),N(0,K)\right)\leq
C2^{-\frac{[N^{\gamma}]}{2}}.$

We finish the proof by plugging this estimate, together with the estimates in
Proposition 1, into (48).

##  4 Discretization of the wavelet variation

We will define an estimator for the Hurst index of the Hermite process based
on the modified wavelet variation $V_{N,M}$ given by (25). In order to obtain
an expression of the estimator which can be numerically computed, we need to
discretize the wavelet coefficient $A_{M}(\ell,N)$.

We will consider the following Riemann approximation: for $M=1,...,d$, $N\geq
1$ and for $\ell\in\mathcal{L}_{N,\gamma}$, we set

$E_{M}(\ell,N)=\sqrt{\frac{1}{M2^{N}}}\frac{1}{2^{N}}\sum_{k=1}^{2^{N}}\Psi\left(\frac{k}{2^{N}}\right)Z_{\frac{k2^{-N}+M\ell
2^{[N^{\beta}]}}{M2^{N}}},$ (52)

and

$\widehat{V}_{N.M}=\frac{1}{\sqrt{|\mathcal{L}_{N,\gamma}|}}\sum_{\ell\in\mathcal{L}_{N,\gamma}}\left[\frac{E_{M}(\ell,N)^{2}}{\mathbf{E}\big{[}A_{M}(\ell,N)^{2}\big{]}}-1\right].$
(53)

We will prove that $V_{N,M}-\widehat{V}_{N,M}$ converges to zero in
$L^{1}(\Omega)$ as $N\to\infty$ and we will deduce that the discretized
wavelet variation $\widehat{V}_{N,M}$ also satisfies a CLT. We start by
evaluating the difference between $A_{M}(\ell,N)$ and its discretized
counterpart $E_{M}(\ell,N)$.

######  Proposition 4.

Let us denote

$t_{N,M}=\frac{A_{M}(\ell,N)}{\sqrt{\mathbf{E}\big{[}A_{M}(\ell,N)^{2}\big{]}}}-\frac{E_{M}(\ell,N)}{\sqrt{\mathbf{E}\big{[}A_{M}(\ell,N)^{2}\big{]}}},\hskip
11.38092ptN\geq 1,M=1,...,d.$ (54)

Then, for $N$ large enough, we have

$\mathbf{E}\big{[}t_{N,M}^{2}\big{]}\leq C2^{-N}2^{[N^{\beta}]2H}.$

Proof:  By (17), we get

$t_{N,M}=\sqrt{\frac{M^{2H+1}}{C_{\Psi}(H)}2^{N(2H+1)}}(A_{M}(\ell,N)-E_{M}(\ell,N)),$

and then

$\mathbf{E}\big{[}t_{N,M}^{2}\big{]}=\frac{M^{2H+1}}{C_{\Psi}(H)}2^{N(2H+1)}(\mathbf{E}\big{[}A_{M}(\ell,N)^{2}\big{]}-2\mathbf{E}\big{[}A_{M}(\ell,N)E_{M}(\ell,N)\big{]}+\mathbf{E}\big{[}E_{M}(\ell,N)^{2}\big{]}).$

By (16) and (52), we write

$\displaystyle\mathbf{E}\big{[}A_{M}(\ell,N)E_{M}(\ell,N)\big{]}$
$\displaystyle=$
$\displaystyle\frac{1}{M2^{2N}}\sum_{k=1}^{2^{N}}\Psi\left(\frac{k}{2^{N}}\right)\int_{\mathbb{R}}dx\Psi(x)\mathbf{E}\left[Z_{\frac{x+M\ell
2^{[N^{\beta}]}}{M2^{N}}}Z_{\frac{k2^{-N}+M\ell
2^{[N^{\beta}]}}{M2^{N}}}\right]$ $\displaystyle=$
$\displaystyle\frac{1}{M2^{2N}}\frac{1}{2(M2^{N})^{2H}}\sum_{k=1}^{2^{N}}\Psi\left(\frac{k}{2^{N}}\right)\int_{\mathbb{R}}dx\Psi(x)\left[(x+M\ell
2^{[N^{\beta}]})^{2H}-|x-k2^{-N}|^{2H}\right],$

where we used the assumption (3). Also, we have

$\displaystyle\mathbf{E}\big{[}E_{M}(\ell,N)^{2}\big{]}$ $\displaystyle=$
$\displaystyle\frac{1}{M2^{3N}}\sum_{k,j=1}^{2^{N}}\Psi\left(\frac{k}{2^{N}}\right)\Psi\left(\frac{j}{2^{N}}\right)\mathbf{E}\left[Z_{\frac{k2^{-N}+M\ell
2^{[N^{\beta}]}}{M2^{N}}}Z_{\frac{j2^{-N}+M\ell
2^{[N^{\beta}]}}{M2^{N}}}\right]$ $\displaystyle=$
$\displaystyle\frac{1}{M2^{3N}}\frac{1}{2(M2^{N})^{2H}}\sum_{k,j=1}^{2^{N}}\Psi\left(\frac{k}{2^{N}}\right)\Psi\left(\frac{j}{2^{N}}\right)$
$\displaystyle\left[(k2^{-N}+M\ell 2^{[N^{\beta}]})^{2H}+(j2^{-N}+M\ell
2^{[N^{\beta}]})^{2H}-|k2^{-N}-j2^{-N}|^{2H}\right].$

Consequently, we express

$\displaystyle\mathbf{E}\big{[}t_{N,M}^{2}\big{]}$ $\displaystyle=$
$\displaystyle\frac{M^{2H+1}}{C_{\Psi}(H)}\left[M^{-(2H+1)}C_{\Psi}(H)-\right.$
$\displaystyle\left.M^{-2H-1}\frac{1}{2^{N}}\sum_{k=1}^{2^{N}}\Psi\left(\frac{k}{2^{N}}\right)\int_{\mathbb{R}}dx\Psi(x)\left[(x+M\ell
2^{[N^{\beta}]})^{2H}-|x-k2^{-N}|^{2H}\right]\right.$
$\displaystyle\left.+\frac{1}{2}M^{-2H-1}\frac{1}{2^{2N}}\sum_{k,j=1}^{2^{N}}\Psi\left(\frac{k}{2^{N}}\right)\Psi\left(\frac{j}{2^{N}}\right)\right.$
$\displaystyle\left.\left[(k2^{-N}+M\ell 2^{[N^{\beta}]})^{2H}+(j2^{-N}+M\ell
2^{[N^{\beta}]})^{2H}-|k2^{-N}-j2^{-N}|^{2H}\right]\right]$ $\displaystyle=$
$\displaystyle
1+\frac{1}{C_{\Psi}(H)}\frac{1}{2^{N}}\sum_{k=1}^{2^{N}}\Psi\left(\frac{k}{2^{N}}\right)\int_{\mathbb{R}}dx\Psi(x)|x-k2^{-N}|^{2H}$
$\displaystyle-\frac{1}{2}\frac{1}{C_{\Psi}(H)}\frac{1}{2^{2N}}\sum_{k,j=1}^{2^{N}}\Psi\left(\frac{k}{2^{N}}\right)\Psi\left(\frac{j}{2^{N}}\right)|k2^{-N}-j2^{-N}|^{2H}$
$\displaystyle-
C_{\Psi}(H)^{-1}\left[\frac{1}{2^{N}}\sum_{k=1}^{2^{N}}\Psi\left(\frac{k}{2^{N}}\right)\int_{\mathbb{R}}dx\Psi(x)(x+M\ell
2^{[N^{\beta}]})^{2H}\right.$
$\displaystyle\left.-\frac{1}{2^{2N}}\sum_{k,j=1}^{2^{N}}\Psi\left(\frac{k}{2^{N}}\right)\Psi\left(\frac{j}{2^{N}}\right)(k2^{-N}+M\ell
2^{[N^{\beta}]})^{2H}\right]$ $\displaystyle=:$ $\displaystyle
1+C_{\Psi}(H)^{-1}(T_{1,N}-\frac{1}{2}T_{2,N})-C_{\Psi}(H)^{-1}Q_{N},$

with

$\displaystyle
T_{1,N}=\frac{1}{2^{N}}\sum_{k=1}^{2^{N}}\Psi\left(\frac{k}{2^{N}}\right)\int_{\mathbb{R}}dx\Psi(x)|x-k2^{-N}|^{2H},$

$T_{2,N}=\frac{1}{2^{2N}}\sum_{k,j=1}^{2^{N}}\Psi\left(\frac{k}{2^{N}}\right)\Psi\left(\frac{j}{2^{N}}\right)|k2^{-N}-j2^{-N}|^{2H},$

and

$\displaystyle Q_{N}$ $\displaystyle=$ $\displaystyle
2^{-N}\sum_{k=1}^{2^{N}}\Psi\left(\frac{k}{2^{N}}\right)\left[\int_{\mathbb{R}}dx\Psi(x)(x+\ell
M2^{[n^{\beta}]})^{2H}-\right.$
$\displaystyle\left.2^{-N}\sum_{j=1}^{2^{N}}\Psi\left(\frac{j}{2^{N}}\right)(k2^{-N}+\ell
M2^{[n^{\beta}]})^{2H}\right]$

We show that

$T_{1,N}\underset{N\to\infty}{\longrightarrow}\int_{0}^{1}\int_{0}^{1}\Psi(x)\Psi(y)|x-y|^{2H-2}dxdy=-2C_{\Psi}(H)$

and

$|T_{1,N}-(-2C_{\Psi}(H))|\leq C2^{-N}.$ (55)

We can decompose $T_{1,N}$ as follows.

$\displaystyle T_{1,N}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\int_{0}^{1}\Psi(x)\Psi(y)|x-y|^{2H-2}dxdy$
$\displaystyle+\sum_{k=1}^{2^{N}}\int_{\frac{k-1}{2^{N}}}^{\frac{k}{2^{N}}}dy\left(\Psi\left(\frac{k}{2^{N}}\right)-\Psi(y)\right)\int_{\mathbb{R}}dx\Psi(x)|x-k2^{-N}|^{2H}$
$\displaystyle+\sum_{k=1}^{2^{N}}\int_{\frac{k-1}{2^{N}}}^{\frac{k}{2^{N}}}dy\Psi(y)\int_{\mathbb{R}}\Psi(x)dx\left(|x+k2^{-N}|^{2H}-|x-y|^{2H}\right)$
$\displaystyle=:$ $\displaystyle-2C_{\Psi}(H)+R_{1,N}+R_{2,N}.$

Let us bound the two rest terms, denoted by $R_{1,N}$ and $R_{2,N}$. We have,
since $\Psi$ is of class $C^{1}$ with support contained in $[0,1]$,

$\displaystyle|R_{1,N}|$ $\displaystyle\leq$ $\displaystyle
C\sum_{k=1}^{2^{N}}\int_{\frac{k-1}{2^{N}}}^{\frac{k}{2^{N}}}dy|y-k2^{-N}|\int_{\mathbb{R}}dx|\Psi(x)||x-k2^{-N}|^{2H}$
(56) $\displaystyle\leq$ $\displaystyle
C2^{-N}\sum_{k=1}^{2^{N}}\int_{\frac{k-1}{2^{N}}}^{\frac{k}{2^{N}}}dy\int_{\mathbb{R}}dx|\Psi(x)|\leq
C2^{-N}.$

Concerning the summand $R_{2,N}$, let us remark that the function

$f:[0,1]\to\mathbb{R}\,:\,a\mapsto|x-a|^{2H}$

satisfies $|f^{\prime}(a)|\leq C$ as soon as $H>\frac{1}{2}$. Thus, we get

$\displaystyle|R_{2,N}|$ $\displaystyle\leq$ $\displaystyle
C\sum_{k=1}^{2^{N}}\int_{\frac{k-1}{2^{N}}}^{\frac{k}{2^{N}}}dy|\Psi(y)||y-k2^{-N}|\int_{\mathbb{R}}|\Psi(x)|dx$
(57) $\displaystyle\leq$ $\displaystyle
C2^{-N}\sum_{k=1}^{2^{N}}\int_{\frac{k-1}{2^{N}}}^{\frac{k}{2^{N}}}dy|\Psi(y)|\int_{\mathbb{R}}|\Psi(x)|dx\leq
C2^{-N}.$

The estimates (56) and (57) imply (55).

In the same way, we can prove that
$T_{2,N}\underset{N\to\infty}{\longrightarrow}-2C_{\Psi}(H)$ and

$\left|T_{2,N}-(-2C_{\Psi}(H))\right|\leq C2^{-N}.$

We now show that $Q_{N}$ converges to zero as $N$ tends to infinity. One has

$\displaystyle Q_{N}$ $\displaystyle=$ $\displaystyle
2^{-N}\sum_{k=1}^{2^{N}}\Psi\left(\frac{k}{2^{N}}\right)\left[\int_{\mathbb{R}}dx\Psi(x)(x+\ell
M2^{[n^{\beta}]})^{2H}-2^{-N}\sum_{j=1}^{2^{N}}\Psi\left(\frac{j}{2^{N}}\right)(j2^{-N}+\ell
M2^{[n^{\beta}]})^{2H}\right]$ $\displaystyle=$ $\displaystyle
2^{-N}\sum_{k=1}^{2^{N}}\Psi\left(\frac{k}{2^{N}}\right)\sum_{j=1}^{2^{N}}\int_{\frac{j-1}{2^{N}}}^{\frac{j}{2^{N}}}dx\left[\Psi(x)(x+\ell
M2^{[n^{\beta}]})^{2H}-\Psi\left(\frac{j}{2^{N}}\right)(j2^{-N}+\ell
M2^{[n^{\beta}]})^{2H}\right].$

Hence

$\displaystyle|Q_{N}|$ $\displaystyle\leq$ $\displaystyle
2^{-N}\sum_{k=1}^{2^{N}}\left|\Psi\left(\frac{k}{2^{N}}\right)\right|\sum_{j=1}^{2^{N}}\int_{\frac{j-1}{2^{N}}}^{\frac{j}{2^{N}}}dx|\Psi(x)|\left|(x+\ell
M2^{[n^{\beta}]})^{2H}-(j2^{-N}+\ell M2^{[n^{\beta}]})^{2H}\right|$
$\displaystyle+2^{-N}\sum_{k=1}^{2^{N}}\left|\Psi\left(\frac{k}{2^{N}}\right)\right|\sum_{j=1}^{2^{N}}\int_{\frac{j-1}{2^{N}}}^{\frac{j}{2^{N}}}dx(j2^{-N}+\ell
M2^{[n^{\beta}]})^{2H}\left|\Psi(x)-\Psi\left(\frac{j}{2^{N}}\right)\right|.$

We use the bounds, for $x\in(\frac{j-1}{2^{N}},\frac{j}{2^{N}})$,

$|(x+\ell M2^{[n^{\beta}]})^{2H}-(j2^{-N}+\ell M2^{[n^{\beta}]})^{2H}|\leq
C2^{-N}2^{(2H-1)[N^{\beta}]}$

and

$\left|\Psi(x)-\Psi\left(\frac{j}{2^{N}}\right)\right|\leq C2^{-N}.$

Thus, we reach

$|Q_{N}|\leq C2^{-N}2^{[N^{\beta}]2H}.$ (58)

From (55) and (58), we obtained the conclusion.

Next, let us compare the wavelet variation (25) and its discretized version
(53).

######  Proposition 5.

For $N$ sufficiently large, we have

$\mathbf{E}\big{[}|V_{N,M}-\widehat{V}_{N,M}|\big{]}\leq
C2^{-N}2^{[N^{\beta}]2H}2^{-\frac{[N^{\gamma}]}{2}}.$

Proof:  By (25) and (53), we can write

$\displaystyle
V_{N,M}-\widehat{V}_{N.M}=\frac{1}{\sqrt{|\mathcal{L}_{N,\gamma}|}}\sum_{\ell\in\mathcal{L}_{N,\gamma}}$
$\displaystyle\left(\frac{A_{M}(\ell,N)}{\sqrt{\mathbf{E}\big{[}A_{M}(\ell,N)^{2}\big{]}}}-\frac{E_{M}(\ell,N)}{\sqrt{\mathbf{E}\big{[}A_{M}(\ell,N)^{2}\big{]}}}\right)\left(\frac{A_{M}(\ell,N)}{\sqrt{\mathbf{E}\big{[}A_{M}(\ell,N)^{2}\big{]}}}+\frac{E_{M}(\ell,N)}{\sqrt{\mathbf{E}\big{[}A_{M}(\ell,N)^{2}\big{]}}}\right)$

and thus, by the Cauchy-Schwarz’s inequality,

$\displaystyle\mathbf{E}\left[\left|V_{N,M}-\widehat{V}_{N.M}\right|\right]$
$\displaystyle\leq$
$\displaystyle\frac{1}{\sqrt{|\mathcal{L}_{N,\gamma}|}}\sum_{\ell\in\mathcal{L}_{N,\gamma}}$
$\displaystyle\mathbf{E}\left[\left|\frac{A_{M}(\ell,N)}{\sqrt{\mathbf{E}A_{M}(\ell,N)^{2}}}-\frac{E_{M}(\ell,N)}{\sqrt{\mathbf{E}A_{M}(\ell,N)^{2}}}\right|\cdot\left|\frac{A_{M}(\ell,N)}{\sqrt{\mathbf{E}A_{M}(\ell,N)^{2}}}+\frac{E_{M}(\ell,N)}{\sqrt{\mathbf{E}A_{M}(\ell,N)^{2}}}\right|\right]$
$\displaystyle\leq$
$\displaystyle\frac{1}{\sqrt{|\mathcal{L}_{N,\gamma}|}}\sum_{\ell\in\mathcal{L}_{N,\gamma}}$
$\displaystyle\left(\mathbf{E}\left[\left|\frac{A_{M}(\ell,N)}{\sqrt{\mathbf{E}A_{M}(\ell,N)^{2}}}-\frac{E_{M}(\ell,N)}{\sqrt{\mathbf{E}A_{M}(\ell,N)^{2}}}\right|^{2}\right]\right)^{\frac{1}{2}}\left(\mathbf{E}\left[\left|\frac{A_{M}(\ell,N)}{\sqrt{\mathbf{E}A_{M}(\ell,N)^{2}}}+\frac{E_{M}(\ell,N)}{\sqrt{\mathbf{E}A_{M}(\ell,N)^{2}}}\right|^{2}\right]\right)^{\frac{1}{2}}.$

Since, clearly,

$\mathbf{E}\left[\left|\frac{A_{M}(\ell,N)}{\sqrt{\mathbf{E}A_{M}(\ell,N)^{2}}}+\frac{E_{M}(\ell,N)}{\sqrt{\mathbf{E}A_{M}(\ell,N)^{2}}}\right|^{2}\right]\leq
C,$

we get from Proposition 4,

$\displaystyle\mathbf{E}\left[\left|V_{N,M}-\widehat{V}_{N.M}\right|\right]$
$\displaystyle\leq$
$\displaystyle\frac{1}{\sqrt{|\mathcal{L}_{N,\gamma}|}}\sum_{\ell\in\mathcal{L}_{N,\gamma}}2^{[N^{\beta}]H}2^{-\frac{N}{2}}$
$\displaystyle\leq$ $\displaystyle
C2^{-\frac{N}{2}}2^{[N^{\beta}]H}2^{\frac{[N^{\gamma}]}{2}}\underset{N\to\infty}{\longrightarrow}0.$

The above result implies that the discretized wavelet variation has the same
limit behavior in distribution as $V_{N,M}$.

######  Corollary 1.

Let $\widehat{V}_{N,M}$ be given by (53) Then the $d$-dimensional random
vector $(\widehat{V}_{N,M},M=1,,,,,d)$ converges in distribution, as $N\to\
\infty$, to the $d$-dimensional Gaussian vector $N(0,K)$, where the matrix $K$
is given by (46) and (47).

Proof:  The result follows from Theorem 1 and Proposition 5.

We finish this section with a short result concerning the almost-sure
convergence to $0$ of the sequence
$(\widehat{V}_{N,M}/\sqrt{|\mathcal{L}_{N,\gamma}|},N\geq 1)$. This fact will
be particularly useful in the next section.

######  Proposition 6.

We have, for all $M\in\\{1,\dots,d\\}$,

$\frac{\widehat{V}_{N,M}}{\sqrt{|\mathcal{L}_{N,\gamma}|}}\underset{N\to\infty}{\longrightarrow}0$
(59)

almost surely, at the fast rate $2^{-N^{a}}$, where $a\in(0,\gamma)$ is
arbitrary and fixed

Proof:  Let us remark that Proposition 5 and Theorem 1 entail that, for
$M=1,\dots,d$, the sequence
$(\mathbf{E}\big{[}|\widehat{V}_{N,M}|\big{]})_{N}$ is bounded. Therefore, as
a consequence of (14), we get, for such $M$,

$\mathbf{E}\big{[}|\widehat{V}_{N,M}|\big{]}\leq C2^{-\frac{[N^{\gamma}]}{2}}$
(60)

The proofs is then obtained by a simple Borel-Cantelli argument. Let
$a\in(0,\gamma)$ be arbitrary and fixed. We deduce from (60) and Markov
inequality that, for $M=1,\dots,d$,

$\sum_{N\geq
1}\mathbf{P}\left(\left|\frac{\widehat{V}_{N,M}}{\sqrt{|\mathcal{L}_{N,\gamma}|}}\right|\geq
2^{N^{a}}\right)\leq\sum_{N\geq
1}2^{N^{a}}2^{-\frac{[N^{\gamma}]}{2}}\mathbf{E}\big{[}|\widehat{V}_{N,M}|\big{]}\leq
C\sum_{N\geq 1}2^{N^{a}}2^{-\frac{[N^{\gamma}]}{2}}<\infty.$

##  5 Estimation of the Hurst parameter

Let us introduce the sequences

$S_{N,M}=\frac{1}{|\mathcal{L}_{N,\gamma}|}\sum_{\ell\in\mathcal{L}_{N,\gamma}}A_{M}(\ell,N)^{2}$
(61)

and

$\widehat{S}_{N,M}=\frac{1}{|\mathcal{L}_{N,\gamma}|}\sum_{\ell\in\mathcal{L}_{N,\gamma}}E_{M}(\ell,N)^{2}$
(62)

where $A_{M}(\ell,N),E_{M}(\ell,N)$ are the wavelet coefficients given by (16)
and (52), respectively. We clearly have, via Proposition 4,

$\mathbf{E}\big{[}|A_{M}(\ell,N)-E_{M}(\ell,N)|^{2}\big{]}\leq
C2^{[N^{\beta}]2H}2^{-N}\mathbf{E}\big{[}A_{M}(\ell,N)^{2}\big{]}\leq
C2^{[N^{\beta}]2H}2^{-N}2^{-N(2H+1)}$

so, by Cauchy-Schwarz’s inequality, we get

$\displaystyle\mathbf{E}\big{[}|S_{N,M}-\widehat{S}_{N,M}|\big{]}$
$\displaystyle\leq$
$\displaystyle\frac{1}{|\mathcal{L}_{N,\gamma}|}\sum_{\ell\in\mathcal{L}_{N,\gamma}}\mathbf{E}\big{[}|A_{M}(\ell,N)^{2}-E_{M}(\ell,N)^{2}|\big{]}$
$\displaystyle\leq$
$\displaystyle\frac{1}{|\mathcal{L}_{N,\gamma}|}\sum_{\ell\in\mathcal{L}_{N,\gamma}}\left(\mathbf{E}\big{[}|A_{M}(\ell,N)-E_{M}(\ell,N)|^{2}\big{]}\right)^{\frac{1}{2}}\left(\mathbf{E}\big{[}|A_{M}(\ell,N)+E_{M}(\ell,N)|^{2}\big{]}\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
C\sqrt{2^{[N^{\beta}]2H}2^{-N}2^{-N(2H+1)}}\sqrt{2^{-N(2H+1)}},$

which entails

$S_{N,M}-\widehat{S}_{N,M}\underset{N\to\infty}{\longrightarrow}0\mbox{ in
}L^{1}(\Omega).$ (63)

We also have, due to (17),

$\mathbf{E}\big{[}S_{N,M}\big{]}=(M2^{N})^{-(2H+1)}C_{\Psi}(H),$

so

$\log\mathbf{E}\big{[}S_{N,M}\big{]}=-(2H+1)\log(M2^{N})+\log C_{\Psi}(H).$
(64)

To construct an estimator for the Hurst index of the Hermite process, we use
the following standard procedure: in (64) we approximate $\mathbf{E}[S_{N,M}]$
by $S_{N,M}$ and thus, due to (63), by $\widehat{S}_{N,M}$. We then have

$\log\widehat{S}_{N,M}\sim-(2H+1)\log(M2^{N})+\log C_{\Psi}(H).$

Next, we make a log-regression of
$\left(\log\widehat{S}_{N,M},M=1,...,d\right)$ on
$\left(\log(M2^{N}),M=1,...,d\right)$ which leads to the estimator
$\widehat{H}_{N}$ given by

$\widehat{H}_{N}=-\frac{1}{2}\frac{\left(\sum_{M=1}^{d}\log\widehat{S}_{N,M}\log
M\right)-\left(\sum_{M=1}^{d}\log\widehat{S}_{N,M}\right)\left(\sum_{M=1}^{d}\log
M\right)}{\left(\sum_{M=1}^{d}(\log M)^{2}\right)-\left(\sum_{M=1}^{d}\log
M\right)^{2}}-\frac{1}{2}$ (65)

We state and prove the limit behavior of the above estimator.

######  Theorem 3.

Let $\widehat{H}_{N}$ be given by (65). Then the estimator $\widehat{H}_{N}$
is strongly consistent, i.e.

$\widehat{H}_{N}\underset{N\to\infty}{\longrightarrow}H\mbox{ almost surely}.$
(66)

Moreover,

$\sqrt{|\mathcal{L}_{N,\gamma}|}(H-\widehat{H}_{N})\underset{N\to\infty}{\longrightarrow^{(d)}}N(0,\sigma^{2}),$
(67)

with
$\sigma^{2}=\frac{1}{4}(L_{d}^{T}L_{d})^{-1}L_{d}KL_{d}^{T}(L_{d}^{T}L_{d})^{-1}$,
where $K$ is the matrix defined in Theorem 2 and $L_{d}$ is the matrix with
$(L_{d})_{M,1}=\log M$ and $(L_{d})_{M,2}=1$, for $M=1,\dots,d$.

Proof:  To obtain the asymptotic properties of the estimator (65), we use the
limit theorems obtained for the discretized wavelet variation
$\widehat{V}_{N,M}$. We notice the following link between $\widehat{S}_{N,M}$
and $\widehat{V}_{N,M}$:

$\frac{\widehat{V}_{N,M}}{\sqrt{|\mathcal{L}_{N,\gamma}|}}+1=\frac{(M2^{N})^{(2H+1)}}{C_{\Psi}(H)}\widehat{S}_{N,M},$

and thus

$\log\widehat{S}_{N,M}=\log\left(\frac{\widehat{V}_{N,M}}{\sqrt{|\mathcal{L}_{N,\gamma}|}}+1\right)-(2H+1)(\log
M+N\log(2))-\log C_{\Psi}(H).$

From this last equality, we deduce

$\displaystyle\widehat{H}_{N}-H=-\frac{1}{2}\frac{\left(\sum_{M=1}^{d}\log\left(\frac{\widehat{V}_{N,M}}{\sqrt{|\mathcal{L}_{N,\gamma}|}}+1\right)\log
M\right)-\left(\sum_{M=1}^{d}\log\left(\frac{\widehat{V}_{N,M}}{\sqrt{|\mathcal{L}_{N,\gamma}|}}+1\right)\right)\left(\sum_{M=1}^{d}\log
M\right)}{\left(\sum_{M=1}^{d}(\log M)^{2}\right)-\left(\sum_{M=1}^{d}\log
M\right)^{2}}$ (68)

The strong consistence (66) is then a straightforward consequence of
Proposition 6.

To prove (67), let us recall that the inequality

$|\log(1+x)-x|\leq x^{2}$ (69)

holds for all $x\in[-1/2,1/2]$. Therefore, writing in (68), for any
$M=1,\dots,d$,

$\log\left(\frac{\widehat{V}_{N,M}}{\sqrt{|\mathcal{L}_{N,\gamma}|}}+1\right)=\log\left(\frac{\widehat{V}_{N,M}}{\sqrt{|\mathcal{L}_{N,\gamma}|}}+1\right)-\frac{\widehat{V}_{N,M}}{\sqrt{|\mathcal{L}_{N,\gamma}|}}+\frac{\widehat{V}_{N,M}}{\sqrt{|\mathcal{L}_{N,\gamma}|}}$

and combining (69) with Proposition (6), we deduce

$\sqrt{|\mathcal{L}_{N,\gamma}|}(H-\widehat{H}_{N})\sim-\frac{1}{2}\frac{\left(\sum_{M=1}^{d}\widehat{V}_{N,M}\log
M\right)-\left(\sum_{M=1}^{d}\widehat{V}_{N,M}\right)\left(\sum_{M=1}^{d}\log
M\right)}{\left(\sum_{M=1}^{d}(\log M)^{2}\right)-\left(\sum_{M=1}^{d}\log
M\right)^{2}}$

as $N\to+\infty$. The limit theorem (67) easily follows, thanks to Corollary
1.

##  6 Appendix

The basic tools from the analysis on Wiener space are presented in this
section. We will focus on some elementary facts about multiple stochastic
integrals. We refer to [45] or [43] for a complete review on the topic.

Consider ${\mathcal{H}}$ a real separable infinite-dimensional Hilbert space
with its associated inner product ${\langle\cdot,\cdot\rangle}_{\mathcal{H}}$,
and $(B(\varphi),\varphi\in{\mathcal{H}})$ an isonormal Gaussian process on a
probability space $(\Omega,{\mathcal{F}},\mathbf{P})$, which is a centered
Gaussian family of random variables such that
$\mathbf{E}\left(B(\varphi)B(\psi)\right)={\langle\varphi,\psi\rangle}_{{\mathcal{H}}}$
for every $\varphi,\psi\in{\mathcal{H}}$. Denote by $I_{q}(q\geq 1)$ the $q$th
multiple stochastic integral with respect to $B$, which is an isometry between
the Hilbert space ${\mathcal{H}}^{\odot q}$ (symmetric tensor product)
equipped with the scaled norm $\sqrt{q!}\,\|\cdot\|_{{\mathcal{H}}^{\otimes
q}}$ and the Wiener chaos of order $q$, which is defined as the closed linear
span of the random variables $H_{q}(B(\varphi))$ where
$\varphi\in{\mathcal{H}},\;\|\varphi\|_{{\mathcal{H}}}=1$ and $H_{q}$ is the
Hermite polynomial of degree $q\geq 1$ defined by :

$H_{q}(x)=\frac{(-1)^{q}}{q!}\exp\left(\frac{x^{2}}{2}\right)\frac{{\mathrm{d}}^{q}}{{\mathrm{d}x}^{q}}\left(\exp\left(-\frac{x^{2}}{2}\right)\right),\;x\in\mathbb{R}.$
(70)

For $q=0$,

$\mathcal{H}_{0}=\mathbb{R}\mbox{ and }I_{0}(x)=x\mbox{ for every
}x\in\mathbb{R}.$ (71)

The isometry property of multiple integrals can be written as follows : for
$p,\;q\geq 0$, $f\in{{\mathcal{H}}^{\otimes p}}$ and
$g\in{{\mathcal{H}}^{\otimes q}}$

$\mathbf{E}\Big{[}I_{p}(f)I_{q}(g)\Big{]}=\left\\{\begin{array}[]{rcl}q!\langle\tilde{f},\tilde{g}\rangle_{{\mathcal{H}}^{\otimes
q}}&&\mbox{if}\;p=q,\\\ \vskip 5.69054pt\cr
0&&\mbox{otherwise,}\end{array}\right.$ (72)

where $\tilde{f}$ stands for the symmetrization of $f$. When
$\mathcal{H}=L^{2}(T)$, with $T$ being an interval of $\mathbb{R}$, we have
the following product formula: for $p,\;q\geq 0$, $f\in{{\mathcal{H}}^{\odot
p}}$ and $g\in{{\mathcal{H}}^{\odot q}}$,

$\displaystyle I_{p}(f)I_{q}(g)$ $\displaystyle=$
$\displaystyle\sum_{r=0}^{p\wedge
q}r!\binom{q}{r}\binom{p}{r}I_{p+q-2r}\left(f\otimes_{r}g\right),$ (73)

where, for $r=0,...,p\wedge q$, the contraction $f\otimes_{r}g$ is the
function in $L^{2}(T^{p+q-2r})$ given by

$(f\otimes_{r}g)(t_{1},...,t_{p+q-2r})=\int_{T^{r}}f(u_{1},...,u_{r},t_{1},...,t_{p-r})g(u_{1},...,u_{r},t_{p-r+1},...,t_{p+q-2r})du_{1}...du_{r}.$
(74)

An useful property of finite sums of multiple stochastic integrals is the
hypercontractivity. Namely, for every fixed real number $p\geq 2$, there
exists a universal deterministic finite constant $C_{p}$, such that, for any
random variable $F$ of the form $F=\sum_{k=0}^{n}I_{k}(f_{k})$ with
$f_{k}\in\mathcal{H}^{\otimes k}$, the following inequality holds:

$\mathbf{E}\big{[}|F|^{p}\big{]}\leq
C_{p}\left(\mathbf{E}\big{[}F^{2}\big{]}\right)^{\frac{p}{2}}.$ (75)

We denote by $D$ the Malliavin derivative operator that acts on cylindrical
random variables of the form $F=g(B(\varphi_{1}),\ldots,B(\varphi_{n}))$,
where $n\geq 1$, $g:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is a smooth function
with compact support and $\varphi_{i}\in{{\mathcal{H}}}$, in the following
way:

$DF=\sum_{i=1}^{n}\frac{\partial g}{\partial
x_{i}}(B(\varphi_{1}),\ldots,B(\varphi_{n}))\varphi_{i}.$

The operator $D$ is closable and it can be extended to $\mathbb{D}^{1,2}$
which denotes the closure of the set of cylindrical random variables with
respect to the norm $\|\cdot\|_{1,2}$ defined as

$\|F\|_{1,2}^{2}:=\mathbf{E}\big{[}|F|^{2}\big{]}+\mathbf{E}\big{[}\|DF\|_{\mathcal{H}}^{2}\big{]}.$

If $F=I_{p}(f)$, where $f\in\mathcal{H}^{\odot p}$ with $\mathcal{H}=L^{2}(T)$
and $p\geq 1$, then

$D_{\ast}F=pI_{p-1}\left(f(\cdot,\ast)\right),$

where $"\cdot"$ stands for $p-1$ variables.

The pseudo inverse $(-L)^{-1}$ of the Ornstein-Uhlenbeck operator $L$ is
defined, for $F=I_{p}(f)$ with $f\in\mathcal{H}^{\odot p}$ and $p\geq 1$, by

$(-L)^{-1}F=\frac{1}{p}I_{p}(f).$

At last notice that in our work, we have $\mathcal{H}=L^{2}(\mathbb{R})$ while
the role of the isonormal process $(B(\varphi),\varphi\in\mathcal{H})$ is
played by the usual Wiener integral on $L^{2}(\mathbb{R})$ associated with the
Brownian motion $(B(y),y\in\mathbb{R})$. In this case, we can provide an
explicit formula for the multiple integral $I_{q}$, ($q\geq 1$). Indeed, if
$f$ is a symmetric function of the form

$f=\sum_{j_{1},\ldots,j_{q}=1}^{n}a_{j_{1},\ldots,j_{q}}\mathbbm{1}_{[s_{j_{1}},t_{j_{1}})}\otimes\cdots\otimes\mathbbm{1}_{[s_{j_{q}},t_{j_{q}})},$
(76)

where, $\otimes$ stands for the tensor product, $a_{j_{1},\ldots,j_{d}}$ are
such that, for all permutation $\sigma$,
$a_{\sigma(j_{1}),\ldots,\sigma_{(}j_{q})}=a_{j_{1},\ldots,j_{q}}$ and
$a_{j_{1},\ldots,j_{q}}=0$ as soon as two indices $j_{1},\ldots,j_{q}$ are
equal and, for all $1\leq\ell\neq\ell^{\prime}\leq q$,
$[s_{j_{\ell}},t_{j_{\ell}})\cap[s_{j_{\ell^{\prime}}},t_{j_{\ell^{\prime}}})=\emptyset$,
then

$I_{q}(f):=\sum_{j_{1},\ldots,j_{q}=1}^{n}a_{j_{1},\ldots,j_{q}}(B(t_{j_{1}})-B(s_{j_{1}}))\times\ldots(B(t_{j_{q}})-B(s_{j_{q}})).$
(77)

It is straightforward that this last random variable belongs to
$L^{2}(\Omega)$. For a general symmetric $f\in L^{2}(\mathbb{R}^{q})$,
$I_{q}(f)$ is then defined using the density of functions of the form (76)
within the set of symmetric square integrable function and by checking that
the corresponding random variables (77) converge in $L^{2}(\Omega)$.

Our main tool to prove the asymptotic normality for random vectors is the
following theorem (Theorem 6.1.1 in [43]).

######  Theorem 4.

Let $m\geq 1$ be an integer number and consider a $m$-dimensional random
vector $F=(F_{1},...,F_{m})$. Assume $F_{i}\in\mathbb{D}^{1,4}$ for every
$i=1,...m$. Let $C\in M_{m}(\mathbb{R})$ be a symmetric and positive definite
matrix and let $Z\sim N(0,C)$. Then

$d_{W}(F,Z)\leq C\sqrt{\sum_{i,j=1}^{m}\mathbf{E}\Big{[}(C_{i,j}-\langle
DF_{i},D(-L)^{-1}F_{j}\rangle)^{2}\Big{]}}.$ (78)

Let us finally mention the fact that, if $F_{i}\in\mathbb{D}^{1,2}$ for every
$i=1,...m$, as a consequence of [43, Theorem 2.9.1],

$\mathbf{E}[\langle DF_{i},D(-L)^{-1}F_{j}\rangle]=\mathbf{E}[F_{i}F_{j}].$
(79)

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