#  Wavelet methods to study the pointwise regularity of the generalized
Rosenblatt process

L. Daw111University of Luxembourg,Department of Mathematics (DMATH), Maison du
Nombre, 6, avenue de la Fonte, L-4364 Esch-sur-Alzette, Grand Duchy Of
Luxembourg. , L. Loosveldt††footnotemark:  222Corresponding author at
<EMAIL_ADDRESS>
###### Abstract

We prove that we can identify three types of pointwise behaviour in the
regularity of the (generalized) Rosenblatt process. This extends to a non
Gaussian setting previous results known for the (fractional) Brownian motion.
On this purpose, fine bounds on the increments of the Rosenblatt process are
needed. Our analysis is essentially based on various wavelet methods.

Keywords: Wiener chaos, Rosenblatt process, Wavelet series, Random Series,
slow/ordinary/rapid points, modulus of continuity

2020 MSC: 60G18, 60G22, 42C40, 26A16, 60G17

##  1 Introduction

Precise study of path behaviour, and in particular regularity, of stochastic
processes is a classical research field, initiated in the 1920s by the works
of Wiener [47]. It lies in between probability and (harmonic) analysis and a
common strategy is to mix probabilistic arguments with analytical tools.
Pioneer works concerned Brownian motion. Among them, one can cite Paley and
Wiener’s expansion [48] using Fourier series, Lévy’s representation [30]
obtained with some techniques of interpolation theory or, more recently,
Kahane’s expansion [25] in the Schauder basis.

In the last decades, the emergence of wavelet analysis allowed to obtain
series expansions for many stochastic processes. Let
$\psi\,:\,\mathbb{R}\to\mathbb{R}$ be a smooth function satisfying the
admissibility condition [35]

$\int_{\mathbb{R}}\frac{|\widehat{\psi}(\xi)|}{|\xi|}\,d\xi<\infty,$ (1)

where $\widehat{\psi}$ is the Fourier transform of $\psi$. As such it
generates an orthonormal basis of $L^{2}(\mathbb{R})$. More precisely, any
function $f\in L^{2}(\mathbb{R})$ can be decomposed as

$f=\sum_{j\in\mathbb{Z}}\sum_{k\in\mathbb{Z}}c_{j,k}\psi(2^{j}\cdot-k),$ (2)

where

$c_{j,k}=2^{j}\int_{\mathbb{R}}f(x)\psi(2^{j}x-k)\,dx.$

It is noteworthy that the expansion (2) holds true in many function spaces. We
refer to the seminal books [15, 35, 34] for more details and proofs of these
facts. Multifractal analysis has demonstrated the efficiency of wavelet
methods to study uniform and pointwise Hölder regularity of functions both
from a theoretical [9, 10, 13, 20, 21, 23] and a practical point of view [2,
3, 11, 16, 19, 22, 37, 45, 46].

Now, let us consider a probability space $(\Omega,\mathcal{A},\mathbb{P})$ and
a real-valued stochastic process $X$ defined on it. If $X$ is smooth enough,
for all $\omega\in\Omega$, one can apply expansion (2) to the simple path
$t\mapsto X(t,\omega)$. This way, one defines a sequence of random wavelet
coefficients $(c_{j,k}(\omega))_{j,k\in\mathbb{Z}}$. For instance, if
$X=B_{H}$ is the fractional Brownian motion of Hurst index $H\in(0,1)$ and if
$\psi$ is a sufficiently regular wavelet, one has [36, 22]

$B_{H}=\sum_{j\in\mathbb{N}}\sum_{k\in\mathbb{Z}}2^{-Hj}\xi_{j,k}\psi_{H+1/2}(2^{j}\cdot-k)+R,$
(3)

where $(R(t,\cdot))_{t\in\mathbb{R}^{+}}$ is a process with alsmost surely
$C^{\infty}$ sample paths, $(\xi_{j,k})_{j\in\mathbb{N},k\in\mathbb{Z}}$ is a
sequence of independent $\mathcal{N}(0,1)$ random variables and $\psi_{H+1/2}$
is a fractional antiderivative of $\psi$, see Section 2 for a precise
definition.

In [18], Esser and Loosveldt undertook a systematic study of Gaussian wavelet
series. Thanks to (3), it applies in particular to the fractional Brownian
motion and leads to the following theorem.

######  Theorem 1.1.

For all $H\in(0,1)$, there exists an event $\Omega_{H}$ of probability $1$
satisfying the following assertions for all $\omega\in\Omega_{H}$ and every
non-empty interval $I$ of $\mathbb{R}$.

  * •

For almost every $t\in I$,

$0<\limsup_{s\to
t}\dfrac{|B_{H}(t,\omega)-B_{H}(s,\omega)|}{|t-s|^{H}\sqrt{\log\log|t-s|^{-1}}}<+\infty.$

Such points are called _ordinary points_.

  * •

There exists a dense set of points $t\in I$ such that

$0<\limsup_{s\to
t}\dfrac{|B_{H}(t,\omega)-B_{H}(s,\omega)|}{|t-s|^{H}\sqrt{\log|t-s|^{-1}}}<+\infty.$

Such points are called _rapid points_.

  * •

There exists a dense set of points $t\in I$ such that

$0<\limsup_{s\to
t}\dfrac{|B_{H}(t,\omega)-B_{H}(s,\omega)|}{|t-s|^{H}}<+\infty.$

Such points are called _slow points_.

Note that Theorem 1.1 extends some well-known results of Kahane concerning the
Brownian motion [25]. The “ordinary”, “rapid” and “slow” terminology is
inspired by them. Let us justify it. In a measure-theoretical point of view,
the modulus of continuity $x\mapsto|x|^{H}\sqrt{\log\log|x|^{-1}}$ is the most
frequent among the points of singles paths. Thus, it is natural to refer it as
ordinary. Now, $|x|^{H}\sqrt{\log\log|x|^{-1}}=o(|x|^{H}\sqrt{\log|x|^{-1}})$
if $x\to 0^{+}$ and thus points for which $x\mapsto|x|^{H}\sqrt{\log|x|^{-1}}$
is the pointwise modulus of continuity are refereed as rapid. On the other
side, points for which $x\mapsto|x|^{H}$ is the pointwise modulus of
continuity are referred as slow because
$|x|^{H}=o(|x|^{H}\sqrt{\log\log|x|^{-1}})$ if $x\to 0^{+}$.

Now, let us turn to the stochastic process we will deal with in this paper.
The Rosenblatt process appears naturally as a limit of normalized sums of
long-range dependent random variables [17]. Like the fractional Brownian
motion, it belongs to the class of Hermite processes, fractional Brownian
motion being of order $1$ while Rosenblatt process is of order $2$. Both are
selfsimilar stochastic processes with stationary increments and are
characterized by a parameter $H$, called the Hurst exponent. However, unlike
the fractional Brownian motion, the Rosenblatt process is not Gaussian. Does
it make a big difference regarding ordinary, rapid and slow points? In other
words, can Theorem 1.1 be extended to cover the non Gaussian Rosenblatt
process?

For the last fifteen years the Rosenblatt process has received a significantly
increasing interest in both theoretical and practical lines of research. Due
to its self-similarity, its applications are numerous across a multitude of
fields, including internet traffic [12] and turbulence [40, 28]. From a
statistical point of view, estimating the value of the Hurst index $H$ is
important for practical applications and various estimators exist, see [8,
43]. Also, from a mathematical point of view the Rosenblatt process has
received a lot of interest since its inception in [39]. Its distribution,
still not known in explicit form, was studied first in [1] and more recently
in [33] and [44].

In this paper, we even consider a generalization of the Rosenblatt process, as
defined and studied in [32]. It depends on two parameters
$H_{1},H_{2}\in(\frac{1}{2},1)$ which are such that $H_{1}+H_{2}>\frac{3}{2}$.
The generalized Rosenblatt process
$\\{R_{H_{1},H_{2}}(t,\cdot)\\}_{t\in\mathbb{R}_{+}}$ is defined as a double
Wiener-Itô integral of a kernel function $K_{H_{1},H_{2}}$ with respect to a
given Brownian motion. More precisely, consider a standard two-sided Brownian
motion $B$, and set

$R_{H_{1},H_{2}}(t,\cdot)=\int_{\mathbb{R}^{2}}^{{}^{\prime}}K_{H_{1},H_{2}}(t,x_{1},x_{2})\,dB(x_{1})dB(x_{2}),$
(4)

where $\int_{\mathbb{R}^{2}}^{{}^{\prime}}$ denotes integration over
$\mathbb{R}^{2}$ excluding the diagonal. The kernel function in (4) is
expressed, for all $(t,x_{1},x_{2})$ on $\mathbb{R}_{+}\times\mathbb{R}^{2}$,
by

$K_{H_{1},H_{2}}(t,x_{1},x_{2})=\frac{1}{\Gamma\left(H_{1}-\frac{1}{2}\right)\Gamma\left(H_{2}-\frac{1}{2}\right)}\int_{0}^{t}(s-x_{1})_{+}^{H_{1}-\frac{3}{2}}(s-x_{2})_{+}^{H_{2}-\frac{3}{2}}\,ds,$

where $\Gamma$ stands for the usual Gamma Euler function, and where for
$(x,\alpha)\in\mathbb{R}^{2}$

$\displaystyle x_{+}^{\alpha}=\left\\{\begin{array}[]{cl}x^{\alpha}&\text{if
}x\geq 0\\\ 0&\text{otherwise.}\end{array}\right.$

Note that the (standard) Rosenblatt process is the process
$\\{R_{H,H}(t,\cdot)\\}_{t\in\mathbb{R}_{+}}$ for $H\in(3/4,1)$. The
generalized Rosenblatt process
$\\{R_{H_{1},H_{2}}(t,\cdot)\\}_{t\in\mathbb{R}_{+}}$ is non-Gaussian, belongs
to the second Wiener chaos, and has the following basic properties:

  1. (1)

Continuity: the trajectories of the Rosenblatt process $R_{H_{1},H_{2}}$ are
continuous.

  2. (2)

Stationary increments:  $R_{H_{1},H_{2}}$ has stationary increments; that is,
the distribution of the process
$\left\\{R_{H_{1},H_{2}}(t+s,\cdot)-R_{H_{1},H_{2}}(s,\cdot)\right\\}_{t\in\mathbb{R}_{+}}$
does not depend on $s\geq 0$.

  3. (3)

Self-similarity:  $R_{H_{1},H_{2}}$ is self-similar with exponent
$H_{1}+H_{2}-1$; that is, the processes
$\\{R_{H_{1},H_{2}}(ct,\cdot)\\}_{t\in\mathbb{R}_{+}}$ and
$\left\\{c^{H_{1}+H_{2}-1}R_{H_{1},H_{2}}(t,\cdot)\right\\}_{t\in\mathbb{R}_{+}}$
have the same distribution for all $c>0$.

In [6], Ayache and Esmili presented a wavelet-type representation of the
generalized Rosenblatt process, very similar to the one given in [36] for
fractional Brownian motion, excepted for the use of integrals of two-
dimensional wavelet bases. This representation is the starting point of this
paper. It is one of our key tools to prove the following Theorem 1.2 which is
the main result of this paper.

######  Theorem 1.2.

For all $H_{1},H_{2}\in(\frac{1}{2},1)$ such that $H_{1}+H_{2}>\frac{3}{2}$,
there exists an event $\Omega_{H_{1},H_{2}}$ of probability $1$ satisfying the
following assertions for all $\omega\in\Omega_{H_{1},H_{2}}$ and every non-
empty interval $I$ of $\mathbb{R}$.

  * •

For almost every $t\in I$,

$0<\limsup_{s\to
t}\dfrac{|R_{H_{1},H_{2}}(t,\omega)-R_{H_{1},H_{2}}(s,\omega)|}{|t-s|^{H_{1}+H_{2}-1}\log\log|t-s|^{-1}}<+\infty.$
(5)

Such points are called _ordinary points_.

  * •

There exists a dense set of points $t\in I$ such that

$0<\limsup_{s\to
t}\dfrac{|R_{H_{1},H_{2}}(t,\omega)-R_{H_{1},H_{2}}(s,\omega)|}{|t-s|^{H_{1}+H_{2}-1}\log|t-s|^{-1}}<+\infty.$
(6)

Such points are called _rapid points_.

  * •

There exists a dense set of points $t\in I$ such that

$\limsup_{s\to
t}\dfrac{|R_{H_{1},H_{2}}(t,\omega)-R_{H_{1},H_{2}}(s,\omega)|}{|t-s|^{H_{1}+H_{2}-1}}<+\infty.$
(7)

Such points are called _slow points_.

Theorem 1.2 shows in particular that slow, ordinary and rapid points are not
specific to Gaussian processes.

######  Remark 1.3.

Let us compare Theorems 1.1 and 1.2. Each type of points is defined in the
same way when considering their pointwise moduli of continuity. Indeed, if $X$
denotes both the fractional Brownian motion or the generalized Rosenblatt
process, we see that the asymptotic behaviour of $|X(t,\omega)-X(s,\omega)|$
is always compared to a modulus of continuity of the form
$|t-s|^{\alpha}\theta(|t-s|)$, with $\alpha$ corresponding to the self-
similarity exponent of $X$ and $\theta$ a potential logarithmic correction.
For the ordinary points, $\theta$ is an iterated logarithm. More precisely,
for the fractional Brownian motion, we have
$\theta(|t-s|)=\sqrt{\log\log|t-s|^{-1}}$ while, for the generalized
Rosenblatt process, $\theta(|t-s|)=\log\log|t-s|^{-1}$. The same feature
appears for the rapid points: in the case of the fractional Brownian motion we
have $\theta(|t-s|)=\sqrt{\log|t-s|^{-1}}$ and for the generalized Rosenblatt
process we have $\theta(|t-s|)=\log|t-s|^{-1}$. Therefore, the only difference
between the corresponding logarithmic corrections is the square root that is
used for the fractional Brownian motion and not for the generalized Rosenblatt
process. It comes from the estimates that can be done on the tails of the
distribution of random variables in the first order Wiener chaos, for the
fractional Brownian motion, or the second order, for the generalized
Rosenblatt process, see Theorems 3.14 and 3.15 below. Concerning the slow
points, there is no logarithmic correction, $\theta=1$ in both case.
Unfortunately, contrary to the fractional Brownian motion, we did not manage
to show the positiveness of the limit in (7). In fact, for that, we would need
to find an almost-sure uniform lower modulus of continuity for the generalized
Rosenblatt process and to be able to judge its optimality, which seems to be a
difficult task. This is discussed in details in Remark 5.1 below, where we
give an almost-sure uniform lower modulus of continuity using the techniques
we use to prove the positiveness of the limits in (5) and (6).

Our strategy to prove Theorem 1.2 is as follows. First, in Section 3 we derive
upper-bounds for the oscillations
$|R_{H_{1},H_{2}}(t,\omega)-R_{H_{1},H_{2}}(s,\omega)|$ that are sharp enough
to imply the finiteness of the limits (5), (6) and (7). This is done by means
of the wavelet-type expansion given in [6], see Theorem 3.3 below. Then, in
Section 4, we give lower bounds for the so-called wavelet-leaders, see Section
2, of the generalized Rosenblatt process on a given compactly supported
wavelet basis. This will prove the positiveness of the limits (5), (6). In
particular, we use different bases depending on whether we deal with the
finiteness of the limits in Theorem 1.2 or with their strict positiveness.
This is very different from [18] where the authors always work with the same
wavelet. The reason is that the expression (19) in Theorem 3.3 below is not a
wavelet series: it involves additional quantities. Therefore, standard
arguments linking wavelet coefficients and regularity of the associated
functions can no longer be used.

There is a priori no obstacles to extend our results in Section 4 to any
Hermite process. On the contrary, extending the results of Section 3 does not
seem obvious at all. This is because a wavelet-type expansion of arbitrary
Hermite process is still missing but also because our strategy relies on
arguments which are specific to the two-dimensional feature of the Rosenblatt
process, see Lemma 2.1 for instance.

Notations used through this paper are rather standard except, maybe, that if
$s,t$ are two real numbers,$\int_{[s,t]}$ stands for $\int_{s}^{t}$ if $s\leq
t$ and $-\int_{s}^{t}=\int_{t}^{s}$ otherwise.

##  2 Some important facts involving wavelets

In this section, we gather all the facts concerning wavelets that we will
strongly use all along this article. First, an immediate but important
consequence of the admissibility condition (1) is that, if the wavelet
$\psi\in L^{1}(\mathbb{R})$, its first moment always vanishes, i.e.

$\int_{\mathbb{R}}\psi(x)dx=0.$

This condition is met for all the wavelets we consider in this paper.

First, while dealing with the upper bounds for the limits in Theorem 1.2, we
will use a wavelet-type expansion of the generalized Rosenblatt process. It is
given in [6] by the mean of the Meyer’s wavelet: $\psi$ belongs to the
Schwartz class $\mathcal{S}(\mathbb{R})$, and its Fourier transform is
compactly supported, see [29]. In particular, for all $H\in(1/2,1)$,
$\psi_{H}$, the fractional antiderivative $\widehat{\psi_{H}}$ of order
$H-1/2$ of $\psi$ is well-defined by means of its Fourier transform as

$\widehat{\psi_{H}}(0)=0\quad\text{and}\quad\widehat{\psi_{H}}(\xi)=(i\xi)^{-(H-\frac{1}{2})}\widehat{\psi}(\xi),\,\forall\,\xi\neq
0.$ (8)

It also belongs to the Schwartz class $\mathcal{S}(\mathbb{R})$, see [4, 6,
41] for instance. Moreover, some standard facts from distribution theory [41,
4] give us the explicit formula

$\psi_{H}(t)=\frac{1}{\Gamma\left(H-\frac{1}{2}\right)}\int_{\mathbb{R}}(t-x)_{+}^{H-\frac{3}{2}}\psi(x)\,dx.$

From (8), we see that
$\operatorname{supp}(\widehat{\psi_{H}})=\operatorname{supp}(\widehat{\psi})$
which is the key fact to establish the following lemma, gathering facts
already proved in [6].

######  Lemma 2.1.

Let $H_{1},H_{2}\in(\frac{1}{2},1)$. If
$(j_{1},j_{2},k_{1},k_{2})\in\mathbb{Z}^{4}$ are such that $|j_{1}-j_{2}|>1$,
then the integral

$I_{j_{1},j_{2}}^{k_{1},k_{2}}:=\int_{\mathbb{R}}\psi_{H_{1}}(2^{j_{1}}x-k_{1})\psi_{H_{2}}(2^{j_{2}}x-k_{2})$

vanishes. Moreover, for all $(j,k_{1},k_{2})\in\mathbb{Z}^{3}$, we have

$\displaystyle I_{j+1,j}^{k_{1},k_{2}}$
$\displaystyle=2^{-j}\int_{\mathbb{R}}e^{-i(k_{1}-2k_{2})\xi}\widehat{\psi_{H_{1}}}(\xi)\overline{\widehat{\psi_{H_{2}}}(2\xi)}\,d\xi,$
(9) $\displaystyle I_{j,j}^{k_{1},k_{2}}$
$\displaystyle=2^{-j}\int_{\mathbb{R}}e^{-i(k_{1}-k_{2})\xi}\widehat{\psi_{H_{1}}}(\xi)\overline{\widehat{\psi_{H_{2}}}(\xi)}\,d\xi,$
(10) $\displaystyle I_{j,j+1}^{k_{1},k_{2}}$
$\displaystyle=2^{-j}\int_{\mathbb{R}}e^{-i(2k_{1}-k_{2})\xi}\widehat{\psi_{H_{1}}}(2\xi)\overline{\widehat{\psi_{H_{2}}}(\xi)}\,d\xi.$
(11)

In addition, for all $L>0$, there exists a constant $C_{L}>0$ such that for
all $(j,k_{1},k_{2})\in\mathbb{Z}^{3}$,

$\displaystyle|I_{j+1,j}^{k_{1},k_{2}}|$ $\displaystyle\leq
C_{L}\frac{2^{-j}}{(3+|k_{1}-2k_{2}|)^{L}},$
$\displaystyle|I_{j,j}^{k_{1},k_{2}}|$ $\displaystyle\leq
C_{L}\frac{2^{-j}}{(3+|k_{1}-k_{2}|)^{L}},$
$\displaystyle|I_{j,j+1}^{k_{1},k_{2}}|$ $\displaystyle\leq
C_{L}\frac{2^{-j}}{(3+|2k_{1}-k_{2}|)^{L}}.$

When dealing with the the lower bounds for the limits in Theorem 1.2, we use
Daubechies compactly supported wavelets [14]. Note that, if
$\operatorname{supp}(\Psi)\subseteq[-N,N]$, for a positive integer $N$, then,
using the first vanishing moment, for all $(j,k)\in\mathbb{N}\times\mathbb{Z}$
and $t\in\mathbb{R}$, one can write

$c_{j,k}=\int_{-N}^{N}\left(f\left(\frac{x+k}{2^{j}}\right)-f\left(t\right)\right)\Psi(x)\,dx$
(12)

Since $\Psi$ is compactly supported, $\Psi(2^{j}\cdot-k)$ is localized around
the dyadic interval

$\lambda_{j,k}:=\left[\frac{k}{2^{j}},\frac{k+1}{2^{j}}\right)$

and it is therefore common to index wavelets these intervals. For simplicity,
we sometimes omit any references to the indices $j$ and $k$ for such intervals
by writing $\lambda=\lambda_{j,k}$, and $k=s(\lambda)$. Similarly,
$c_{\lambda}$ refers to the quantity $c_{j,k}$. The notation $\Lambda_{j}$
stands for the set of dyadic intervals $\lambda$ of $\mathbb{R}$ with side
length $2^{-j}$. The unique dyadic interval from $\Lambda_{j}$ containing the
point $t\in\mathbb{R}$ is denoted $\lambda_{j}(t)$. The set of dyadic
intervals is $\Lambda:=\cup_{j\in\mathbb{N}}\Lambda_{j}$. Two dyadic intervals
$\lambda$ and $\lambda^{\prime}$ are adjacent if there exist $j\in\mathbb{N}$
such that $\lambda,\lambda^{\prime}\in\Lambda_{j}$ and $\mbox{\rm
dist}(\lambda,\lambda^{\prime})=0$. The set of dyadic intervals adjacent to
$\lambda$ is denoted by $3\lambda$. In this setting, one defines the wavelet
leader [21] of $f$ at $t$ and of scale $j$ by

$d_{j}(t_{0})=\max_{\lambda\in
3\lambda_{j}(t_{0})}\sup_{\lambda^{\prime}\subseteq\lambda}|c_{\lambda}^{\prime}|.$
(13)

Then, if $\operatorname{supp}(\psi)\subseteq[-N,N]$, from (12), one can write

$d_{j}(t)\leq
2N\sup_{s\in(t_{0}-2^{-j}(N+2),t_{0}+2^{-j}(N+2))}|f(s)-f(t)|\|\psi\|_{L^{\infty}}.$
(14)

When we study stochastic processes, the wavelet leaders are random variables
$d_{j}(t,\omega)$. Inequality (14) with some easy computations implies that in
order to obtain the positiveness of the limit (5), it suffices to show that
for all $\omega\in\Omega_{H_{1},H_{2}}$ and all open intervals
$I\subseteq\mathbb{R}^{+}$, for almost every $t\in I$,

$0<\limsup_{j\to+\infty}\frac{d_{j}(t,\omega)}{2^{-j(H_{1}+H_{2}-1)}\log(j)}.$
(15)

Similarly, to prove the positiveness of the limit (6), we just have to show
that for all $\omega\in\Omega_{H_{1},H_{2}}$ and all open intervals
$I\subseteq\mathbb{R}^{+}$, there exists a dense set of points $t\in I$ such
that

$0<\limsup_{j\to+\infty}\frac{d_{j}(t,\omega)}{2^{-j(H_{1}+H_{2}-1)}j}.$ (16)

######  Remark 2.2.

Let us mention that wavelet leaders can not be used to prove the finiteness of
the limits in Theorem 1.2 because they do not precisely characterize the
pointwise regularity, see for instance [27, 31] for more details.

##  3 Upper bounds for oscillations

Starting from now and until the end of the paper, we fix
$H_{1},H_{2}\in(\frac{1}{2},1)$ such that $H_{1}+H_{2}>\frac{3}{2}$. In this
section, we show the finiteness of the limits (5), (6) and (7). Concerning the
rapid points, we will in fact show a stronger result, obtaining an almost sure
uniform modulus of continuity for the generalized Rosenblatt process.

We use a wavelet-type expansion of the generalized Rosenblatt process. It
relies on the following random variables.

######  Definition 3.1.

For all $(j_{1},j_{2},k_{1},k_{2})\in\mathbb{Z}^{4}$, let
$\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}$ be the second order Wiener chaos
random variable defined by

$2^{\frac{j_{1}+j_{2}}{2}}\int_{\mathbb{R}^{2}}^{\prime}\psi(2^{j_{1}}x_{1}-k_{1})\psi(2^{j_{2}}x_{2}-k_{2})\,dB(x_{1})dB(x_{2}).$

######  Remark 3.2.

For all $(j_{1},j_{2},k_{1},k_{2})\in\mathbb{Z}^{4}$, we have ([6, Proposition
2.3])

$\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}=\left(2^{\frac{j_{1}}{2}}\int_{\mathbb{R}}\psi(2^{j_{1}}x-k_{1})dB(x)\right)\left(2^{\frac{j_{2}}{2}}\int_{\mathbb{R}}\psi(2^{j_{2}}x-k_{2})dB(x)\right)$
(17)

for $j_{1}\neq j_{2}$ or $k_{1}\neq k_{2}$, and

$\varepsilon_{j_{1},j_{1}}^{k_{1},k_{1}}=\left(2^{\frac{j_{1}}{2}}\int_{\mathbb{R}}\psi(2^{j_{1}}x-k_{1})dB(x)\right)^{2}-1$
(18)

for $j_{1}=j_{2}$ and $k_{1}=k_{2}$. Using the fact that
$(2^{j/2}\psi(2^{j}\cdot-k))_{(j,k)\in\mathbb{Z}^{2}}$ forms an orthonormal
basis of $L^{2}(\mathbb{R})$, and elementary properties of Wiener integral, we
know that
$(2^{j/2}\int_{\mathbb{R}}\psi(2^{j}x-k)\,dB(x))_{(j,k)\in\mathbb{Z}^{2}}$ is
a family of iid $\mathcal{N}(0,1)$ random variables. So the random variables
$\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}$ and
$\varepsilon_{j_{1}^{\prime},j_{2}^{\prime}}^{k_{1}^{\prime},k_{2}^{\prime}}$
are independent as soon as

$\\{(j_{1},k_{1}),(j_{2},k_{2})\\}\cap\\{(j_{1}^{\prime},k_{1}^{\prime}),(j_{2}^{\prime},k_{2}^{\prime})\\}=\cmemptyset.$

The following theorem, proved in [6], gives the wavelet-type expansion we use
in this section.

######  Theorem 3.3.

Let $\psi$ be the Meyer wavelet and $I$ be any compact interval of
$\mathbb{R}_{+}$. Almost surely, the random series

$\sum_{(j_{1},j_{2},k_{1},k_{2})\in\mathbb{Z}^{4}}2^{j_{1}(1-H_{1})+j_{2}(1-H_{2})}\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}\int_{0}^{t}\psi_{H_{1}}(2^{j_{1}}x-k_{1})\psi_{H_{2}}(2^{j_{2}}x-k_{2})\,dx$
(19)

converges uniformly to $R_{H_{1},H_{2}}$ on the interval $I$.

######  Remark 3.4.

Any open interval in $\mathbb{R}$ can be written as a countable union of
dyadic intervals $(\lambda_{j,k})_{j\in\mathbb{N},k\in\mathbb{Z}}$. Then, to
prove Theorem 1.2, it is sufficient to show that, for all
$j\in\mathbb{N},k\in\mathbb{Z}$, there exist an event $\Omega_{j,k}$ of
probability $1$ such that, for all $\omega\in\Omega_{j,k}$, almost every
$t\in\lambda_{j,k}$ is ordinary and there exist $t_{r}\in\lambda_{j,k}$ which
is rapid and $t_{s}\in\lambda_{j,k}$ which is slow. For the sake of simpleness
in notation, we will only do the proofs in full details for
$\lambda_{0,0}=[0,1)$. In fact, after dilatation and translation, our proofs
hold true for any arbitrary dyadic interval.

###  3.1 Rapid points

Let us first focus on rapid points. we prove that
$x\mapsto|x|^{H_{1}+H_{2}-1}\log|x|^{-1}$ is almost surely a uniform modulus
of continuity for $R_{H_{1},H_{2}}$.

######  Proposition 3.5.

There exists an event $\Omega_{\text{rap}}$ of probability $1$ such that for
all $\omega\in\Omega_{\text{rap}}$ there exists $C_{R}(\omega)>0$ such that,
for all $t,s\in(0,1)$, we have

$|R_{H_{1},H_{2}}(t,\omega)-R_{H_{1},H_{2}}(s,\omega)|\leq
C_{R}(\omega)|t-s|^{H_{1}+H_{2}-1}\log|t-s|^{-1}.$ (20)

Let us set, for all $s,t\in(0,1)$ and
$(j_{1},j_{2},k_{1},k_{2})\in\mathbb{Z}^{4}$,

$I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]=\int_{[t,s]}\psi_{H_{1}}(2^{j_{1}}x-k_{1})\psi_{H_{2}}(2^{j_{2}}x-k_{2})\,dx.$

All along this section, if $s,t\in(0,1)$ are given, $n$ always refers to the
unique positive integer such that

$2^{-n-1}<|t-s|\leq 2^{-n}.$ (21)

Our proof consists in writing

$|R_{H_{1},H_{2}}(t,\cdot))-R_{H_{1},H_{2}}(s,\cdot)|=\left|\sum_{(j_{1},j_{2},k_{1},k_{2})\in\mathbb{Z}^{4}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]\right|$
(22)

and to split the sum in the right-hand side in subsums determined according to
the position of $j_{1}$ and $j_{2}$ with respect to $n$. To bound from above
some of these subsums the following lemma is key.

######  Lemma 3.6.

[6, Lemma 2.4.] There exist an event $\Omega^{*}$ of probability $1$ and a
positive random variable $C_{1}$ with finite moment of any order, such that,
for all $\omega\in\Omega^{*}$ and for each
$(j_{1},j_{2},k_{1},k_{2})\in\mathbb{Z}^{4}$,

$|\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}(\omega)|\leq
C_{1}(\omega)\sqrt{\log(3+|j_{1}|+|k_{1}|)}\sqrt{\log(3+|j_{2}|+|k_{2}|)}.$
(23)

In view of Lemma 3.6, we set

$L_{j_{1},j_{2}}^{k_{1},k_{2}}=\sqrt{\log(3+|j_{1}|+|k_{1}|)}\sqrt{\log(3+|j_{2}|+|k_{2}|)}.$

As a first step, Lemmata 3.7 to 3.12 are devoted to bound some deterministic
series whose general term is

$2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}L_{j_{1},j_{2}}^{k_{1},k_{2}}|I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]|.$

This first lemma will be useful to bound the subsums in the right-hand side of
(22) for $j_{1}<n$ and $j_{2}<n$.

######  Lemma 3.7.

There exists a deterministic constant $C>0$ such that, for all $t,s\in(0,1)$,
we have

$\sum_{j_{1}<n}\sum_{j_{2}<n}\sum_{(k_{1},k_{2})\in\mathbb{Z}^{2}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}L_{j_{1},j_{2}}^{k_{1},k_{2}}\left|I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]\right|\leq
C|t-s|^{H_{1}+H_{2}-1}\log|t-s|^{-1}.$

###### Proof.

Let us start by considering, for all $(j_{1},j_{2})\in\mathbb{Z}^{2}$, the
series

$\displaystyle R_{j_{1},j_{2}}\,:\,t$
$\displaystyle\mapsto\sum_{(k_{1},k_{2})\in\mathbb{Z}^{2}}L_{j_{1},j_{2}}^{k_{1},k_{2}}\int_{0}^{t}|\psi_{H_{1}}(2^{j_{1}}x-k_{1})\psi_{H_{2}}(2^{j_{2}}x-k_{2})|\,dx\text{
and}$ $\displaystyle{R^{\prime}}_{j_{1},j_{2}}\,:\,t$
$\displaystyle\mapsto\sum_{(k_{1},k_{2})\in\mathbb{Z}^{2}}L_{j_{1},j_{2}}^{k_{1},k_{2}}|\psi_{H_{1}}(2^{j_{1}}t-k_{1})\psi_{H_{2}}(2^{j_{2}}t-k_{2})|.$

The fast decay of the fractional antiderivatives of $\psi$ allows us to write,
for all $H\in\\{H_{1},H_{2}\\}$ and for all $x\in\mathbb{R}$

$|\psi_{H}(x)|\leq C(1+|x|)^{-4}.$ (24)

Moreover, according to [6, Lemma 4.2] for all $L>1$ there exists $C>0$ such
that, for all $j\in\mathbb{Z}$ and $x\in\mathbb{R}$

$\sum_{k\in\mathbb{Z}}\frac{\sqrt{\log(3+|j|+k)}}{(3+|2^{j}x-k|)^{L}}\leq
C\sqrt{\log(3+|j|+2^{j}|x|)}.$ (25)

Therefore, if $K$ is any compact set of $\mathbb{R}_{+}$, if $s=\sup_{K}$, for
all $t\in K$, we have

$\displaystyle|R_{j_{1},j_{2}}(t)|$ $\displaystyle\leq
C\int_{0}^{t}\sqrt{\log(3+|j_{1}|+2^{j_{1}}|x|)}\sqrt{\log(3+|j_{2}|+2^{j_{2}}|x|)}\,dx$
$\displaystyle\leq
Cs\sqrt{\log(3+|j_{1}|+2^{j_{1}}s)}\sqrt{\log(3+|j_{2}|+2^{j_{2}}s)}.$

The same arguments can be applied to ${R^{\prime}}_{j_{1},j_{2}}$, which means
that both series converge uniformly on any compact set of $\mathbb{R}_{+}$.
From this, we can use mean value theorem: for all
$(j_{1},j_{2})\in\mathbb{Z}^{2}$ there is $\xi(j_{1},j_{2})\in[s,t]$ such that

$\displaystyle\sum_{(k_{1},k_{2})\in\mathbb{Z}^{2}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}L_{j_{1},j_{2}}^{k_{1},k_{2}}\left|I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]\right|$
$\displaystyle\leq|t-s|\sum_{(k_{1},k_{2})\in\mathbb{Z}^{2}}L_{j_{1},j_{2}}^{k_{1},k_{2}}|\psi_{H_{1}}(2^{j_{1}}\xi-
k_{1})\psi_{H_{2}}(2^{j_{2}}\xi-k_{2})|.$ (26)

Now, we use the fast decay of the fractional antiderivatives of $\psi$ (24)
and inequality (25) to bound (26) from above: for all $j_{1},j_{2}<n$,

$\displaystyle\sum_{(k_{1},k_{2})\in\mathbb{Z}^{2}}$ $\displaystyle
L_{j_{1},j_{2}}^{k_{1},k_{2}}|\psi_{H_{1}}(2^{j_{1}}\xi-
k_{1})\psi_{H_{2}}(2^{j_{2}}\xi-k_{2})|$ $\displaystyle\leq
C\left(\sum_{k_{1}\in\mathbb{Z}}\frac{\sqrt{\log(3+|j_{1}|+|k_{1}|)}}{(3+|2^{j_{1}}\xi-
k_{1}|)^{4}}\right)\left(\sum_{k_{2}\in\mathbb{Z}}\frac{\sqrt{\log(3+|j_{2}|+|k_{2}|)}}{(3+|2^{j_{2}}\xi-
k_{2}|)^{4}}\right)$ $\displaystyle\leq
C\sqrt{\log(3+|j_{1}|+2^{j_{1}}|\xi|)}\sqrt{\log(3+|j_{2}|+2^{j_{2}}|\xi|)}$
$\displaystyle\leq
C\sqrt{\log(3+|j_{1}|+2^{j_{1}})}\sqrt{\log(3+|j_{2}|+2^{j_{2}})},$

as $\xi\in(0,1)$. Let us then remark that

$\displaystyle\sum_{j_{1}<n}$ $\displaystyle
2^{j_{1}(1-H_{1})}\sqrt{\log(3+|j_{1}|+2^{j_{1}})}$
$\displaystyle=\sum_{j_{1}\leq
0}2^{j_{1}(1-H_{1})}\sqrt{\log(3+|j_{1}|+2^{j_{1}})}+\sum_{j_{1}=0}^{n-1}2^{j_{1}(1-H_{1})}\sqrt{\log(3+|j_{1}|+2^{j_{1}})}$
$\displaystyle\leq
C+\sum_{j_{1}=0}^{n-1}2^{j_{1}(1-H_{1})}\sqrt{\log(3+|j_{1}|+2^{j_{1}})}$
$\displaystyle\leq C2^{n(1-H_{1})}\sqrt{n},$ (27)

as $1-H_{1}>0$. The same can be applied to the sum over $j_{2}$ and we finally
get

$\displaystyle\sum_{j_{1}<n}\sum_{j_{2}<n}\sum_{(k_{1},k_{2})\in\mathbb{Z}^{2}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}L_{j_{1},j_{2}}^{k_{1},k_{2}}\left|I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]\right|$
$\displaystyle\leq
C|t-s|\sum_{j_{1}<n}\sum_{j_{2}<n}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}\sqrt{\log(3+|j_{1}|+2^{j_{1}})}\sqrt{\log(3+|j_{2}|+2^{j_{2}})}$
$\displaystyle\leq C|t-s|2^{n(2-H_{1}-H_{2})}n$ $\displaystyle\leq
C|t-s|^{H_{1}+H_{2}-1}\log|t-s|^{-1}.$

∎

Lemmata 3.10 and 3.11 will help finding an upper bound for the subsums in the
right-hand side of (22) with $j_{1}<n\leq j_{2}$ or $j_{2}<n\leq j_{1}$ as
well as the ones where $n\leq j_{1}\leq j_{2}$ and $n\leq j_{2}\leq j_{1}$.
Let us define the following partition of $\mathbb{Z}$, which determines the
relative positions of $[k_{2}2^{-j_{2}},(k_{2}+1)2^{-j_{2}})$ and $[s,t]$.

######  Definition 3.8.

For all $j_{2}\in\mathbb{N}$, we set

$\displaystyle\mathbb{Z}_{j_{2}}^{<}(t,s)$
$\displaystyle=\\{k_{2}\in\mathbb{Z}\,:\,k_{2}2^{-j_{2}}<\min\\{t,s\\}\\},$
$\displaystyle\mathbb{Z}_{j_{2}}^{>}(t,s)$
$\displaystyle=\\{k_{2}\in\mathbb{Z}\,:\,k_{2}2^{-j_{2}}>\max\\{t,s\\}\\},$
$\displaystyle\text{and }\mathbb{Z}_{j_{2}}[t,s]$
$\displaystyle=\mathbb{Z}\setminus(\mathbb{Z}_{j_{2}}^{<}(t,s)\cup\mathbb{Z}_{j_{2}}^{>}(t,s)).$

######  Remark 3.9.

Note that we have $\\#\mathbb{Z}_{j_{2}}[t,s]\leq 2^{j_{2}-n}+1$.

Let us also observe that for all $a,b>0$,

$\log(3+a+b)\leq\log(3+a)\log(3+b).$ (28)

######  Lemma 3.10.

There exists a deterministic constant $C>0$ such that, for all $t,s\in(0,1)$
and $j_{1}\leq j_{2}$, the quantities

$\sum_{k_{1}\in\mathbb{Z}}\sum_{k_{2}\in\mathbb{Z}_{j_{2}}^{<}(t,s)}L_{j_{1},j_{2}}^{k_{1},k_{2}}\left|I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]\right|$
(29)
$\sum_{k_{1}\in\mathbb{Z}}\sum_{k_{2}\in\mathbb{Z}_{j_{2}}^{>}(t,s)}L_{j_{1},j_{2}}^{k_{1},k_{2}}\left|I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]\right|$
(30)

are bounded from above by

$C\sqrt{\log(3+|j_{1}|+2^{j_{1}})}\sqrt{\log(3+|j_{2}|+2^{j_{2}})}2^{-j_{2}}.$

###### Proof.

Let us bound (29), the proof for (30) being similar. From the fast decay of
the fractional antiderivatives of $\psi$ (24), inequalities (25) and (28) for
$j_{1}\leq j_{2}$, we have

$\displaystyle\eqref{eqn:r<>3}$ $\displaystyle\leq
C\int_{[s,t]}\left(\sum_{k_{1}\in\mathbb{Z}}\frac{\sqrt{\log(3+|j_{1}|+|k_{1}|)}}{(3+|2^{j_{1}}x-k_{1}|)^{4}}\right)\left(\sum_{k_{2}\in\mathbb{Z}^{<}_{j_{2}}(t,s)}\frac{\sqrt{\log(3+|j_{2}|+|k_{2}|)}}{(3+|2^{j_{2}}x-k_{2}|)^{4}}\right)\,dx$
$\displaystyle\leq
C\sqrt{\log(3+|j_{1}|+2^{j_{1}})}\sqrt{\log(3+|j_{2}|+2^{j_{2}})}$
$\displaystyle\quad\int_{[s,t]}\sum_{k_{2}\in\mathbb{Z}^{<}_{j_{2}}(t,s)}\frac{\sqrt{\log(3+|2^{j_{2}}x-k_{2}|)}}{(3+|2^{j_{2}}x-k_{2}|)^{4}}\,dx.$

For all $x\in[s,t]$ the mapping $y\mapsto
2+2^{j_{2}}x-2^{j_{2}}\min\\{s,t\\}+y)^{-3}$ is decreasing and thus

$\displaystyle\int_{[s,t]}$
$\displaystyle\sum_{k_{2}\in\mathbb{Z}^{<}_{j_{2}}(t,s)}\frac{\sqrt{\log(3+|2^{j_{2}}x-k_{2}|)}}{(3+|2^{j_{2}}x-k_{2}|)^{4}}\,dx$
$\displaystyle\leq\int_{[s,t]}\sum_{k_{2}\in\mathbb{Z}^{<}_{j_{2}}(t,s)}\frac{dx}{(3+2^{j_{2}}x-k_{2})^{3}}$
$\displaystyle\leq\int_{[s,t]}\sum_{m=0}^{+\infty}\frac{dx}{(3+2^{j_{2}}x-2^{j_{2}}\min\\{s,t\\}+m)^{3}}$
$\displaystyle\leq\int_{[s,t]}\int_{0}^{+\infty}\frac{dxdy}{(2+2^{j_{2}}x-2^{j_{2}}\min\\{s,t\\}+y)^{3}}$
$\displaystyle\leq C2^{-j_{2}}.$ (31)

This bound leads to

$\eqref{eqn:r<>3}\leq
C\sqrt{\log(3+|j_{1}|+2^{j_{1}})}\sqrt{\log(3+|j_{2}|+2^{j_{2}})}2^{-j_{2}}.$
(32)

∎

######  Lemma 3.11.

There exists a deterministic constant $C>0$ such that, for all $t,s\in(0,1)$
and $j_{1}\leq j_{2}$, the quantities

$\sum_{k_{1}\in\mathbb{Z}}\sum_{k_{2}\in\mathbb{Z}_{j_{2}}[t,s]}L_{j_{1},j_{2}}^{k_{1},k_{2}}\left|\int_{-\infty}^{\min\\{s,t\\}}\psi_{H_{1}}(2^{j_{1}}x-k_{1})\psi_{H_{2}}(2^{j_{2}}x-k_{2})\,dx\right|$
$\sum_{k_{1}\in\mathbb{Z}}\sum_{k_{2}\in\mathbb{Z}_{j_{2}}[t,s]}L_{j_{1},j_{2}}^{k_{1},k_{2}}\left|\int_{\max\\{s,t\\}}^{+\infty}\psi_{H_{1}}(2^{j_{1}}x-k_{1})\psi_{H_{2}}(2^{j_{2}}x-k_{2})\,dx\right|$

are bounded from above by

$C\sqrt{\log(3+|j_{1}|+2^{j_{2}})}\sqrt{\log(3+|j_{2}|+2^{j_{2}})}2^{-j_{2}}.$

###### Proof.

Let us assume that $s\leq t$, the argument for $t<s$ being similar. As
$j_{2}\geq j_{1}$, we have, by inequality (25),

$\displaystyle\int_{-\infty}^{s}$
$\displaystyle\left(\sum_{k_{1}\in\mathbb{Z}}\frac{\sqrt{\log(3+|j_{1}|+|k_{1}|)}}{(3+|2^{j_{1}}x-k_{1}|)^{4}}\right)\left(\sum_{k_{2}\in\mathbb{Z}_{j_{2}}[t,s]}\frac{\sqrt{\log(3+|j_{2}|+|k_{2}|)}}{(3+|2^{j_{2}}x-k_{2}|)^{4}}\right)\,dx$
$\displaystyle\leq
C_{L}\int_{-\infty}^{s}\sqrt{\log(3+|j_{1}|+2^{j_{1}}|x|)}\left(\sum_{k_{2}\in\mathbb{Z}_{j_{2}}[t,s]}\frac{\sqrt{\log(3+|j_{2}|+|k_{2}|)}}{(3+|2^{j_{2}}x-k_{2}|)^{4}}\right)\,dx$
$\displaystyle\leq
C_{L}\int_{-\infty}^{s}\sqrt{\log(3+|j_{1}|+2^{j_{2}}|x|)}\left(\sum_{k_{2}\in\mathbb{Z}_{j_{2}}[t,s]}\frac{\sqrt{\log(3+|j_{2}|+|k_{2}|)}}{(3+|2^{j_{2}}x-k_{2}|)^{4}}\right)\,dx.$

For all $k_{2}\in\mathbb{Z}_{j_{2}}[t,s]$, $|k_{2}|\leq 2^{j_{2}}$, we have,
using (28),

$\displaystyle\log$
$\displaystyle(3+|j_{1}|+2^{j_{2}}|x|)\leq\log(3+|j_{1}|+2^{j_{2}})\log(3+|2^{j_{2}}x-k_{2}|)\text{
and}$ $\displaystyle\log$
$\displaystyle(3+|j_{2}|+|k_{2}|)\leq\log(3+j_{2}+2^{j_{2}})\log(3+|2^{j_{2}}x-k_{2}|).$

Thus, it only remains us to deal with

$\displaystyle\int_{-\infty}^{s}\sum_{k_{2}\in\mathbb{Z}_{j_{2}}[t,s]}\frac{dx}{(3+|2^{j_{2}}x-k_{2}|)^{3}}.$

But, for all $x\leq s$ and $k_{2}\in\mathbb{Z}_{j_{2}}[t,s]$,
$|2^{j_{2}}x-k_{2}|=k_{2}-2^{j_{2}}x$ and then, using the same method as in
(31), we get

$\displaystyle\int_{-\infty}^{s}\sum_{k_{2}\in\mathbb{Z}_{j_{2}}[t,s]}\frac{dx}{(3+k_{2}-2^{j_{2}}x)^{3}}\leq
C2^{-j_{2}}$ (33)

which finally leads to

$\displaystyle\sum_{k_{1}\in\mathbb{Z}}\sum_{k_{2}\in\mathbb{Z}_{j_{2}}[t,s]}L_{j_{1},j_{2}}^{k_{1},k_{2}}\left|\int_{-\infty}^{s}\psi_{H_{1}}(2^{j_{1}}x-k_{1})\psi_{H_{2}}(2^{j_{2}}x-k_{2})\,dx\right|$
$\displaystyle\quad\leq
C\sqrt{\log(3+|j_{1}|+2^{j_{2}})}\sqrt{\log(3+j_{2}+2^{j_{2}})}2^{-j_{2}}.$
(34)

We get in the same way,

$\displaystyle\sum_{k_{1}\in\mathbb{Z}}\sum_{k_{2}\in\mathbb{Z}_{j_{2}}[t,s]}L_{j_{1},j_{2}}^{k_{1},k_{2}}\left|\int_{t}^{+\infty}\psi_{H_{1}}(2^{j_{1}}x-k_{1})\psi_{H_{2}}(2^{j_{2}}x-k_{2})\,dx\right|$
$\displaystyle\quad\leq
C\sqrt{\log(3+|j_{1}|+2^{j_{2}})}\sqrt{\log(3+j_{2}+2^{j_{2}})}2^{-j_{2}}.$

∎

Next Lemma will be used to bound the subsums of (22) with $j_{1}<n\leq j_{2}$
or $j_{2}<n\leq j_{1}$.

######  Lemma 3.12.

There exists a deterministic constant $C>0$ such that, for all $t,s\in(0,1)$,
the quantities

$R^{<\geq n}[t,s]:=\sum_{j_{1}<n}\sum_{j_{2}\geq
n}\sum_{(k_{1},k_{2})\in\mathbb{Z}^{2}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}L_{j_{1},j_{2}}^{k_{1},k_{2}}|I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]|$
$R^{\geq<n}[t,s]:=\sum_{j_{1}\geq
n}\sum_{j_{2}<n}\sum_{(k_{1},k_{2})\in\mathbb{Z}^{2}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}L_{j_{1},j_{2}}^{k_{1},k_{2}}|I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]|$

are bounded from above by

$C|t-s|^{H_{1}+H_{2}-1}\log|t-s|^{-1}.$

###### Proof.

As $R^{<\geq n}[t,s]$ and $R^{\geq<n}[t,s]$ can clearly be treated
symmetrically, we restrict our attention to $R^{<\geq n}[t,s]$. One sees that

$\displaystyle\sum_{j_{1}<n}\sum_{j_{2}\geq
n}\sum_{(k_{1},k_{2})\in\mathbb{Z}^{2}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}L_{j_{1},j_{2}}^{k_{1},k_{2}}I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]$
$\displaystyle=\sum_{j_{1}<n}\sum_{j_{2}\geq
n}\sum_{k_{1}\in\mathbb{Z}}\sum_{k_{2}\in\mathbb{Z}_{j_{2}}^{<}(t,s)}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}L_{j_{1},j_{2}}^{k_{1},k_{2}}I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]$
(35) $\displaystyle+\sum_{j_{1}<n}\sum_{j_{2}\geq
n}\sum_{k_{1}\in\mathbb{Z}}\sum_{k_{2}\in\mathbb{Z}_{j_{2}}^{>}(t,s)}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}L_{j_{1},j_{2}}^{k_{1},k_{2}}I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]$
(36) $\displaystyle+\sum_{j_{1}<n}\sum_{j_{2}\geq
n}\sum_{k_{1}\in\mathbb{Z}}\sum_{k_{2}\in\mathbb{Z}_{j_{2}}[t,s]}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}L_{j_{1},j_{2}}^{k_{1},k_{2}}I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s].$
(37)

For (35), we use Lemma 3.10 to get

$\displaystyle|\eqref{eqn:lemma3121}|$ $\displaystyle\leq
C\sum_{j_{1}<n}2^{j_{1}(1-H_{1})}\sqrt{\log(3+|j_{1}|+2^{j_{1}})}\sum_{j_{2}\geq
n}2^{-j_{2}H_{2}}\sqrt{\log(3+|j_{2}|+2^{j_{2}})}.$

The sum over over $j_{1}$ is bounded just as in (27) while, for the sum over
$j_{2}$, we have

$\displaystyle\sum_{j_{2}\geq
n}2^{-j_{2}H_{2}}\sqrt{\log(3+|j_{2}|+2^{j_{2}})}$
$\displaystyle\leq\sum_{j_{2}\geq n}2^{-j_{2}H_{2}}\sqrt{\log(3+2^{j_{2}+1})}$
$\displaystyle\leq C2^{-nH_{2}}\sqrt{n}.$ (38)

We bound (36) in exactly the same way.

For (37), let us again assume $s\leq t$, then we write

$\displaystyle I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]$
$\displaystyle=\int_{\mathbb{R}}\psi_{H_{1}}(2^{j_{1}}x-k_{1})\psi_{H_{2}}(2^{j_{2}}x-k_{2})\,dx$
$\displaystyle-\int_{-\infty}^{s}\psi_{H_{1}}(2^{j_{1}}x-k_{1})\psi_{H_{2}}(2^{j_{2}}x-k_{2})\,dx$
$\displaystyle-\int_{t}^{+\infty}\psi_{H_{1}}(2^{j_{1}}x-k_{1})\psi_{H_{2}}(2^{j_{2}}x-k_{2})\,dx.$
(39)

Since $j_{1}<n$ and $j_{2}\geq n$, recalling Lemma 2.1, the sum

$\sum_{k_{1}\in\mathbb{Z}}\sum_{k_{2}\in\mathbb{Z}_{j_{2}}[t,s]}L_{j_{1},j_{2}}^{k_{1},k_{2}}\int_{\mathbb{R}}\psi_{H_{1}}(2^{j_{1}}x-k_{1})\psi_{H_{2}}(2^{j_{2}}x-k_{2})\,dx,$

vanishes except maybe when $(j_{1},j_{2})=(n-1,n)$. In this case, note that
$\\#\mathbb{Z}_{n}[t,s]\leq 2$ and, for all $k_{2}\in\mathbb{Z}_{n}[t,s]$,
$|k_{2}|\leq 2^{n}$.Then, by Lemma 2.1 and inequality (25), we get

$\displaystyle\sum_{k_{1}\in\mathbb{Z}}\sum_{k_{2}\in\mathbb{Z}_{n}[t,s]}L_{n-1,n}^{k_{1},k_{2}}\left|I_{n-1,n}^{k_{1},k_{2}}\right|$
$\displaystyle\quad\leq
C2^{-n}\sum_{k_{1}\in\mathbb{Z}}\sum_{k_{2}\in\mathbb{Z}_{n}[t,s]}\frac{\sqrt{\log(3+n-1+|k_{1}|)}\sqrt{\log(3+n+|k_{2}|}}{(3+|2k_{1}-k_{2}|)^{4}}$
$\displaystyle\quad\leq
C2^{-n}\sum_{k_{2}\in\mathbb{Z}_{n}[t,s]}\sqrt{\log(3+n-1+|\frac{k_{2}}{2}|)}\sqrt{\log(3+n+|k_{2}|)}$
$\displaystyle\quad\leq C2^{-n}n$

Now, using Lemma 3.11, we also get

$\displaystyle\left|\sum_{j_{1}<n}\sum_{j_{2}\geq
n}\sum_{k_{1}\in\mathbb{Z}}\sum_{k_{2}\in\mathbb{Z}_{j_{2}}[t,s]}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}L_{j_{1},j_{2}}^{k_{1},k_{2}}\int_{-\infty}^{s}\psi_{H_{1}}(2^{j_{1}}x-k_{1})\psi_{H_{2}}(2^{j_{2}}x-k_{2})\,dx\right|$
$\displaystyle\leq\sum_{j_{1}<n}\sum_{j_{2}\geq
n}2^{j_{1}(1-H_{1})}2^{-j_{2}H_{2}}\sqrt{\log(3+|j_{1}|+2^{j_{2}})}\sqrt{\log(3+|j_{2}|+2^{j_{2}})}$
$\displaystyle\leq\sum_{j_{1}<0}\sum_{j_{2}\geq
n}2^{j_{1}(1-H_{1})}2^{-j_{2}H_{2}}\sqrt{\log(3+|j_{1}|)}\log(3+2^{j_{2}+1})$
$\displaystyle\quad+\sum_{j_{1}=0}^{n-1}\sum_{j_{2}\geq
n}2^{j_{1}(1-H_{1})}2^{-j_{2}H_{2}}\sqrt{\log(3+2^{j_{2}+1})}\sqrt{\log(3+2^{j_{2}+1})}$
$\displaystyle\leq C2^{n(1-H_{1}-H_{2})}n$ (40)

The series

$\left|\sum_{j_{1}<n}\sum_{j_{2}\geq
n}\sum_{k_{1}\in\mathbb{Z}}\sum_{k_{2}\in\mathbb{Z}_{j_{2}}[t,s]}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}L_{j_{1},j_{2}}^{k_{1},k_{2}}\int_{-\infty}^{s}\psi_{H_{1}}(2^{j_{1}}x-k_{1})\psi_{H_{2}}(2^{j_{2}}x-k_{2})\,dx\right|$

is bounded in exactly the same way and the conclusion follows. ∎

It remains us to bound the subsums of (22) with $j_{1}\geq n$ and $j_{2}\geq
n$. For this, let us define some random variables associated with dyadic
intervals.

######  Definition 3.13.

If $\lambda$ is a dyadic interval of scale $n$, we define, for all $j\geq n$,
the indexation sets

$\displaystyle S_{j}^{0}(\lambda)$
$\displaystyle:=\\{(k^{(1)},K^{(1)},k^{(2)},K^{(2)})\in\mathbb{Z}^{4}\,:\,\frac{k^{(1)}}{2^{j}},\frac{K^{(1)}}{2^{j}},\frac{k^{(2)}}{2^{j}},\frac{K^{(2)}}{2^{j}}\in\lambda\\},$
$\displaystyle S_{j}^{1}(\lambda)$
$\displaystyle:=\\{(k^{(1)},K^{(1)},k^{(2)},K^{(2)})\in\mathbb{Z}^{4}\,:\,\frac{k^{(1)}}{2^{j+1}},\frac{K^{(1)}}{2^{j+1}},\frac{k^{(2)}}{2^{j}},\frac{K^{(2)}}{2^{j}}\in\lambda\\},$
$\displaystyle S_{j}^{2}(\lambda)$
$\displaystyle:=\\{(k^{(1)},K^{(1)},k^{(2)},K^{(2)})\in\mathbb{Z}^{4}\,:\,\frac{k^{(1)}}{2^{j}},\frac{K^{(1)}}{2^{j}},\frac{k^{(2)}}{2^{j+1}},\frac{K^{(2)}}{2^{j+1}}\in\lambda\\}$

and consider the random variables, for $(k^{(1)},K^{(1)},k^{(2)},K^{(2)})\in
S_{j}^{0}(\lambda)$,

$\sideset{{}_{j}^{0}}{{}_{k^{(1)},K^{(1)}}^{k^{(2)},K^{(2)}}}{\sum}:=\sum_{k^{(1)}\leq
k_{1}\leq K^{(1)}}\sum_{k^{(2)}\leq k_{2}\leq
K^{(2)}}\varepsilon_{j,j}^{k_{1},k_{2}}I_{j,j}^{k_{1},k_{2}}$ (41)

, for $(k^{(1)},K^{(1)},k^{(2)},K^{(2)})\in S_{j}^{1}(\lambda)$,

$\sideset{{}_{j}^{1}}{{}_{k^{(1)},K^{(1)}}^{k^{(2)},K^{(2)}}}{\sum}:=\sum_{k^{(1)}\leq
k_{1}\leq K^{(1)}}\sum_{k^{(2)}\leq k_{2}\leq
K^{(2)}}\varepsilon_{j+1,j}^{k_{1},k_{2}}I_{j+1,j}^{k_{1},k_{2}}$ (42)

and, for $(k^{(1)},K^{(1)},k^{(2)},K^{(2)})\in S_{j}^{2}(\lambda)$,

$\sideset{{}_{j}^{2}}{{}_{k^{(1)},K^{(1)}}^{k^{(2)},K^{(2)}}}{\sum}:=\sum_{k^{(1)}\leq
k_{1}\leq K^{(1)}}\sum_{k^{(2)}\leq k_{2}\leq
K^{(2)}}\varepsilon_{j,j+1}^{k_{1},k_{2}}I_{j,j+1}^{k_{1},k_{2}}.$ (43)

The idea behind the definition of these random variables is, as $|t-s|\leq
2^{-n}$, $s\in 3\lambda_{n}(t)$ and thus any sum of the form

$\sum_{k_{1}\in\mathbb{Z}_{j}[t,s]}\sum_{k_{2}\in\mathbb{Z}_{\ell}[t,s]}\varepsilon_{j,\ell}^{k_{1},k_{2}}I_{j,\ell}^{k_{1},k_{2}}$
(44)

for $\ell\in\\{j,j+1\\}$ can be written as the sum of random variables (41),
(42) or (43) for some $(k^{(1)},K^{(1)},k^{(2)},K^{(2)})$ belonging to at most
two $S_{j}^{\ell}(\lambda)$ $(\ell\in\\{0,1,2\\}$) with
$\lambda\in\lambda_{n}(t)$. Indeed,

  * •

if $t$ and $s$ both belong to $\lambda_{n}(t)$ then we only need to rewrite
(44) in the form (41), (42) or (43) for $(k^{(1)},K^{(1)},k^{(2)},K^{(2)})\in
S_{j}^{\ell}(\lambda_{n}(t))$;

  * •

if $s\in\lambda$ with $\lambda\in 3\lambda_{n}(t)\setminus\lambda_{n}(t)$ then
we need to consider a first sum indexed by a quadruple of
$S_{j}^{\ell}(\lambda_{n}(t))$ and a second indexed by a quadruple of
$S_{j}^{\ell}(\lambda)$.

The reason why we decide to put $\lambda$ instead of $3\lambda$ in the
definition of the sets $S_{j}^{\ell}(\lambda)$ is that if, for all
$n\in\mathbb{N}$ and for all $\lambda\in\Lambda_{n}$ and $j\geq n$, we define
the random variable

$\Xi_{j}(\lambda)=\max_{\ell\in\\{0,1,2\\}}\sup_{(k^{(1)},K^{(1)},k^{(2)},K^{(2)})\in
S_{j}^{\ell}(\lambda))}\frac{\left|\sideset{{}_{j}^{\ell}}{{}_{k^{(1)},K^{(1)}}^{k^{(2)},K^{(2)}}}{\sum}\right|}{\left\|\sideset{{}_{j}^{\ell}}{{}_{k^{(1)},K^{(1)}}^{k^{(2)},K^{(2)}}}{\sum}\right\|_{L^{2}(\Omega)}},$
(45)

we want $\Xi_{j}(\lambda)$ to be independent of $\Xi_{j}(\lambda^{\prime})$ as
soon as $\lambda\cap\lambda^{\prime}=\cmemptyset$. Moreover, from the
definitions of the random variables (42), (41) and (43), the remarks below
Theorem 3.3 and the explicit expressions (9), (10) and (11), the law of
$\Xi_{j}(\lambda)$ does not depend on $\lambda\in\Lambda_{n}$but only on
$j-n$.

The key results to estimate the random variables $\Xi_{j}$ are [24, Theorem
6.7 and Theorem 6.12] that we recall here.

######  Theorem 3.14.

There exists a strictly positive universal deterministic constant
$\overset{\star}{C}$ such that, for every random variable $X$ belonging to the
second order Wiener chaos and for each real number $y\geq 2$, one has

$\mathbb{P}(|X|\geq y\|X\|_{L^{2}(\Omega)})\leq\exp(-\overset{\star}{C}y).$

######  Theorem 3.15.

If $X$ is a random variable belonging to the second order Wiener chaos, there
exist $a,b,y_{0}>0$ such that, for all $y\geq y_{0}$,

$\exp(-ay)\leq\mathbb{P}(|X|\geq y)\leq\exp(-by).$

######  Remark 3.16.

As stated in [24], the constants $a,b$ in Theorem 3.15 are not universal and
depend on the law of $X$. Note that $b$ can be recovered from Theorem 3.14 and
thus is universal on the unit sphere in $L^{2}(\Omega)$.

######  Lemma 3.17.

There exists a deterministic constant $C>0$ such that, for all
$n\in\mathbb{N}$, $\lambda\in\Lambda_{n}$, $j\geq n$, $\ell\in\\{0,1,2\\}$ and
$(k^{(1)},K^{(1)},k^{(2)},K^{(2)})\in S_{j}^{\ell}(\lambda)$, we have

$\left\|\sideset{{}_{j}^{\ell}}{{}_{k^{(1)},K^{(1)}}^{k^{(2)},K^{(2)}}}{\sum}\right\|_{L^{2}(\Omega)}\leq
C2^{\frac{-j-n}{2}}.$

###### Proof.

Following an idea from [6, Lemma 2.21], we write

$\left\|\sideset{{}_{j}^{\ell}}{{}_{k^{(1)},K^{(1)}}^{k^{(2)},K^{(2)}}}{\sum}\right\|_{L^{2}(\Omega)}\leq\sum_{\mathcal{R}\in\\{<,>,=\\}}\left\|\sideset{{}_{j}^{\ell}}{{}_{k^{(1)},K^{(1)}}^{k^{(2)},K^{(2)}}}{\sum}_{\mathcal{R}}\right\|_{L^{2}(\Omega)}$

where

$\sideset{{}_{j}^{\ell}}{{}_{k^{(1)},K^{(1)}}^{k^{(2)},K^{(2)}}}{\sum}_{\mathcal{R}}$

is the subsum of (42), (41) or (43) in which $k_{1}\mathcal{R}k_{2}$. By doing
so, we make sure that two random variables
$\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}$ and
$\varepsilon_{j_{1}^{\prime},j_{2}^{\prime}}^{k_{1}^{\prime},k_{2}^{\prime}}$
appearing in this subsum are uncorrelated except when
$(k_{1},k_{2})=(k_{1}^{\prime},k_{2}^{\prime})$. Then from Lemma 2.1, we have
for $\ell=0$ (the argument being the same for $\ell=1$ or $\ell=2$), for all
$\mathcal{R}\in\\{<,>,=\\}$,

$\displaystyle\left\|\sideset{{}_{j}^{\ell}}{{}_{k^{(1)},K^{(1)}}^{k^{(2)},K^{(2)}}}{\sum}_{\mathcal{R}}\right\|_{L^{2}(\Omega)}^{2}$
$\displaystyle=\sum_{k^{(1)}\leq k_{1}\leq K^{(1)}}\sum_{k^{(2)}\leq k_{2}\leq
K^{(2)},k_{1}\mathcal{R}k_{2}}\mathbb{E}[(\varepsilon_{j,j}^{k_{1},k_{2}})^{2}](I_{j,j}^{k_{1},k_{2}})^{2}$
$\displaystyle\leq\sum_{k^{(1)}\leq k_{1}\leq
K^{(1)}}\sum_{k_{2}\in\mathbb{Z}}\frac{2^{-2j}}{(3+|k_{1}-k_{2}|)^{8}}.$

Since $\\#\\{k_{1}\in\mathbb{Z}\,:\,k^{(1)}\leq k_{1}\leq K^{(1)}\\}\leq
2^{j-n}$, we conclude that

$\left\|\sideset{{}_{j}^{\ell}}{{}_{k^{(1)},K^{(1)}}^{k^{(2)},K^{(2)}}}{\sum}\right\|_{L^{2}(\Omega)}\leq
C2^{\frac{-j-n}{2}}.$ (46)

∎

######  Lemma 3.18.

There exist an event $\widetilde{\Omega}$ of probability $1$ and a positive
random variable $C_{2}$ with finite moment of any order such that, on
$\widetilde{\Omega}$

$\forall n\in\mathbb{N},\leavevmode\nobreak\
\forall\,\lambda\subseteq[0,1],\leavevmode\nobreak\
\lambda\in\Lambda_{n},\leavevmode\nobreak\ \forall j\geq
n,\leavevmode\nobreak\ \Xi_{j}(\lambda)\leq C_{2}\,(j-n+1)n.$ (47)

###### Proof.

Let us take $\theta>0$ and consider, for all $n\in\mathbb{N}$ the event

$A_{n}:=\left\\{\forall\,\lambda\subseteq[0,1],\lambda\in\Lambda_{n},\,\forall
j\geq n,\,\Xi_{j}(\lambda)\leq\theta(j-n+1)n\right\\}.$

If $A_{n}^{c}$ stands for the complementary set of $A_{n}$ in $\Omega$, we
have, of course,

$\displaystyle\mathbb{P}(A_{n}^{c})=\mathbb{P}(\exists\lambda\subseteq[0,1],\lambda\in\Lambda_{n}\,:\,\exists
j\geq n\text{ s. t. }\Xi_{j}(\lambda)\geq\theta(j-n+1)n).$

But, for all $\lambda\subseteq[0,1],\lambda\in\Lambda_{n}$, $j\geq n$,
$\ell\in\\{0,1,2\\}$ and $(k^{(1)},K^{(1)},k^{(2)},K^{(2)})\in
S_{j}^{\ell}(\lambda)$ we have, by Theorem 3.14, if $\theta\geq 2$,

$\displaystyle\mathbb{P}\left(\frac{\left|\sideset{{}_{j}^{\ell}}{{}_{k^{(1)},K^{(1)}}^{k^{(2)},K^{(2)}}}{\sum}\right|}{\left\|\sideset{{}_{j}^{\ell}}{{}_{k^{(1)},K^{(1)}}^{k^{(2)},K^{(2)}}}{\sum}\right\|_{L^{2}(\Omega)}}\geq\theta(j-n+1)n\right)\leq\exp(-\overset{\star}{C}\theta(j-n+1)n).$

As, for all $j\geq n$, $\\#S_{j}^{\ell}(\lambda)\leq 2^{4(j-n)}$ and
$\\#\\{\lambda\subseteq[0,1]\,:\,\lambda\in\Lambda_{n}\\}=2^{n}$, we get

$\displaystyle\mathbb{P}(A_{n}^{c})$ $\displaystyle\leq C2^{n}\sum_{j\geq
n}2^{4(j-n)}\exp(-\overset{\star}{C}\theta(j-n+1)n)$ $\displaystyle\leq
C2^{n}\exp(-\overset{\star}{C}\theta n)\sum_{j\geq
n}2^{4(j-n)}\exp(-\overset{\star}{C}\theta(j-n))$

for a deterministic constant $C>0$. Therefore, if we take
$\theta>4\log(2)/\overset{\star}{C}$, the conclusion follows from Borel-
Cantelli Lemma. ∎

######  Lemma 3.19.

Let $\Omega^{*}$ and $\widetilde{\Omega}$ be the events of probability $1$
given by Lemmata 3.6 and 3.18 respectively. There exists a positive random
variable $C_{3}$ with finite moment of any order such that, on
$\Omega^{*}\cap\widetilde{\Omega}$, for all $t,s\in(0,1)$ the random variable

$\left|\sum_{j_{1}\geq
n}\sum_{j_{2}>n}\sum_{(k_{1},k_{2})\in\mathbb{Z}^{2}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]\right|\\\
$ (48)

is bounded from above by

$C_{3}|t-s|^{H_{1}+H_{2}-1}\log|t-s|^{-1}.$

###### Proof.

We start by splitting the sums in (48) in two parts:

$\displaystyle\sum_{j_{1}\geq n}\sum_{j_{2}\geq
j_{1}}\sum_{(k_{1},k_{2})\in\mathbb{Z}^{2}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]\text{
and}$ $\displaystyle\sum_{j_{2}\geq
n}\sum_{j_{1}>j_{2}}\sum_{(k_{1},k_{2})\in\mathbb{Z}^{2}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s].$
(49)

We only focus on the first sums, as the argument is symmetric in $j_{1}$ and
$j_{2}$. As in Lemma 3.12 we write

$\displaystyle\sum_{j_{1}\geq n}\sum_{j_{2}\geq
j_{1}}\sum_{(k_{1},k_{2})\in\mathbb{Z}^{2}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]$
$\displaystyle=\sum_{j_{1}\geq n}\sum_{j_{2}\geq
j_{1}}\sum_{k_{1}\in\mathbb{Z}}\sum_{k_{2}\in\mathbb{Z}_{j_{2}}^{<}(t,s)}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]$
(50) $\displaystyle+\sum_{j_{1}\geq n}\sum_{j_{2}\geq
j_{1}}\sum_{k_{1}\in\mathbb{Z}}\sum_{k_{2}\in\mathbb{Z}_{j_{2}}^{>}(t,s)}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]$
(51) $\displaystyle+\sum_{j_{1}\geq n}\sum_{j_{2}\geq
j_{1}}\sum_{k_{1}\in\mathbb{Z}}\sum_{k_{2}\in\mathbb{Z}_{j_{2}}[t,s]}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s].$
(52)

To bound (50), we use inequality (23) and Lemma 3.10 to get

$\displaystyle|\eqref{eqn:lemma3191}|$ $\displaystyle\leq
CC_{1}\sum_{j_{1}\geq n}\sum_{j_{2}\geq
j_{1}}2^{j_{1}(1-H_{1})}2^{-j_{2}H_{2}}\sqrt{\log(3+|j_{1}|+2^{j_{1}})}\sqrt{\log(3+|j_{2}|+2^{j_{2}})}$
$\displaystyle\leq CC_{1}\sum_{j_{1}\geq
n}2^{j_{1}(1-H_{1}-H_{2})}\sqrt{j_{1}}$ $\displaystyle\leq
CC_{1}2^{n(1-H_{1}-H_{2})}n,$

by applying twice inequality (38). The sum (51) is bounded in exactly the same
way.

To bound (52), we use once again the equality (39). First we have, by
inequality (23) and Lemma 3.11,

$\displaystyle\left|\sum_{j_{1}\geq n}\sum_{j_{2}\geq
j_{1}}\sum_{k_{1}\in\mathbb{Z}}\sum_{k_{2}\in\mathbb{Z}_{j_{2}}[t,s]}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}\int_{-\infty}^{s}\psi_{H_{1}}(2^{j_{1}}x-k_{1})\psi_{H_{2}}(2^{j_{2}}x-k_{2})\,dx\right|$
$\displaystyle\leq CC_{1}\sum_{j_{1}\geq n}\sum_{j_{2}\geq
j_{1}}2^{j_{1}(1-H_{1})}2^{-j_{2}H_{2}}\sqrt{\log(3+|j_{1}|+2^{j_{2}})}\sqrt{\log(3+|j_{2}|+2^{j_{2}})}$
$\displaystyle\leq CC_{1}2^{n(1-H_{1}-H_{2})}n.$ (53)

We bound

$\displaystyle\left|\sum_{j_{1}\geq n}\sum_{j_{2}\geq
j_{1}}\sum_{k_{1}\in\mathbb{Z}}\sum_{k_{2}\in\mathbb{Z}_{j_{2}}[t,s]}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}\int_{t}^{+\infty}\psi_{H_{1}}(2^{j_{1}}x-k_{1})\psi_{H_{2}}(2^{j_{2}}x-k_{2})\,dx\right|$

in the same way.

It only remains us to find an estimate for

$\sum_{j_{1}\geq n}\sum_{j_{2}\geq
j_{1}}\sum_{k_{1}\in\mathbb{Z}}\sum_{k_{2}\in\mathbb{Z}_{j_{2}}[t,s]}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}I_{j_{1},j_{2}}^{k_{1},k_{2}}$

and thus, recalling Lemma 2.1, we reduce the problem to first bound, for
$j\geq n$ and $\ell\in\\{j,j+1\\}$, the sums

$\sum_{k_{1}\in\mathbb{Z}_{j}^{<}(t,s)}\sum_{k_{2}\in\mathbb{Z}_{\ell}[t,s]}\varepsilon_{j,\ell}^{k_{1},k_{2}}I_{j,\ell}^{k_{1},k_{2}},$
(54)
$\sum_{k_{1}\in\mathbb{Z}_{j}^{>}(t,s)}\sum_{k_{2}\in\mathbb{Z}_{\ell}[t,s]}\varepsilon_{j,\ell}^{k_{1},k_{2}}I_{j,\ell}^{k_{1},k_{2}},$
(55)
$\sum_{k_{1}\in\mathbb{Z}_{j}[t,s]}\sum_{k_{2}\in\mathbb{Z}_{\ell}[t,s]}\varepsilon_{j,\ell}^{k_{1},k_{2}}I_{j,\ell}^{k_{1},k_{2}}$
(56)

on $\Omega^{*}\cap\widetilde{\Omega}$. Let us consider (54) with $\ell=j$, the
argument for $\ell=j+1$ and (55) being similar. Using again Lemmata 2.1 and
3.6, we have on $\Omega^{*}\cap\widetilde{\Omega}$, since for all
$k_{2}\in\mathbb{Z}_{j}[t,s]$, $|k_{2}|\leq 2^{j}$, for $j\geq n$,

$\displaystyle|\eqref{eqn:r>>6}|$ $\displaystyle\leq
CC_{1}\sum_{k_{1}\in\mathbb{Z}_{j}^{<}(t,s)}\sum_{k_{2}\in\mathbb{Z}_{j}[t,s]}\frac{2^{-j}}{(3+|k_{1}-k_{2}|)^{4}}\sqrt{\log(3+j+|k_{1}|)}\sqrt{\log(3+j+|k_{2}|)}$
$\displaystyle\leq
CC_{1}\sum_{k_{1}\in\mathbb{Z}_{j}^{<}(t,s)}\sum_{k_{2}\in\mathbb{Z}_{j}[t,s]}\frac{2^{-j}\sqrt{j}}{(3+k_{2}-k_{1})^{4}}\sqrt{\log(3+j+|k_{1}|)}$
$\displaystyle\leq
CC_{1}\sum_{k_{1}\in\mathbb{Z}_{j}^{<}(t,s)}\sum_{m=0}^{+\infty}\frac{2^{-j}\sqrt{j}}{(3+2^{j}\min\\{s,t\\}+m-k_{1})^{4}}\sqrt{\log(3+j+|k_{1}|)}$
$\displaystyle\leq
CC_{1}\sum_{k_{1}\in\mathbb{Z}_{j}^{<}(t,s)}2^{-j}\sqrt{j}\int_{0}^{+\infty}\frac{dy}{(2+2^{j}\min\\{s,t\\}+y-k_{1})^{4}}\sqrt{\log(3+j+|k_{1}|)}$
$\displaystyle\leq
CC_{1}2^{-j}\sqrt{j}\sum_{k_{1}\in\mathbb{Z}_{j}^{<}(t,s)}\frac{\sqrt{\log(3+j+|k_{1}|)}}{(2+2^{j}\min\\{s,t\\}-k_{1})^{3}}$
$\displaystyle\leq CC_{1}2^{-j}\sqrt{j}\sqrt{\log(3+j+2^{j}\min\\{s,t\\})}$
$\displaystyle\leq CC_{1}2^{-j}j.$ (57)

It follows that

$\left|\sum_{j\geq
n}2^{j(2-H_{1}-H_{2})}\sum_{\ell=j}^{j+1}\sum_{k_{1}\in\mathbb{Z}_{j}^{<}(t,s)}\sum_{k_{2}\in\mathbb{Z}_{\ell}[t,s]}\varepsilon_{j,\ell}^{k_{1},k_{2}}I_{j,\ell}^{k_{1},k_{2}}\right|\leq
CC_{1}2^{n(1-H_{1}-H_{2})}n$

and, similarly,

$\left|\sum_{j\geq
n}2^{j(2-H_{1}-H_{2})}\sum_{\ell=j}^{j+1}\sum_{k_{1}\in\mathbb{Z}_{j}^{>}(t,s)}\sum_{k_{2}\in\mathbb{Z}_{\ell}[t,s]}\varepsilon_{j,\ell}^{k_{1},k_{2}}I_{j,\ell}^{k_{1},k_{2}}\right|\leq
CC_{1}2^{n(1-H_{1}-H_{2})}n.$

The bound for (56) is obtained using (47) and (46) which lead to

$\displaystyle\left|\sum_{j\geq
n}2^{j(2-H_{1}-H_{2})}\sum_{\ell=j}^{j+1}\sum_{k_{1}\in\mathbb{Z}_{j}[t,s]}\sum_{k_{2}\in\mathbb{Z}_{\ell}[t,s]}\varepsilon_{j,\ell}^{k_{1},k_{2}}I_{j,\ell}^{k_{1},k_{2}}\right|$
$\displaystyle\leq CC_{2}\sum_{j\geq
n}2^{j(\frac{3}{2}-H_{1}-H_{2})}2^{-\frac{n}{2}}(j-n+1)n$ $\displaystyle\leq
CC_{2}2^{n(\frac{3}{2}-H_{1}-H_{2})}2^{-\frac{n}{2}}n$
$\displaystyle=CC_{2}2^{n(1-H_{1}-H_{2})}n,$

as $\frac{3}{2}<H_{1}+H_{2}$.

Putting all of these together we get that (48) is bounded from above by

$C\max\\{C_{1},C_{2}\\}|t-s|^{H_{1}+H_{2}-1}\log|t-s|^{-1}$

on $\Omega^{*}\cap\widetilde{\Omega}$. ∎

We now prove the main result of this subsection.

###### Proof of Proposition 3.5.

Let us consider $\omega$ in the event $\Omega^{*}\cap\widetilde{\Omega}$ of
probability $1$, where $\Omega^{*}$ and $\widetilde{\Omega}$ are given by
Lemmata 3.6 and 3.18 respectively.

If $t,s\in(0,1)$, we write

$\displaystyle\begin{split}|R_{H_{1},H_{2}}&(t,\omega)-R_{H_{1},H_{2}}(s,\omega)|\\\
&\leq\left|\sum_{j_{1}<n}\sum_{j_{2}<n}\sum_{(k_{1},k_{2})\in\mathbb{Z}^{2}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}(\omega)I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]\right|\\\
\quad&+\left|\sum_{j_{1}<n}\sum_{j_{2}\geq
n}\sum_{(k_{1},k_{2})\in\mathbb{Z}^{2}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}(\omega)I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]\right|\\\
&+\left|\sum_{j_{1}\geq
n}\sum_{j_{2}<n}\sum_{(k_{1},k_{2})\in\mathbb{Z}^{2}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}(\omega)I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]\right|\\\
&+\left|\sum_{j_{1}\geq n}\sum_{j_{2}\geq
n}\sum_{(k_{1},k_{2})\in\mathbb{Z}^{2}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}(\omega)I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]\right|.\end{split}$

The first sum is bounded from above by Lemmata 3.6 and 3.7, the second and the
third one are bounded from above by Lemmata 3.6 and 3.12 and the last one is
bounded from above by Lemma 3.19. ∎

######  Remark 3.20.

Starting from now and until the end of this section, one can reduce our
attention to the process

$\left\\{R_{H_{1},H_{2}}^{\prime}(t)=\sum_{j_{1}=0}^{+\infty}\sum_{j_{2}=0}^{+\infty}\sum_{(k_{1},k_{2})\in\mathbb{Z}^{2}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}I_{j_{1},j_{2}}^{k_{1},k_{2}}[0,\cdot]\right\\}$

because almost surely, it is the most irregular part of $R_{H_{1},H_{2}}$.
Indeed, using different estimates obtained in this subsection, one can see
that, almost surely, there exists a constant $C>0$ such that, for all
$s,t\in(0,1)$,

$\displaystyle\left|\sum_{j_{1}<0}\sum_{j_{2}<0}\sum_{(k_{1},k_{2})\in\mathbb{Z}^{2}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]\right|$
$\displaystyle\leq C|t-s|,$
$\displaystyle\left|\sum_{j_{1}<0}\sum_{j_{2}>0}\sum_{(k_{1},k_{2})\in\mathbb{Z}^{2}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]\right|$
$\displaystyle\leq C|t-s|^{-H_{2}}\log|t-s|^{-1}$
$\displaystyle\left|\sum_{j_{1}>0}\sum_{j_{2}<0}\sum_{(k_{1},k_{2})\in\mathbb{Z}^{2}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]\right|$
$\displaystyle\leq C|t-s|^{-H_{1}}\log|t-s|^{-1}$

and we conclude because $H_{1}+H_{2}-1<\min\\{H_{1},H_{2}\\}<1$.

###  3.2 Ordinary points

Let us now go to the almost sure finiteness of the limit (6) for almost every
point. The main idea behind our method is that wavelets which contribute the
most in $|R_{H_{1},H_{2}}(t,\cdot)-R_{H_{1},H_{2}}(s,\cdot)|$ are the ones
with associated dyadic intervals “close” to the interval $[t,s]$. Thus, we aim
at proving the following Proposition.

######  Proposition 3.21.

There exists an event $\Omega_{\text{ord}}$ of probability $1$ such that for
all $\omega\in\Omega_{\text{ord}}$, for almost every $t\in(0,1)$,

$\limsup_{s\to
t}\dfrac{|R_{H_{1},H_{2}}(t,\omega)-R_{H_{1},H_{2}}(s,\omega)|}{|t-s|^{H_{1}+H_{2}-1}\log\log|t-s|^{-1}}<+\infty.$

As in [18], for all $j\in\mathbb{N}$, we denote by $k_{j}(t)$ the unique
integer such that $t\in[k_{j}(t)2^{-j},(k_{j}(t)+1)2^{-j})$. In other words,
$k_{j}(t)=s(\lambda_{j}(t))$. If $t\in(0,1)$ is fixed, applying Lemma 3.6 to
the sequence of random variables
$(\xi_{j_{1},j_{2}}^{k_{1}^{\prime},k_{2}^{\prime}})_{(j_{1},j_{2},k_{1}^{\prime},k_{2}^{\prime})\in\mathbb{Z}^{4}}$
defined by

$\xi_{j_{1},j_{2}}^{k_{1}^{\prime},k_{2}^{\prime}}=\varepsilon_{j_{1},j_{2}}^{k_{1}^{\prime}+k_{j_{1}(t)},k_{2}^{\prime}+k_{j_{2}(t)}}$

we deduce the existence of $\Omega_{t}^{*}$, an event of probability $1$, and
$C_{t,1}$, a positive random variable with finite moment of any order, such
that, for all $\omega\in\Omega_{t}^{*}$ and for each
$(j_{1},j_{2},k_{1},k_{2})\in\mathbb{Z}^{4}$, one has

$|\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}(\omega)|\leq
C_{t,1}(\omega)\sqrt{\log(3+|j_{1}|+|k_{1}-k_{j_{1}}(t)|)}\sqrt{\log(3+|j_{2}|+|k_{2}-k_{j_{2}}(t)|)}.$
(58)

In view of this fact, let us set, for $t\in(0,1)$ and
$(j_{1},j_{2},k_{1},k_{2})\in\mathbb{N}^{2}\times\mathbb{Z}^{2}$

$L_{j_{1},j_{2}}^{k_{1},k_{2}}(t)=\sqrt{\log(3+j_{1}+|k_{1}-k_{j_{1}}(t)|)}\sqrt{\log(3+j_{2}+|k_{2}-k_{j_{2}}(t)|)}.$

In what follows, we show how to modify Lemmata 3.7 to 3.19 from the previous
subsection, using $L_{j_{1},j_{2}}^{k_{1},k_{2}}(t)$ instead of
$L_{j_{1},j_{2}}^{k_{1},k_{2}}$. Before all, we need the following Lemma which
is inspired by results from [18] that can be extended in our case.

######  Lemma 3.22.

For all $L>2$ there exists a constant $C_{L}>0$ such that, for all
$n\in\mathbb{N}$ and $t,s\in(0,1)$ such that $2^{-n-1}<|t-s|\leq 2^{-n}$, for
all $x\in[s,t]$

  1. 1.

For all $0\leq j<n$

$\sum_{k\in\mathbb{Z}}\frac{\sqrt{\log(3+j+|k-k_{j}(t)|}}{(3+|2^{j}x-k|)^{L}}\leq
C_{L}\sqrt{\log(3+j)}.$

  2. 2.

For all $j\geq n$

$\sum_{k\in\mathbb{Z}}\frac{\sqrt{\log(3+j+|k-k_{j}(t)|}}{(3+|2^{j}x-k|)^{L}}\leq
C_{L}\sqrt{j-n+1}\sqrt{\log(3+j)}.$

###### Proof.

For all $j\in\mathbb{N}$, $k\in\mathbb{Z}$ and $x\in[s,t]$, observe that

$|k-k_{j}(t)|\leq|k-2^{j}x|+|2^{j}x-2^{j}t|+|2^{j}t-k_{j}(t)|\leq|k-2^{j}x|+2^{j-n}+1.$
(59)

If $0\leq j<n$, then it follows from (59) that $|k-k_{j}(t)|\leq|2^{j}x-k|+2$
which allow us to write, thanks to inequality (28),

$\displaystyle\frac{\sqrt{\log(3+j+|k-k_{j}(t)|}}{(3+|2^{j}x-k|)}$
$\displaystyle\leq\sqrt{\log(3+j)}\frac{\sqrt{\log(5+|2^{j}x-k|)}}{|2^{j}x-k|+3}$
$\displaystyle\leq C\sqrt{\log(3+j)}.$

where $C:=\sup_{x\geq 0}\left(\frac{\sqrt{\log(5+x)}}{x+3}\right)$ and we
conclude using the boundedness of the function

$\xi\mapsto\sum_{k\in\mathbb{Z}}\frac{1}{(1+|\xi-k|)^{M}}$ (60)

for all $M>1$.

Now, if $j\geq n$, from (59) we get $|k-k_{j}(t)|\leq|2^{j}x-k|+2^{j-n+1}$ and
thus, again by inequality (28),

$\displaystyle\frac{\sqrt{\log(3+j+|k-k_{j}(t)|}}{(3+|2^{j}x-k|)}$
$\displaystyle\leq\sqrt{\log(3+2^{j-n+1})}\sqrt{\log(3+j)}\frac{\sqrt{\log(3+|2^{j}x-k|)}}{|2^{j}x-k|+3}$
$\displaystyle\leq C^{\prime}\sqrt{j-n+1}\sqrt{\log(3+j)}.$

where $C^{\prime}:=\sqrt{3}\sup_{x\geq
0}\left(\frac{\sqrt{\log(3+x)}}{x+3}\right)$ and the conclusion comes again
from the boundedness of the function in (60) for all $M>1$ ∎

######  Lemma 3.23.

There exists a deterministic constant $C>0$ such that, for all $t,s\in(0,1)$
we have

$\displaystyle\sum_{0\leq j_{1}<n}$ $\displaystyle\sum_{0\leq
j_{2}<n}\sum_{(k_{1},k_{2})\in\mathbb{Z}^{2}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}L_{j_{1},j_{2}}^{k_{1},k_{2}}(t)\left|I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]\right|$
$\displaystyle\leq C|t-s|^{H_{1}+H_{2}-1}\log\log|t-s|^{-1}.$

###### Proof.

If $\xi\in[s,t]$, we get from the fast decay of the fractional antiderivatives
of $\psi$ (24) and inequality (58), for $0\leq j_{1},j_{2}<n$,

$\displaystyle\sum_{(k_{1},k_{2})\in\mathbb{Z}^{2}}$ $\displaystyle
L_{j_{1},j_{2}}^{k_{1},k_{2}}(t)|\psi_{H_{1}}(2^{j_{1}}\xi-
k_{1})\psi_{H_{2}}(2^{j_{2}}\xi-k_{2})|$ $\displaystyle\leq
CC_{1}\left(\sum_{k_{1}\in\mathbb{Z}}\frac{\sqrt{\log(3+j_{1}+|k_{1}-k_{j_{1}}(t)|)}}{(3+|2^{j_{1}}\xi-
k_{1}|)^{4}}\right)$
$\displaystyle\qquad\qquad\left(\sum_{k_{2}\in\mathbb{Z}}\frac{\sqrt{\log(3+j_{2}+|k_{2}-k_{j_{2}}(t)|)}}{(3+|2^{j_{2}}\xi-
k_{2}|)^{4}}\right).$

These last two sums are bounded by the first point of Lemma 3.22. Using

$\sum_{j_{1}=0}^{n-1}2^{j_{1}(1-H_{1})}\sqrt{\log(3+j_{1})}\leq
C2^{n(1-H_{1})}\sqrt{\log(n)}$ (61)

instead of (27), we conclude, just as in Lemma 3.7, that the desired
inequality holds. ∎

######  Lemma 3.24.

There exists a deterministic constant $C>0$ such that, for all $t,s\in(0,1)$
and $0\leq j_{1}<n\leq j_{2}$, the quantities

$\sum_{k_{1}\in\mathbb{Z}}\sum_{k_{2}\in\mathbb{Z}_{j_{2}}^{<}(t,s)}L_{j_{1},j_{2}}^{k_{1},k_{2}}(t)\left|I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]\right|$
(62)
$\sum_{k_{1}\in\mathbb{Z}}\sum_{k_{2}\in\mathbb{Z}_{j_{2}}^{>}(t,s)}L_{j_{1},j_{2}}^{k_{1},k_{2}}(t)\left|I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]\right|$
(63)

are bounded from above by

$C\sqrt{j_{2}-n+1}\sqrt{\log(3+j_{1})}\sqrt{\log(3+j_{2})}2^{-j_{2}}.$

###### Proof.

Let us prove the bound for (62), the argument for (63) being similar. We have,
by the first part of Lemma 3.22, for $0\leq j_{1}<n\leq j_{2}$,

$\displaystyle\eqref{eqn:r<>3t}\leq
C\sqrt{\log(3+j_{1})}\int_{[s,t]}\sum_{k_{2}\in\mathbb{Z}_{j_{2}}^{<}(t,s)}\frac{\sqrt{\log(3+j_{2}+|k_{2}-k_{j_{2}}(t)|)}}{(3+|2^{j_{2}}x-k_{2}|)^{4}}\,dx$

and, as for all $k_{2}\in\mathbb{Z}_{j_{2}}^{<}(t,s)$ and $x\in[s,t]$ we have

$|k_{2}-k_{j_{2}}(t)|\leq|2^{j_{2}}x-k_{2}|+|k_{j_{2}}(t)-2^{j_{2}}x|\leq|2^{j_{2}}x-k_{2}|+2^{j_{2}-n}+1$

and, by inequality (28),

$\sqrt{\log(3+j_{2}+|k_{2}-k_{j_{2}}(t)|)}\leq
C\sqrt{j_{2}-n+1}\sqrt{\log(3+j_{2})}\sqrt{\log(3+|2^{j_{2}}x-k_{2}|)}$

it just remains us to use the bound (31) to write

$\eqref{eqn:r<>3t}\leq
C\sqrt{j_{2}-n+1}\sqrt{\log(3+j_{1})}\sqrt{\log(3+j_{2})}2^{-j_{2}}.$ (64)

∎

######  Lemma 3.25.

There exists a deterministic constant $C>0$ such that, for all $t,s\in(0,1)$
and $0\leq j_{1}<n\leq j_{2}$, the quantities

$\sum_{k_{1}\in\mathbb{Z}}\sum_{k_{2}\in\mathbb{Z}_{j_{2}}[t,s]}L_{j_{1},j_{2}}^{k_{1},k_{2}}(t)\left|\int_{-\infty}^{\min\\{s,t\\}}\psi_{H_{1}}(2^{j_{1}}x-k_{1})\psi_{H_{2}}(2^{j_{2}}x-k_{2})\,dx\right|$
(65)
$\sum_{k_{1}\in\mathbb{Z}}\sum_{k_{2}\in\mathbb{Z}_{j_{2}}[t,s]}L_{j_{1},j_{2}}^{k_{1},k_{2}}(t)\left|\int_{\max\\{s,t\\}}^{+\infty}\psi_{H_{1}}(2^{j_{1}}x-k_{1})\psi_{H_{2}}(2^{j_{2}}x-k_{2})\,dx\right|$
(66)

are bounded from above by

$C\sqrt{\log(3+j_{1})}\sqrt{\log(3+j_{2})}\sqrt{j_{2}-n+1}2^{-j_{2}}.$

###### Proof.

Again we assume $s\leq t$. First, using the fast decay of the fractional
antiderivatives of $\psi$ (24), (65) is bounded from above by

$\displaystyle\int_{-\infty}^{s}\sum_{k_{1}\in\mathbb{Z}}\sum_{k_{2}\in\mathbb{Z}_{j_{2}}[t,s]}\frac{\sqrt{\log(3+j_{1}+|k_{1}-k_{j_{1}}(t)|)}}{(3+|2^{j_{1}}x-k_{1}|)^{4}}\frac{\sqrt{\log(3+j_{2}+|k_{2}-k_{j_{2}}(t)|)}}{(3+|2^{j_{2}}x-k_{2}|)^{4}}\,dx.$
(67)

Observe that, for all $k_{1}\in\mathbb{Z}$, $k_{2}\in\mathbb{Z}_{j_{2}}[t,s]$
and $x\in(-\infty,s]$, we have, as $j_{1}<n\leq j_{2}$,

$\displaystyle|2^{j_{1}}x-k_{j_{1}}(t)|$
$\displaystyle\leq|2^{j_{1}}x-2^{j_{1}-j_{2}}k_{2}|+|2^{j_{1}-j_{2}}k_{2}-2^{j_{1}}t|+|2^{j_{1}}t-k_{j_{1}}(t)|$
$\displaystyle\leq|2^{j_{2}}x-k_{2}|+2$

and therefore

$|k_{1}-k_{j_{1}}(t)|\leq|2^{j_{1}}x-k_{1}|+|2^{j_{2}}x-k_{2}|+2$

while

$|k_{2}-k_{j_{2}}(t)|\leq|k_{2}-2^{j_{2}}t|+|2^{j_{2}}t-k_{j_{2}}(t)|\leq
2^{j_{2}-n}+1.$

it allows to write, thanks to inequality (28), the boundedness of the function
(60) and inequality (33)

$\displaystyle|\eqref{eqn:demoordistep2}|$ $\displaystyle\leq
C\sqrt{\log(3+j_{1})}\sqrt{\log(3+j_{2})}\sqrt{j_{2}-n+1}\int_{-\infty}^{s}\sum_{k_{2}\in\mathbb{Z}_{j_{2}}[t,s]}\frac{dx}{(3+|2^{j_{2}}x-k_{2}|)^{3}}$
$\displaystyle\leq
C\sqrt{\log(3+j_{1})}\sqrt{\log(3+j_{2})}\sqrt{j_{2}-n+1}2^{-j_{2}}.$

We bound the second sums in the same way. ∎

######  Lemma 3.26.

There exists a deterministic constant $C>0$ such that, for all $t,s\in(0,1)$,
the quantities

$\sum_{0\leq j_{1}<n}\sum_{j_{2}\geq
n}\sum_{(k_{1},k_{2})\in\mathbb{Z}^{2}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}L_{j_{1},j_{2}}^{k_{1},k_{2}}(t)|I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]|$
$\sum_{j_{1}\geq n}\sum_{0\leq
j_{2}<n}\sum_{(k_{1},k_{2})\in\mathbb{Z}^{2}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}L_{j_{1},j_{2}}^{k_{1},k_{2}}(t)|I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]|$

are bounded from above by

$C|t-s|^{H_{1}+H_{2}-1}\log\log|t-s|^{-1}.$

###### Proof.

The proof is exactly the same as the one of Lemma 3.12 excepted that we use
Lemmata 3.24 and 3.25 instead of Lemmata 3.10 and 3.11 respectively and that
we conclude using again (61) instead of (27) and

$\sum_{j_{2}=n}^{+\infty}2^{-j_{2}H_{2}}\sqrt{j_{2}-n+1}\sqrt{\log(3+j_{2})}\leq
C^{\prime}2^{-nH_{2}}\sqrt{\log(n)}.$ (68)

instead of (38). ∎

######  Lemma 3.27.

There exists a deterministic constant $C>0$ such that, for all $t,s\in(0,1)$
and $n\leq j_{1}\leq j_{2}$, the quantities (62) and (63) are bounded from
above by

$C\sqrt{j_{2}-n+1}\sqrt{j_{1}-n+1}\sqrt{\log(3+j_{1})}\sqrt{\log(3+j_{2})}2^{-j_{2}}.$

###### Proof.

The proof is exactly the same as for Lemma 3.24 except that, here, we use the
second part of Lemma 3.22 instead of the first one. ∎

######  Lemma 3.28.

There exists a deterministic constant $C>0$ such that, for all $t,s\in(0,1)$
and $n\leq j_{1}\leq j_{2}$ the quantities (65) and (66) are bounded from
above by

$C\sqrt{j_{1}-n+1}\sqrt{j_{2}-n+1}\sqrt{\log(3+j_{1})}\sqrt{\log(3+j_{2})}2^{-j_{2}}.$

###### Proof.

The proof is exactly the same as for Lemma 3.25 except that, here, we use the
second part of Lemma 3.22 instead of the first one. ∎

Just as we did for the rapid points, it remains us to bound the random
variables $\Xi_{j}(\lambda)$. Here, we don’t want anymore to show the
existence of an uniform modulus but only a pointwise modulus of continuity at
a fixed point of interest $t$. Therefore, we just have to bound, for all
$n\in\mathbb{N}$ the random variables $\Xi_{j}(\lambda)$ for $j\geq n$ and
$\lambda\in 3\lambda_{n}(t)$. We thus have the following result.

######  Lemma 3.29.

For all $t\in(0,1)$, there exist an event $\widetilde{\Omega_{t}}$ of
probability $1$ and a positive random variable $C_{t,2}$ with finite moment of
any order such that, on $\widetilde{\Omega_{t}}$,

$\forall n\in\mathbb{N},\leavevmode\nobreak\ \forall\lambda\in
3\lambda_{n}(t),\leavevmode\nobreak\ \forall j\geq n,\leavevmode\nobreak\
\Xi_{j}(\lambda)\leq C_{t,2}\,(j-n+1)\log(n).$ (69)

###### Proof.

If $t\in(0,1)$ is fixed and $\theta>0$, let us define the event

$A_{n}(t)=\\{\forall\lambda\in 3\lambda_{n}(t)\leavevmode\nobreak\ \forall
j\geq n,\leavevmode\nobreak\ \Xi_{j}(\lambda)\leq\theta(j-n+1)\log(n)\\}.$

Similarly to Lemma 3.18, we get

$\displaystyle\mathbb{P}(A_{n}(t)^{c})$ $\displaystyle\leq C\sum_{j\geq
n}2^{4(j-n)}\exp(-\overset{\star}{C}\theta(j-n+1)\log(n))$ $\displaystyle\leq
C\exp(-\overset{\star}{C}\theta\log(n))\sum_{j\geq
n}2^{4(j-n)}\exp(-\overset{\star}{C}\theta(j-n)),$

for a determistic constant $C>0$. Therefore, if we take again
$\theta>4\log(2)/\overset{\star}{C}$ then Borel-Cantelli Lemma implies the
existence of an event $\widetilde{\Omega_{t}}$ of probability $1$ and
$C_{t,2}$ a positive random variable of finite moment of any order such that,
on $\widetilde{\Omega_{t}}$, assertion (69) holds. ∎

######  Lemma 3.30.

If $t\in(0,1)$, let $\Omega_{t}^{*}$ be the event of probability $1$ where
inequality (58) holds and $\widetilde{\Omega_{t}}$ be the event of probability
$1$ given by Lemma 3.29. There exists a positive random variable $C_{t,3}$
with finite moment of any order such that, on
$\Omega_{t}^{*}\cap\widetilde{\Omega_{t}}$, for all $s\in(0,1)$ the random
variable

$\left|\sum_{j_{1}\geq
n}\sum_{j_{2}>n}\sum_{(k_{1},k_{2})\in\mathbb{Z}^{2}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]\right|$
(70)

is bounded from above by

$C_{t,3}|t-s|^{H_{1}+H_{2}-1}\log|t-s|^{-1}.$

###### Proof.

Again, we use the split (3.1) and we only do the details for the first sum. We
deal with the series (50) and (51) in the same way that in Lemma 3.19 but
using inequality (58) and Lemmata 3.27 and 3.28 and finally inequality (68).

For (52), first, by Lemma 3.28 and inequality (58), we have, on
$\Omega_{t}^{*}\cap\widetilde{\Omega_{t}}$

$\displaystyle\left|\sum_{j_{1}\geq n}\sum_{j_{2}\geq
j_{1}}\sum_{k_{1}\in\mathbb{Z}}\sum_{k_{2}\in\mathbb{Z}_{j_{2}}[t,s]}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}\int_{-\infty}^{s}\psi_{H_{1}}(2^{j_{1}}x-k_{1})\psi_{H_{2}}(2^{j_{2}}x-k_{2})\,dx\right|$
$\displaystyle\leq CC_{t,1}\sum_{j_{1}\geq n}\sum_{j_{2}\geq
j_{1}}2^{j_{1}(1-H_{1})}2^{-j_{2}H_{2}}\sqrt{j_{1}-n+1}\sqrt{j_{2}-n+1}\sqrt{\log(3+j_{1})}\sqrt{\log(3+j_{2})}$
$\displaystyle\leq CC_{1}2^{n(1-H_{1}-H_{2})}\log(n).$ (71)

We bound

$\displaystyle\left|\sum_{j_{1}\geq n}\sum_{j_{2}\geq
j_{1}}\sum_{k_{1}\in\mathbb{Z}}\sum_{k_{2}\in\mathbb{Z}_{j_{2}}[t,s]}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}\int_{-\infty}^{s}\psi_{H_{1}}(2^{j_{1}}x-k_{1})\psi_{H_{2}}(2^{j_{2}}x-k_{2})\,dx\right|$

on $\Omega_{t}^{*}\cap\widetilde{\Omega_{t}}$ exactly in the same way.

To finish the proof, again, we have to bound (54), (55) and (56) for
$\ell\in\\{j,j+1\\}$ (with $j\geq n$) on
$\Omega_{t}^{*}\cap\widetilde{\Omega_{t}}$. For (54), in the case $\ell=j$,
one can note that, for all $k_{2}\in\mathbb{Z}_{j}[t,s]$,
$|k_{2}-k_{j}(t)|\leq 2^{j-n}+1$ and, for all
$k_{1}\in\mathbb{Z}_{j}^{<}(t,s)$,
$|k_{1}-k_{j}(t)|\leq|2^{j}\min\\{t,s\\}-k_{1}|+2^{j-n}+1$. Using the same
tricks as in (57), we get, on $\Omega_{t}^{*}\cap\widetilde{\Omega_{t}}$

$\displaystyle|\eqref{eqn:r>>6}|$ $\displaystyle\leq
CC_{t,1}(j-n+1)\log(3+j)2^{-j}\sum_{k_{1}\in\mathbb{Z}_{j}^{<}(t,s)}\frac{1}{(2+|2^{j}\min\\{s,t\\}-k_{1}|)^{3}}$
$\displaystyle\leq C(j-n+1)\log(3+j)2^{-j}.$

The bounds for (55) and in the case $\ell=j+1$ are obtained in the same way.
Finally to bound (56), we use (69) and (46) and get on
$\Omega_{t}^{*}\cap\widetilde{\Omega_{t}}$

$\displaystyle\left|\sum_{j\geq
n}2^{j(2-H_{1}-H_{2})}\sum_{\ell=j}^{j+1}\sum_{k_{1}\in\mathbb{Z}_{j}[t,s]}\sum_{k_{2}\in\mathbb{Z}_{\ell}[t,s]}\varepsilon_{j,\ell}^{k_{1},k_{2}}I_{j,\ell}^{k_{1},k_{2}}\right|$
$\displaystyle\leq CC_{t,2}\sum_{j\geq
n}2^{j(\frac{3}{2}-H_{1}-H_{2})}2^{-\frac{n}{2}}(j-n+1)\log(n)$
$\displaystyle\leq CC_{t,2}2^{n(1-H_{1}-H_{2})}\log(n).$

We conclude that (70) is bounded from above by

$C\max\\{C_{t,1},C_{t,2}\\}|t-s|^{H_{1}+H_{2}-1}\log|t-s|^{-1}$

on $\Omega_{t}^{*}\cap\widetilde{\Omega_{t}}$. ∎

We can now prove Proposition 3.21.

###### Proof of Proposition 3.21.

Let us fix $t\in(0,1)$ and consider
$\omega\in\Omega_{t}^{*}\cap\widetilde{\Omega_{t}}$. For all $s\in(0,1)$, we
write333We recall that $R_{H_{1},H_{2}}^{\prime}$ is defined in Remark 3.20.

$\displaystyle\begin{split}|R_{H_{1},H_{2}}^{\prime}&(t,\omega)-R_{H_{1},H_{2}}^{\prime}(s,\omega)|\\\
&\leq\left|\sum_{0\leq j_{1}<n}\sum_{0\leq
j_{2}<n}\sum_{(k_{1},k_{2})\in\mathbb{Z}^{2}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}(\omega)I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]\right|\\\
\quad&+\left|\sum_{0\leq j_{1}<n}\sum_{j_{2}\geq
n}\sum_{(k_{1},k_{2})\in\mathbb{Z}^{2}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}(\omega)I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]\right|\\\
&+\left|\sum_{j_{1}\geq n}\sum_{0\leq
j_{2}<n}\sum_{(k_{1},k_{2})\in\mathbb{Z}^{2}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}(\omega)I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]\right|\\\
&+\left|\sum_{j_{1}\geq n}\sum_{j_{2}\geq
n}\sum_{(k_{1},k_{2})\in\mathbb{Z}^{2}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}(\omega)I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]\right|.\end{split}$

We bound from above the first sum by inequality (58) and Lemma 3.23, the
second and the third sums by inequality (58) and Lemma 3.26 and the last sum
by Lemma 3.30.

Using inequalities (68) and (71) and Remark 3.20, one can finally write that
for all $t\in(0,1)$, for all $\omega$ in the event of probability $1$
$\Omega_{t}^{*}\cap\widetilde{\Omega_{t}}$

$\limsup_{s\to
t}\dfrac{|R_{H_{1},H_{2}}(t,\omega)-R_{H_{1},H_{2}}(s,\omega)|}{|t-s|^{H_{1}+H_{2}-1}\log\log|t-s|^{-1}}<+\infty$

and we conclude by Fubini Theorem. ∎

###  3.3 Slow points

In this section, we aim at showing that the generalized Rosenblatt process
admits slow points: we prove the following Proposition.

######  Proposition 3.31.

There exists an event $\Omega_{\text{slo}}$ of probability $1$ such that for
all $\omega\in\Omega_{\text{slo}}$ there exist $t\in(0,1)$ such that

$\limsup_{s\to
t}\dfrac{|R_{H_{1},H_{2}}(t,\omega)-R_{H_{1},H_{2}}(s,\omega)|}{|t-s|^{H_{1}+H_{2}-1}}<+\infty.$
(72)

In [25], Kahane described a procedure to insure the existence of slow points
for the Brownian motion. This procedure was then generalized in [18] to fit
for any arbitrary fractional Brownian motion. It consists in showing that for
any $m>0$, almost surely, there exist $\mu>0$ and $t\in(0,1)$ such that, if
one sets

$\Lambda_{j}^{0}(t)=\\{\lambda\in\Lambda_{j}:\,|s(\lambda(t))-s(\lambda)|\leq
1\\}$ (73)

and, for all $1\leq l$

$\Lambda_{j}^{l}(t)=\\{\lambda\in\Lambda_{j},:\,2^{m(l-1)}<|s(\lambda(t))-s(\lambda)|\leq
2^{ml}\\},$ (74)

then, for all $\lambda\in\Lambda_{j}^{l}(t)$ we have

$|\varepsilon_{\lambda}|\leq 2^{l}\mu,$ (75)

where $\varepsilon_{\lambda}$ is the random variable

$2^{\frac{j}{2}}\int_{\mathbb{R}}\psi_{\lambda}(x)\,dB(x).$

In this procedure, if $\mu\in\mathbb{N}$,for all $j,l\in\mathbb{N}_{0}$ and
$\lambda\in\Lambda_{j}$, $\lambda\subseteq[0,1]$, we define

$\Lambda_{j,l}(\lambda)=\\{\lambda^{\prime}\in\Lambda_{j},\,:\,|s(\lambda)-s(\lambda^{\prime})|\leq
2^{ml}\\}$

and the random set

$S_{j,l}^{\mu}=\\{\lambda^{\prime}\in\Lambda_{j},\,:\,2^{l}\mu<|\varepsilon_{\lambda^{\prime}}|\leq
2^{l+1}\mu\\}.$

Finally we consider the random set

$I_{j}^{\mu}=\\{\lambda\in\Lambda_{j},\lambda\subseteq[0,1]:\,\forall
l\in\mathbb{N}_{0},\,\Lambda_{j,l}(\lambda)\cap S_{j,l}^{\mu}=\cmemptyset\\},$

and show that almost surely, there exists $\mu\in\mathbb{N}$ such that

$S_{\text{low}}^{\mu}=\bigcap_{j\in\mathbb{N}_{0}}\bigcup_{\lambda\in
I_{j}^{\mu}}\overline{\lambda}\neq\cmemptyset$

which is equivalent to the fact that, for any $J$

$S_{\text{low},J}^{\mu}=\bigcap_{j\leq J}\bigcup_{\lambda\in
I_{j}^{\mu}}\overline{\lambda}\neq\cmemptyset$

as $(S_{\text{low},J}^{\mu})_{J}$ is a decreasing sequence of compact sets. To
do so, let us denote by $2S_{\text{low},J}^{\mu}$ the sets of dyadic intervals
of scale $J+1$ obtained by cutting in two the remaining intervals444The
interval $[k2^{-j},(k+1)2^{-j}]$ is cut into
$[(2k)2^{-(j+1)},(2k+1)2^{-(j+1)}]$ and $[(2k+1)2^{-(j+1)},(2k+2)2^{-(j+1)}]$.
in $S_{\text{low},J}^{\mu}$ and remark that $S_{\text{low},J+1}^{\mu}$ is
obtained from $2S_{\text{low},J}^{\mu}$ by removing the dyadic intervals
$\lambda$ such that $\Lambda_{J+1,l}(\lambda)\cap
S_{J+1,l}^{\mu}\neq\cmemptyset$ for a $l\in\mathbb{N}_{0}$. But now, if
$\xi\sim\mathcal{N}(0,1)$, we set, for all such a $l$

$p_{l}(\mu)=\mathbb{P}(2^{l}\mu<|\xi|\leq 2^{l+1}\mu).$

and note that, if $N$ is the number of intervals of $S_{\text{low},J}^{\mu}$,
counting the number of intervals in $2S_{\text{low},J}^{\mu}\cap
S_{J+1,l}^{\mu}$ is a binomial random variable of parameter $2N$ and
$p_{l}(\mu)$ and this number is thus bounded by

$2N(p_{l}(\mu)+(l+1)\sqrt{p_{l}(\mu)(1-p_{l}(\mu))})$

on an event of probability $1-(l+1)^{-2}N^{-1}$. Therefore, to pass from
$S_{\text{low},J}^{\mu}$ to $S_{\text{low},J+1}^{\mu}$ we remove at most

$2N\sum_{l=0}^{+\infty}(2^{ml+1}+1)(p_{l}(\mu)+(l+1)\sqrt{p_{l}(\mu)(1-p_{l}(\mu))})$

intervals with probability greater than $1-N^{-1}$. But if $\mu$ is large
enough, as $p_{l}(\mu)$ is of order $\frac{e^{-(2^{l}\mu)^{2}}}{2^{l}\mu}$,
one can make sure that this last term is bounded by $\frac{N}{2}$. So, if
$N_{J}^{\mu}$ is the random variable counting the number of subintervals of
$S_{\text{low},J}^{\mu}$, we have

$\mathbb{P}(N_{J+1}^{\mu}\geq\frac{3}{2}N_{J}^{\mu}|N_{J}^{\mu}=N)\geq
1-N^{-1}$

which leads to the recursive formula

$\mathbb{P}(N_{J+1}^{\mu}\geq(\frac{3}{2})^{J+1})\geq(1-(\frac{2}{3})^{J})\mathbb{P}(N_{J}^{\mu}\geq(\frac{3}{2})^{J}),\quad\forall
J\in\mathbb{N}_{0},$

see [18, Lemma 3.6 and Theorem 3.7.]. Finally, we deduce

$\mathbb{P}\big{(}\bigcup_{\mu}\bigcap_{J\in\mathbb{N}_{0}}(N_{J}^{\mu}\geq
1)\big{)}=1.$ (76)

Moreover, we can show that, in this case,
$S_{\text{low}}^{\mu}\cap(0,1)\neq\cmemptyset$. If $\alpha>0$, applying this
procedure with $\frac{1}{m}<\alpha$ gives us that any point $t\in
S_{\text{low}}^{\mu}\cap(0,1)$ is a slow point of the fractional Brownian
motion of exponent $\alpha$.

From formulas (17) and (18), we see that this procedure is also useful to
bound the random variables appearing in the expansion (19) of the generalized
Rosenblatt process. But, from the proofs of Propositions 3.5 and 3.21 we know
that this is not sufficient and we also need to give a bound for the random
variables $\Xi_{j}(\lambda)$, for $\lambda\in 3\lambda_{n}(t),n\in\mathbb{N}$
and $j\geq n$. Such dyadic intervals are precisely the ones in the set
$\Lambda_{n,0}(\lambda_{n}(t))$ and this fact forces us to consider the
following modification of the procedure. For all $j\in\mathbb{N}$, if $l\neq
0$, the sets $S_{j,l}^{\mu}$ remain untouched as well as its associated
probability $p_{l}(\mu)$ while for $l=0$ we set

$S_{j,0}^{\mu}=\\{\lambda^{\prime}\in\Lambda_{j},\lambda^{\prime}\subseteq[0,1]\,:\,\exists
j^{\prime}\geq j\,\Xi_{j^{\prime}}(\lambda^{\prime})>(j^{\prime}-j+1)\mu\\},$

with associated probability (which only depends on $\mu$)

$p_{0}(\mu)=\mathbb{P}(\exists j^{\prime}\geq
j\,\Xi_{j^{\prime}}(\lambda)>(j^{\prime}-j+1)\mu).$

As $\Xi_{j^{\prime}}(\lambda_{1})$ is independent of
$\Xi_{j^{\prime}}(\lambda_{2})$ as soon as
$\lambda_{1}\cap\lambda_{2}=\cmemptyset$, for all $J\in\mathbb{N}$, if $N$ is
again the number of dyadic intervals of $S_{\text{low},J}^{\mu}$, the number
of such intervals in $2S_{\text{low},J}^{\mu}\cap S_{J+1,0}^{\mu}$ is still a
binomial random variable of parameter $2N$ and $p_{0}(\mu)$. Therefore If
$\mu$ is large enough, using Theorems 3.14 and 3.15, one can still affirm

$2N\sum_{l=0}^{+\infty}(2^{ml+1}+1)(p_{l}(\mu)+(l+1)\sqrt{p_{l}(\mu)(1-p_{l}(\mu))})\leq\frac{N}{2}$

and the end of the procedure is saved: equality (76) still holds. Now, if
$t\in S_{\text{low}}^{\mu}\cap(0,1)$ we know that

$\forall\,n\in\mathbb{N},\,\forall\,\lambda\in 3\lambda_{n}(t)\,,\forall j\geq
n\,,\Xi_{j}(\lambda)\leq(j-n+1)\mu.$ (77)

Let us remark that, as for all $\lambda\in\Lambda_{n}$,
$|\varepsilon_{\lambda}^{2}|\leq 2\Xi_{n}(\lambda)+1$, we still have, in this
case, for all $\lambda\in 3\lambda_{n}(t)$, $|\varepsilon_{\lambda}|\leq
C\mu$, for a deterministic constant $C>0$. .

Starting from now we take $m$ such that $1/m<\min\\{H_{1},H_{2}\\}$ and
$2/m<1-H_{1}-H_{2}$

In order to use notations (73) and (74), here after $\lambda_{1}$ (resp.
$\lambda_{2}$) will always stand for the dyadic interval
$[k_{1}2^{-j_{1}},(k_{1}+1)2^{-j_{1}})$ (resp.
$[k_{2}2^{-j_{2}},(k_{2}+1)2^{-j_{2}})$) and $\psi_{\lambda_{1}}$ (resp.
$\psi_{\lambda_{2}}$) will be the associated antiderivative of wavelet
$\psi_{H_{1}}(2^{j_{1}}\cdot-k_{1})$ (resp. $\psi_{H_{2}}(2^{j_{2}}\cdot-
k_{2})$) and $I_{\lambda_{1},\lambda_{2}}[t,s]$ will stand for
$I_{j_{1},j_{2}}^{k_{1},k_{2}}[t,s]$. Finally,
$\varepsilon_{\lambda_{1},\lambda_{2}}$ will stand for
$\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}$. If $t\in(0,1)$, let
$(y_{\lambda}(t))_{\lambda\Lambda}$ be the sequence defined by

$y_{\lambda}(t)=2^{l}\text{ if }\lambda\in\Lambda_{j}^{l}(t).$

Note that, if we apply the preceding procedure, we find $\Omega_{\text{slo}}$
an event of probability $1$ such that, for all $\omega\in\Omega_{\text{slo}}$,
there exists $\mu$ for which $S_{\text{low}}^{\mu}\cap(0,1)\neq\cmemptyset$.
Then, if $t$ belong to this set, we have, thanks to inequality (75) and
equalities (17) and (18)

$|\varepsilon_{\lambda_{1},\lambda_{2}}(\omega)|\leq
C\mu^{2}y_{\lambda_{1}}(t)y_{\lambda_{2}}(t),$ (78)

for a deterministic constant $C>0$. Again, we need to adapt the Lemmata from
previous sections with this alternative upper bound.

######  Lemma 3.32.

There exists a deterministic constant $C>0$ such that, for all $t,s\in(0,1)$
we have

$\displaystyle\sum_{0\leq j_{1}<n}$ $\displaystyle\sum_{0\leq
j_{2}<n}\sum_{\lambda_{1}\in\Lambda_{j_{1}},\lambda_{2}\in\Lambda_{j_{2}}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}y_{\lambda_{1}}(t)y_{\lambda_{2}}(t)\left|I_{\lambda_{1},\lambda_{2}}[t,s]\right|$
$\displaystyle\leq C|t-s|^{H_{1}+H_{2}-1}.$

###### Proof.

If $\xi\in[s,t]$ and $\lambda\in\lambda_{j}^{l}(t)$, for $0\leq j<n$ and
$l\geq 1$,

$|2^{j}\xi-s(\lambda)|\geq|s(\lambda(t))-s(\lambda)|-2>2^{m(l-1)}-2$

and so, using the fast decay of the fractional antiderivatives of $\psi$ (24)
and the definition of $(y_{\lambda})_{\lambda\in\Lambda}$ , we get for $0\leq
j_{1},j_{2}<n$

$\displaystyle\sum_{\lambda_{1}\in\Lambda_{j_{1}},\lambda_{2}\in\Lambda_{j_{2}}}y_{\lambda_{1}}(t)y_{\lambda_{2}}(t)|\psi_{\lambda_{1}}(\xi)\psi_{\lambda_{2}}(\xi)|$
$\displaystyle\quad=\sum_{(l_{1},l_{2})\in\mathbb{N}_{0}^{2}}\sum_{\lambda_{1}\Lambda_{j_{1}}^{l_{1}}(t)}\sum_{\lambda_{2}\Lambda_{j_{2}}^{l_{2}}(t)}y_{\lambda_{1}}(t)y_{\lambda_{2}}(t)|\psi_{\lambda_{1}}(\xi)\psi_{\lambda_{2}}(\xi)|$
$\displaystyle\quad\leq
C\sum_{(l_{1},l_{2})\in\mathbb{N}_{0}}\sum_{\lambda_{1}\Lambda_{j_{1}}^{l_{1}}(t)}\sum_{\lambda_{2}\Lambda_{j_{2}}^{l_{2}}(t)}\frac{2^{l_{1}+l_{2}}}{(3+|2^{j_{1}}\xi-
k_{1}|)^{4}(3+|2^{j_{2}}\xi-k_{2}|)^{4}}$ $\displaystyle\quad\leq
C\sum_{(l_{1},l_{2})\in\mathbb{N}_{0}}\sum_{\lambda_{1}\Lambda_{j_{1}}^{l_{1}}(t)}\sum_{\lambda_{2}\Lambda_{j_{2}}^{l_{2}}(t)}\frac{2^{l_{1}+l_{2}}2^{-m(l_{1}+l_{2})}}{(3+|2^{j_{1}}\xi-
k_{1}|)^{3}(3+|2^{j_{2}}\xi-k_{2}|)^{3}}$ $\displaystyle\leq
C\sum_{k_{1}\in\mathbb{Z}}\frac{1}{(3+|2^{j_{1}}\xi-
k_{1}|)^{3}}\sum_{k_{2}\in\mathbb{Z}}\frac{1}{(3+|2^{j_{2}}\xi-k_{2}|)^{3}}$
$\displaystyle\leq C.$ (79)

It leads, just as in Lemmata 3.7 and 3.23, to the desired estimate. ∎

In what follows, we use these notations instead of the one given in Definition
3.8:

$\Lambda_{j_{2}}^{<}(t,s)=\\{\lambda_{2}\in\Lambda_{j_{2}}\,:\,s(\lambda_{2})\in\mathbb{Z}_{j_{2}}^{<}(t,s)\\},$
$\Lambda_{j_{2}}^{>}(t,s)=\\{\lambda_{2}\in\Lambda_{j_{2}}\,:\,s(\lambda_{2})\in\mathbb{Z}_{j_{2}}^{>}(t,s)\\},$
$\Lambda_{j_{2}}[t,s]=\\{\lambda_{2}\in\Lambda_{j_{2}}\,:\,s(\lambda_{2})\in\mathbb{Z}_{j_{2}}[t,s]\\}.$

######  Lemma 3.33.

There exists a deterministic constant $C>0$ such that, for all $t,s\in(0,1)$
and $0\leq j_{1}<n\leq j_{2}$, the quantities

$\sum_{\lambda_{1}\in\Lambda_{j_{1}}}\sum_{\lambda_{2}\in\Lambda_{j_{2}}^{<}(t,s)}y_{\lambda_{1}}(t)y_{\lambda_{2}}(t)\left|I_{\lambda_{1},\lambda_{2}}[t,s]\right|$
(80)
$\sum_{\lambda_{1}\in\Lambda_{j_{1}}}\sum_{\lambda_{2}\in\Lambda_{j_{2}}^{>}(t,s)}y_{\lambda_{1}}(t)y_{\lambda_{2}}(t)\left|I_{\lambda_{1},\lambda_{2}}[t,s]\right|$
(81)

are bounded by

$C2^{\frac{1}{m}(j_{2}-n)}2^{-j_{2}}.$

###### Proof.

Again, we prove the bound for (80), the reasoning for (81) being similar. Let
us remark that, if $j_{2}\geq n$ $x\in[s,t]$ and
$\lambda_{j_{2}}(x)\in\Lambda_{j_{2}}^{l}(t)$ then, the construction and the
definition of $(y_{\lambda}(t))_{\lambda\in\Lambda}$ gives that

  * •

$l\leq\frac{1}{m}(j_{2}-n)$, as $|s-t|\leq 2^{-n}$,

  * •

if $\lambda\in\Lambda_{j_{2}}^{l_{2}}(x)$ then $|y_{\lambda}|\leq
2^{l_{2}}2^{l+1}\mu$ while, by definition, if $l_{2}\geq 1$

$3+|2^{j_{2}}x-s(\lambda)|\geq 2+2^{m(l_{2}-1)}.$

Therefore, if we set

$D_{j_{2}}^{l}(t)=\bigcup_{\lambda\in\Lambda_{j_{2}}^{l}(t)}\lambda,$

we have

$\displaystyle\begin{split}\eqref{eqn:r<>3s}&\leq\sum_{\lambda_{1}\in\Lambda_{j_{1}}}\sum_{\lambda_{2}\in\Lambda_{j_{2}}^{<}(t,s)}y_{\lambda_{1}}(t)y_{\lambda_{2}}(t)\int_{[s,t]}|\psi_{\lambda_{1}}(x)\psi_{\lambda_{2}}(x)|\,dx\\\
&\leq\sum_{0\leq
l\leq\frac{1}{m}(j_{2}-n)}\sum_{\lambda_{1}\in\Lambda_{j_{1}}}\sum_{\lambda_{2}\in\Lambda_{j_{2}}^{<}(t,s)}y_{\lambda_{1}}(t)y_{\lambda_{2}}(t)\int_{D_{j_{2}}^{l}(t)}|\psi_{\lambda_{1}}(x)\psi_{\lambda_{2}}(x)|\,dx.\end{split}$
(82)

But, for all $x\in D_{j_{2}}^{l}$, using the same method as in (79), but
splitting the sums according to the set $\Lambda_{j_{1}}^{l_{1}}(x)$ and
$\Lambda_{j_{2}}^{l_{2}}(x)$ on which
$y_{\lambda_{1}}(t)y_{\lambda_{2}}(t)\leq 2^{l+l_{1}+l_{2}+1}$ we get

$\displaystyle\begin{split}\sum_{\lambda_{1}\in\Lambda_{j_{1}}}&\sum_{\lambda_{2}\in\Lambda_{j_{2}}^{<}(t,s)}|\varepsilon_{\lambda_{1},\lambda_{2}}||\psi_{\lambda_{1}}(x)\psi_{\lambda_{2}}(x)|\\\
&\leq
C2^{l+1}\sum_{\lambda_{1}\in\Lambda_{j_{1}}}\frac{1}{(3+|2^{j_{1}}x-k_{1}|)^{3}}\sum_{\lambda_{2}\in\Lambda_{j_{2}}^{<}(t,s)}\frac{1}{(3+|2^{j_{2}}x-k_{2}|)^{3}}\\\
&\leq
C2^{l+1}\sum_{\lambda_{2}\in\Lambda_{j_{2}}^{<}(t,s)}\frac{1}{(3+|2^{j_{2}}x-k_{2}|)^{3}}.\end{split}$
(83)

Finally, using the techniques in (31), we get

$\eqref{eqn:r<>3s}\leq C2^{\frac{1}{m}(j_{2}-n)}2^{-j_{2}}.$

∎

######  Lemma 3.34.

There exists a deterministic constant $C>0$ such that, for all $t,s\in(0,1)$
and $0\leq j_{1}<n\leq j_{2}$, the quantities

$\sum_{\lambda_{1}\in\Lambda_{j_{1}}}\sum_{\lambda_{2}\in\Lambda_{j_{2}}[t,s]}y_{\lambda_{1}}(t)y_{\lambda_{2}}(t)\left|\int_{-\infty}^{\min\\{s,t\\}}\psi_{H_{1}}(2^{j_{1}}x-k_{1})\psi_{H_{2}}(2^{j_{2}}x-k_{2})\,dx\right|$
(84)
$\sum_{\lambda_{1}\in\Lambda_{j_{1}}}\sum_{\lambda_{2}\in\Lambda_{j_{2}}[t,s]}y_{\lambda_{1}}(t)y_{\lambda_{2}}(t)\left|\int_{\max\\{s,t\\}}^{+\infty}\psi_{H_{1}}(2^{j_{1}}x-k_{1})\psi_{H_{2}}(2^{j_{2}}x-k_{2})\,dx\right|$
(85)

are bounded by

$C2^{\frac{1}{m}(j_{2}-n)}2^{-j_{2}}.$

###### Proof.

Again, we assume $s\leq t$. If $x\in(-\infty,s]$ is such that
$\lambda_{j_{1}}(x)\in\Lambda_{j_{1}}^{l}(s)$, we have, for all
$\lambda_{1}\in\Lambda_{j_{1}}^{l_{1}}(x)$ and
$\lambda_{2}\in\Lambda_{j_{2}}[t,s]\cap\Lambda_{j_{2}}^{l_{2}}(s)$ (with
$j_{1}<n\leq j_{2})$,

$\displaystyle\begin{split}\frac{y_{\lambda_{1}}(t)y_{\lambda_{2}}(t)}{(3+|2^{j_{1}}x-k_{1}|)^{4}(3+|2^{j_{2}}x-k_{2}|)^{4}}&\leq
C\frac{2^{\frac{1}{m}(j_{2}-n)+l+l_{1}+l_{2}+1}\mu^{2}}{(3+|2^{j_{1}}x-k_{1}|)^{4}(3+|2^{j_{2}}x-k_{2}|)^{5}}\\\
&\leq
C\frac{2^{\frac{1}{m}(j_{2}-n)+l+1}}{(3+|2^{j_{1}}x-k_{1}|)^{3}(3+|2^{j_{2}}x-k_{2}|)^{4}}\\\
&\leq
C\frac{2^{\frac{1}{m}(j_{2}-n)}}{(3+|2^{j_{1}}x-k_{1}|)^{3}(3+|2^{j_{2}}x-k_{2}|)^{3}}\end{split}$
(86)

because

$3+|2^{j_{2}}x-k_{2}|=3+k_{2}-2^{j_{2}}x\geq 2+2^{j_{1}}(s-x)\geq 2^{m(l-1)}.$

Thus we get, using the fast decay of the fractional antiderivatives of the
wavelet before splitting the integral over $(-\infty,s]$ into the integral
over the sets $(-\infty,s]\cap D_{j_{1}}^{l}(s)$, in the same way as in (82),
using (86) and finally the boundedness of the function (60) for $M=3$ and
inequality (33)

$\displaystyle\sum_{\lambda_{1}\in\Lambda_{j_{1}}}$
$\displaystyle\sum_{\lambda_{2}\in\Lambda_{j_{2}}[t,s]}\int_{-\infty}^{s}|\psi_{\lambda_{1}}(x)\psi_{\lambda_{2}}(x)|\,dx$
$\displaystyle\leq
C2^{\frac{1}{m}(j_{2}-n)}\int_{-\infty}^{s}\sum_{\lambda_{1}\in\Lambda_{j_{1}}}\sum_{\lambda_{2}\in\Lambda_{j_{2}}[t,s]}\frac{dx}{(3+|2^{j_{1}}x-k_{1}|)^{3}(3+|2^{j_{2}}x-k_{2}|)^{3}}$
$\displaystyle\leq
C2^{\frac{1}{m}(j_{2}-n)}\int_{-\infty}^{s}\sum_{\lambda_{2}\in\Lambda_{j_{2}}[t,s]}\frac{dx}{(3+|2^{j_{2}}x-k_{2}|)^{3}}$
$\displaystyle\leq C2^{\frac{1}{m}(j_{2}-n)}2^{-j_{2}}.$

In the same way we get

$\sum_{\lambda_{1}\in\Lambda_{j_{1}}}\sum_{\lambda_{2}\in\Lambda_{j_{2}}[t,s]}y_{\lambda_{1}}(t)y_{\lambda_{2}}(t)\left|\int_{\max\\{t,s\\}}^{+\infty}\psi_{H_{1}}(2^{j_{1}}x-k_{1})\psi_{H_{2}}(2^{j_{2}}x-k_{2})\,dx\right|\leq
C2^{-j_{2}}.$ (87)

∎

######  Lemma 3.35.

There exists a deterministic constant $C>0$ such that, for all $t,s\in(0,1)$,
the quantities

$\sum_{0\leq j_{1}<n}\sum_{j_{2}\geq
n}\sum_{\lambda_{1}\in\Lambda_{j_{1}},\lambda_{2}\in\Lambda_{j_{2}}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}y_{\lambda_{1}}(t)y_{\lambda_{2}}(t)|I_{\lambda_{1},\lambda_{2}}[t,s]|$
$\sum_{j_{1}\geq n}\sum_{0\leq
j_{2}<n}\sum_{\lambda_{1}\in\Lambda_{j_{1}},\lambda_{2}\in\Lambda_{j_{2}}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}y_{\lambda_{1}}(t)y_{\lambda_{2}}(t)|I_{\lambda_{1},\lambda_{2}}[t,s]|$

are bounded by

$C|t-s|^{H_{1}+H_{2}-1}.$

###### Proof.

The proof is exactly the same as the one of Lemma 3.12 excepted that we use
Lemmata 3.33 and 3.34 instead of Lemmata 3.10 and 3.11 respectively. It leads
on one side us to consider the sums

$\left(\left(\sum_{j_{1}=0}^{n-1}2^{j_{1}(1-H_{1})}\sum_{j_{2}=n}^{+\infty}2^{\frac{1}{m}(j_{2}-n)}2^{-j_{2}H_{2}}\right)+2^{n(1-H_{1}-H_{2})}\right)$

which are bounded by

$C2^{n(1-H_{1}-H_{2})}\leq C|t-s|^{H_{1}+H_{2}-1}$

because $\frac{1}{m}<H_{2}$. On the other side, if we write
$I_{\lambda_{1},\lambda_{2}}$ for $I_{j_{1},j_{2}}^{k_{1},k_{2}}$ in Lemma
2.1, we have, from it,

$\displaystyle\begin{split}&\left|\sum_{\lambda_{1}\in\Lambda_{n-1}}\sum_{\lambda_{2}\in\Lambda_{n}[t,s]}y_{\lambda_{1}}(t)y_{\lambda_{2}}(t)I_{\lambda_{1},\lambda_{2}}\right|\\\
&\quad\leq
C2^{-n}\left|\sum_{l_{1}=0}^{+\infty}\sum_{\lambda_{1}\in\Lambda_{n-1}^{l_{1}}(t)}\sum_{\lambda_{2}\in\Lambda_{n}[t,s]}\frac{2^{l_{1}}}{(3+|2k_{1}-k_{2}|)^{4}}\right|\\\
&\quad\leq
C2^{-n}\left|\sum_{\lambda_{1}\in\Lambda_{n-1}}\sum_{\lambda_{2}\in\Lambda_{n}[t,s]}\frac{1}{(3+|2k_{1}-k_{2}|)^{3}}\right|\\\
&\quad\leq C2^{-n}.\end{split}$ (88)

∎

######  Lemma 3.36.

There exists a deterministic constant $C>0$ such that, for all $t,s\in(0,1)$
and $n\leq j_{1}\leq j_{2}$ the quantities (80) and (81) are bounded by

$C2^{\frac{1}{m}(j_{1}-n)}2^{\frac{1}{m}(j_{2}-n)}2^{-j_{2}}.$

###### Proof.

The proof is essentially the same as for Lemma 3.33 excepted that, now, as
$n\leq j_{1}\leq j_{2}$, we remark that if $x\in D_{j_{2}}^{l}(t)$ for a
$0\leq l\leq\frac{1}{m}(j_{2}-n)$ then $x\in D_{j_{1}}^{l^{\prime}}(t)$ for a
$0\leq l^{\prime}\leq\frac{1}{m}(j_{1}-n)$. ∎

######  Lemma 3.37.

There exists a deterministic constant $C>0$ such that, for all $t,s\in(0,1)$
and $n\leq j_{1}\leq j_{2}$ the quantities (84) and (85) are bounded by

$C2^{\frac{1}{m}(j_{1}-n)}2^{\frac{1}{m}(j_{2}-n)}2^{-j_{2}}.$

###### Proof.

The proof is essentially the same as for Lemma 3.34 and the only modification
is the same as in the proof of Lemma 3.36. ∎

This time, the bound for the random variables $\Xi_{j}(\lambda)$ are already
considered in the construction and we can directly go to the proof of the main
Proposition of this subsection.

###### Proof of Proposition 3.31.

If we apply the procedure with $m$ such that $1/m<\min\\{H_{1},H_{2}\\}$ and
$2/m<1-H_{1}-H_{2}$, we find an event $\Omega_{\text{slo}}$ of probability $1$
such that, for all $\omega\in\Omega_{\text{slo}}$, there is $\mu\in\mathbb{N}$
for which $S_{\text{low}}^{\mu}\cap(0,1)\neq\cmemptyset$. Then, if
$\omega\in\Omega_{\text{slo}}$ and $t\in
S_{\text{low}}^{\mu}(\omega)\cap(0,1)$ and $s\in(0,1)$, we write

$\displaystyle\begin{split}|R_{H_{1},H_{2}}^{\prime}(&t,\omega)-R_{H_{1},H_{2}}^{\prime}(s,\omega)|\\\
&\leq\left|\sum_{0\leq j_{1}<n}\sum_{0\leq
j_{2}<n}\sum_{\lambda_{1}\in\Lambda_{j_{1}},\lambda_{2}\in\Lambda_{j_{2}}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}\varepsilon_{\lambda_{1},\lambda_{2}}(\omega)I_{\lambda_{1},\lambda_{2}}[t,s]\right|\\\
\quad&+\left|\sum_{0\leq j_{1}<n}\sum_{j_{2}\geq
n}\sum_{\lambda_{1}\in\Lambda_{j_{1}},\lambda_{2}\in\Lambda_{j_{2}}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}\varepsilon_{\lambda_{1},\lambda_{2}}(\omega)I_{\lambda_{1},\lambda_{2}}[t,s]\right|\\\
&+\left|\sum_{j_{1}\geq n}\sum_{0\leq
j_{2}<n}\sum_{\lambda_{1}\in\Lambda_{j_{1}},\lambda_{2}\in\Lambda_{j_{2}}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}\varepsilon_{j_{1},j_{2}}^{k_{1},k_{2}}(\omega)I_{\lambda_{1},\lambda_{2}}[t,s]\right|\\\
&+\left|\sum_{j_{1}\geq n}\sum_{j_{2}\geq
n}\sum_{\lambda_{1}\in\Lambda_{j_{1}},\lambda_{2}\in\Lambda_{j_{2}}}2^{j_{1}(1-H_{1})}2^{j_{2}(1-H_{2})}\varepsilon_{\lambda_{1},\lambda_{2}}(\omega)I_{\lambda_{1},\lambda_{2}}[t,s]\right|.\end{split}$
(89)

As inequality (78) holds, we use Lemma 3.32 to bound the first sum, and Lemma
3.35 to bound the second and the third one. For the last sum, from inequality
(78) and Lemmata 3.36 and 3.37, it just remains us to find bound for the
random variables (54), (55) and (56)with $\ell\in\\{j,j+1\\}$ on
$\Omega_{\text{slo}}$. For (54) with $\ell=j$ , we have, as in (88) and then
(57)

$\displaystyle\left|\sum_{\lambda_{1}\in\Lambda_{j}^{<}(t,s)}\sum_{\lambda_{2}\in\Lambda_{j}[t,s]}\varepsilon_{\lambda_{1},\lambda_{2}}(\omega)I_{\lambda_{1},\lambda_{2}}\right|$
$\displaystyle\quad\leq
C2^{-j}2^{\frac{2}{m}(j-n)}\mu^{2}\left|\sum_{\lambda_{1}\in\Lambda_{j}^{<}(t,s)}\sum_{\lambda_{2}\in\Lambda_{j}[t,s]}\frac{1}{(3+|2k_{1}-k_{2}|)^{3}}\right|$
$\displaystyle\quad\leq C2^{-j}2^{\frac{2}{m}(j-n)}\mu^{2}.$

The same bound holds when we consider the sums over
$\lambda_{1}\in\Lambda_{j}^{>}(t,s)$ or $\lambda_{2}\in\Lambda_{j+1}[t,s]$,
i.e. for(54) and (55). Finally the construction and especially (77) insures us
that

$\left|\sum_{\lambda_{1}\in\Lambda_{j}[t,s]}\sum_{\lambda_{2}\in\Lambda_{j}[t,s]}\varepsilon_{\lambda_{1},\lambda_{2}}(\omega)I_{\lambda_{1},\lambda_{2}}\right|\leq
C(j-n+1)2^{\frac{-j-n}{2}}\mu.$

Therefore, the last term in (89) is bounded from above by

$\displaystyle C\mu^{2}\left(\sum_{j_{1}\geq
n}2^{j_{1}(1-H_{1})}2^{\frac{1}{m}(j_{1}-n)}\sum_{j_{2}\geq
j_{1}}2^{-j_{2}H_{2}}2^{\frac{1}{m}(j_{2}-n)}+\sum_{j\geq
n}2^{j(\frac{3}{2}-H_{1}-H_{2})}(j-n+1)2^{-\frac{n}{2}}\right)$
$\displaystyle\leq C\mu^{2}\left(\sum_{j_{1}\geq
n}2^{j_{1}(1-H_{1}-H_{2})}2^{\frac{2}{m}(j_{1}-n)}+2^{n(\frac{3}{2}-H_{1}-H_{2})}2^{-\frac{n}{2}}\right)$
$\displaystyle\leq C\mu^{2}2^{n(1-H_{1}-H_{2})}$ $\displaystyle\leq
C\mu^{2}|t-s|^{H_{1}+H_{2}-1}$

and thus inequality (72) holds. ∎

##  4 Lower bounds for wavelet leaders

In this section, we show that the limits (5) and (6) are strictly positive. In
[7], the authors used the independence of the increments of the Brownian
motion to bound from below its wavelet leaders. But, for the (generalized)
Rosenblatt process this nice feature is not met anymore. Nevertheless,
following an idea by Ayache in a close but different context555In [5], Ayache
does not consider wavelets at all but directly work on Wiener-Itô integrals
[5], we decompose the wavelet coefficients of the generalized Rosenblatt
process in two parts. We gain some independence properties in the first part
while the second is, in some sense, negligible compared to the first, see
Proposition 4.7 below. All along this section, $C$ stands for a deterministic
constant whose value may change from a line to another but does not depend on
any relevant quantities, and in order to ease notations we set

$C_{H_{1},H_{2}}:=\frac{1}{\Gamma\left(H_{1}-\frac{1}{2}\right)\Gamma\left(H_{2}-\frac{1}{2}\right)}$

and for $s,x_{1},x_{2}\in\mathbb{R}$

$f_{H_{1},H_{2}}(s,x_{1},x_{2})=\left(s-x_{1}\right)_{+}^{H_{1}-3/2}\left(s-x_{2}\right)_{+}^{H_{2}-3/2}$

Let $\Psi$ be a wavelet with compact support included in $[-N,N]$. Using
formula (12) at $t=k/2^{j}$, the wavelet coefficient $c_{j,k}$ of the
generalized Rosenblatt process is given by

$\displaystyle c_{j,k}=$
$\displaystyle\,\int_{-N}^{N}\left[R_{H_{1},H_{2}}\left(\frac{x+k}{2^{j}}\right)-R_{H_{1},H_{2}}\left(\frac{k}{2^{j}}\right)\right]\Psi(x)dx$
$\displaystyle=$
$\displaystyle\,c_{H_{1},H_{2}}\int_{-N}^{N}\Psi(x)\int_{\mathbb{R}^{2}}^{\prime}\int_{\frac{k}{2^{j}}}^{\frac{x+k}{2^{j}}}f_{H_{1},H_{2}}(s,x_{1},x_{2})\,ds\,dB(x_{1})\,dB(x_{2})\,dx$
$\displaystyle=$
$\displaystyle\,c_{H_{1},H_{2}}\int_{\mathbb{R}^{2}}^{\prime}\int_{-N}^{N}\Psi(x)\int_{\frac{k}{2^{j}}}^{\frac{x+k}{2^{j}}}f_{H_{1},H_{2}}(s,x_{1},x_{2})\,ds\,dx\,dB(x_{1})\,dB(x_{2})$
$\displaystyle=$
$\displaystyle\,\,c_{H_{1},H_{2}}\int_{A}^{\prime}\int_{-N}^{N}\Psi(x)\int_{\frac{k}{2^{j}}}^{\frac{x+k}{2^{j}}}f_{H_{1},H_{2}}(s,x_{1},x_{2})\,ds\,dx\,dB(x_{1})\,dB(x_{2})$

where $A:=\left]-\infty,\frac{k+N}{2^{j}}\right]^{2}$, because, as soon as
$x\in[-N,N]$ and $s\in[k2^{-J},(k+N)2^{-j}]$, $f(s,x_{1},x_{2})$ vanishes for
all $x_{1},x_{2}$ outside of $A$.

######  Definition 4.1.

Given an integer $M\geq 0$, $c_{j,k}$ can be written as following

$c_{j,k}=\widetilde{c_{j,k}}^{M}+\widecheck{c_{j,k}}^{M}$

where

$\displaystyle\widetilde{c_{j,k}}^{M}=c_{H_{1},H_{2}}\int_{\lambda_{j,k}^{M}}^{\prime}\int_{-N}^{N}\Psi(x)\int_{\frac{k}{2^{j}}}^{\frac{x+k}{2^{j}}}f_{H_{1},H_{2}}(s,x_{1},x_{2})\,ds\,dx\,dB(x_{1})\,dB(x_{2})$
(90)

with

$\lambda_{j,k}^{M}:=\left]\frac{k-NM}{2^{j}},\frac{k+N}{2^{j}}\right]^{2}$

and

$\widecheck{c_{j,k}}^{M}=c_{H_{1},H_{2}}\int_{A\setminus\lambda_{j,k}^{M}}^{\prime}\int_{-N}^{N}\Psi(x)\int_{\frac{k}{2^{j}}}^{\frac{x+k}{2^{j}}}f_{H_{1},H_{2}}(s,x_{1},x_{2})\,ds\,dx\,dB(x_{1})\,dB(x_{2}).$

######  Remark 4.2.

Let us highlight the fact that using time change of variable for Wiener-Itô
integrals [38, Theorem 8.5.7], for all $j,k$, we have
$\widetilde{c_{j,k}}^{M}$ is equal in law to the random variable

$c_{H_{1},H_{2}}2^{-j(H_{1}+H_{2}-1)}\int_{I_{M}}^{\prime}\int_{-N}^{N}\psi(x)\int_{0}^{x}f_{H_{1},H_{2}}(s,x_{1},x_{2})\,dsdx\,dB(x_{1})dB(x_{2})$

with $I_{M}=(-MN,N]^{2}$, while $\widecheck{c_{j,k}}^{M}$ is equal in law to
the random variable

$c_{H_{1},H_{2}}2^{-j(H_{1}+H_{2}-1)}\int_{I_{M}^{\prime}}^{\prime}\int_{-N}^{N}\psi(x)\int_{0}^{x}f_{H_{1},H_{2}}(s,x_{1},x_{2})\,dsdx\,dB(x_{1})dB(x_{2})$

with $I^{\prime}_{M}=(-\infty,N]^{2}\setminus(-MN,N]^{2}$.

######  Definition 4.3.

For all $(j,k)\in\mathbb{N}\times\mathbb{Z}$ and $M\in\mathbb{N}$ we define
the random variables

$\displaystyle\widetilde{\varepsilon_{j,k}}^{M}:=\dfrac{\widetilde{c_{j,k}}^{M}}{2^{-j(H_{1}+H_{2}-1)}}\text{
and
}\widecheck{\varepsilon_{j,k}}^{M}:=\dfrac{\widecheck{c_{j,k}}^{M}}{2^{-j(H_{1}+H_{2}-1)}}.$

######  Remark 4.4.

Note that $\widetilde{\varepsilon_{j,k}}^{M}$ and
$\widetilde{\varepsilon_{j^{\prime},k^{\prime}}}^{M}$ are independent when

$\displaystyle\lambda_{j,k}^{M}\cap\lambda_{j^{\prime},k^{\prime}}^{M}=\cmemptyset.$
(91)

Indeed, if $(f_{j})_{j}$ is a sequence of real-valued step functions on
$\mathbb{R}^{2}\setminus\\{(x,x)\,:\,x\in\mathbb{R}\\}$ which converge to the
integrand with respect to $dB(x_{1})dB(x_{2})$ in (90) then
$\int_{\mathbb{R}^{2}}^{\prime}f_{j}(x_{1},x_{2})\,dB(x_{1})dB(x_{2})$ is a
polynomial function of a finite number of increments $B(t_{2})-B(t_{1})$ of
the Brownian motion for some $t_{1},t_{2}\in\lambda_{j,k}^{M}$. Thus
$\widetilde{\varepsilon_{j,k}}^{M}$ is measurable with respect to the
$\sigma$-algebra generated by these increments

$\sigma_{j,k}^{M}:=\sigma\left(\\{B(t_{2})-B(t_{1})\,:\,t_{1},t_{2}\in\lambda_{j,k}^{M}\\}\right).$

Using the independence of the increments of the Brownian motion, one concludes
that $\sigma_{j,k}^{M}$ and $\sigma_{j^{\prime},k^{\prime}}^{M}$ are
independent as soon as condition (91) is met and so the same holds for
$\widetilde{\varepsilon_{j,k}}^{M}$ and
$\widetilde{\varepsilon_{j,k^{\prime}}}^{M}$. Moreover,
$\widetilde{\varepsilon_{j_{1},k_{1}}}^{M},\dots,\widetilde{\varepsilon_{j_{n},k_{n}}}^{M}$
are independent when the following condition is satisfied

$\displaystyle\
\lambda_{j_{i},k_{i}}^{M}\cap\lambda_{j_{l},k_{l}}^{M}=\cmemptyset\mbox{ for
all }1\leq i<l\leq n.$ (92)

This leads to defining the following condition.

######  Definition 4.5.

Let $n\geq 2$. We say $\lambda_{j_{1},k_{1}},\dots,\lambda_{j_{n},k_{n}}$
satisfy condition $(C_{M})$ if (92) is satisfied.

From Remark 4.2, we know that
$(\widetilde{\varepsilon_{j,k}}^{M})_{\lambda\in\Lambda}$ is a family of
identically distributed second order Wiener chaos random variables. Moreover,
$\widetilde{\varepsilon_{j_{1},k_{1}}}^{M},\dots,\widetilde{\varepsilon_{j_{n},k_{n}}}^{M}$
are independent as soon as $\lambda_{j_{1},k_{1}},\dots,\lambda_{j_{n},k_{n}}$
satisfies $(C_{M})$. The following proposition provides a lower bound
(independent of $M$) for the tail behavior of the random variable
$\widetilde{\varepsilon_{j,k}}^{M}$.

######  Proposition 4.6.

Let $M\in\mathbb{N}$ and $y\in\mathbb{R}^{+}$. If $M$ and $y$ are large
enough, then the exists a deterministic constant $c_{2}>0$ (independent of
$M$) such that

$\mathbb{P}\left(|\widetilde{\varepsilon_{j,k}}^{M}|>y\right)\geq\exp{\left(-c_{2}y\right)}$
(93)

for all $(j,k)\in\mathbb{N}\times\mathbb{Z}$

###### Proof.

Fix $y\in\mathbb{R}^{+}$ (large enough). Our aim is to prove the existence of
lower bound for
$\mathbb{P}\left(|\widetilde{\varepsilon_{\lambda}}^{M}|>y\right)$ which is
independent of $M$. To this end, we start by proving the following lemma

######  Lemma 4.7.

There exist three strictly positive deterministic constants
$C_{\Psi,H_{1},H_{2}}$, $C^{\prime}_{\Psi,H_{1},H_{2}}$ and
$C^{*}_{\Psi,H_{1},H_{2}}$ such that for all
$(j,k)\in\mathbb{N}\times\mathbb{Z}$ and $M\geq 2$ one has

$\displaystyle C_{\Psi,H_{1},H_{2}}2^{-j(H_{1}+H_{2}-1)}\leq$
$\displaystyle\left\lVert\widetilde{c_{j,k}}^{M}\right\rVert_{L^{2}(\Omega)}\leq
C^{\prime}_{\Psi,H_{1},H_{2}}2^{-j(H_{1}+H_{2}-1)}$
$\displaystyle\left\lVert\widecheck{c_{j,k}}^{M}\right\rVert_{L^{2}(\Omega)}\leq
C^{*}_{\Psi,H_{1},H_{2}}2^{-j(H_{1}+H_{2}-1)}M^{\max\\{H_{1},H_{2}\\}-1}$

###### Proof.

Let us assume, w.l.o.g. that $H_{1}\geq H_{2}$. We define the functions

$\Phi_{1}\,:\,(x_{1},x_{2})\mapsto\int_{-N}^{N}\Psi(x)\int_{0}^{x}f_{H_{1},H_{2}}(s,x_{1},x_{2})ds\,dx,$
$\Phi_{2}\,:\,(x_{1},x_{2})\mapsto\int_{-N}^{N}\Psi(x)\int_{0}^{x}f_{H_{1},H_{2}}(s,x_{2},x_{1})ds\,dx,$

and the symmetric function666The function $\Phi$ is in the fact the
symmetrization of $\Phi_{1}$.

$\Phi=\frac{1}{2}\left(\Phi_{1}+\Phi_{2}\right).$

By Remark 4.2 we have, using the “Wiener isometry” 777For $f$ a symmetric
function in $L^{2}(\mathbb{R}^{2})$ , and $I_{2}(f)$ the second order Wiener-
Itô integral of $f$. One has
$\mathbb{E}(I_{m}(f))^{2}=2!||f||_{L^{2}(\mathbb{R}^{2})}$. [42, Section 5],

$\displaystyle\left\lVert\widetilde{c_{j,k}}^{M}\right\rVert_{L^{2}(\Omega)}=\sqrt{2}c_{H_{1},H_{2}}2^{-j(H_{1}+H_{2}-1)}\left\|\Phi\right\|_{L^{2}(I_{M})}$

and thus it suffices to take

$\displaystyle
C_{\Psi,H_{1},H_{2}}:=\sqrt{2}c_{H_{1},H_{2}}\left\|\Phi\right\|_{L^{2}([-N,N]^{2})}$
$\displaystyle
C^{\prime}_{\Psi,H_{1},H_{2}}:=\sqrt{2}c_{H_{1},H_{2}}\left\|\Phi\right\|_{L^{2}((-\infty,N]^{2})}$

Now, still using Remark 4.2 and “Wiener isometry” we have

$\displaystyle\left\lVert\widecheck{c_{j,k}}^{M}\right\rVert_{L^{2}(\Omega)}$
$\displaystyle=\sqrt{2}c_{H_{1},H_{2}}2^{-j(H_{1}+H_{2}-1)}\left\|\Phi\right\|_{L^{2}(I_{M}^{\prime})}$
$\displaystyle\leq\sqrt{2}c_{H_{1},H_{2}}2^{-j(H_{1}+H_{2}-1)}\left\|\Phi_{1}\right\|_{L^{2}(I_{M}^{\prime})}.$

Also as

$\displaystyle
I^{\prime}_{M}=(-\infty,N]^{2}\setminus(-MN,N]^{2}\subset\mathbb{R}\times\left(-\infty,-MN\right]\bigcup\left(-\infty,-MN\right]\times\mathbb{R},$

we write

$\displaystyle\left\|\Phi_{1}\right\|_{L^{2}(I_{M}^{\prime})}^{2}$
$\displaystyle=\int_{I_{M}^{\prime}}\left|\int_{-N}^{N}\Psi(x)\int_{0}^{x}(s-x_{1})_{+}^{H_{1}-\frac{3}{2}}(s-x_{2})_{+}^{H_{2}-\frac{3}{2}}ds\,dx\right|^{2}dx_{1}\,dx_{2}$
$\displaystyle\leq\int_{I_{M}^{\prime}}\left(\int_{-N}^{N}|\Psi(x)|\int_{[0,x]}(s-x_{1})_{+}^{H_{1}-\frac{3}{2}}(s-x_{2})_{+}^{H_{2}-\frac{3}{2}}ds\,dx\right)^{2}dx_{1}\,dx_{2}$
$\displaystyle\leq\int_{\mathbb{R}}\int_{-\infty}^{-MN}\left(\int_{-N}^{N}|\Psi(x)|\int_{[0,x]}(s-x_{1})_{+}^{H_{1}-\frac{3}{2}}(s-x_{2})_{+}^{H_{2}-\frac{3}{2}}ds\,dx\right)^{2}dx_{1}\,dx_{2}$
$\displaystyle+\int_{\mathbb{R}}\int_{-\infty}^{-MN}\left(\int_{-N}^{N}|\Psi(x)|\int_{[0,x]}(s-x_{1})_{+}^{H_{1}-\frac{3}{2}}(s-x_{2})_{+}^{H_{2}-\frac{3}{2}}ds\,dx\right)^{2}dx_{2}\,dx_{1}.$

Let us deal with the first term in the last sum, the second one can be treated
similarly by permuting the roles of $H_{1}$ and $H_{2}$ as well as $x_{1}$ and
$x_{2}$. As the function $y\mapsto y^{H_{1}-3/2}$ is decreasing, one gets

$\displaystyle\int_{\mathbb{R}}\int_{-\infty}^{-MN}\left(\int_{-N}^{N}|\Psi(x)|\int_{[0,x]}(s-x_{1})_{+}^{H_{1}-\frac{3}{2}}(s-x_{2})_{+}^{H_{2}-\frac{3}{2}}ds\,dx\right)^{2}dx_{1}\,dx_{2}$
$\displaystyle\leq\left(\int_{-\infty}^{-MN}(-N-x_{1})^{2H_{1}-3}\,dx_{1}\right)\times\int_{\mathbb{R}}\left(\int_{-N}^{N}|\Psi(x)|\left|\int_{[0,x]}\left(s-x_{2}\right)_{+}^{H_{2}-3/2}ds\right|dx\right)^{2}dx_{2}.$

Concerning the first integral, we have, as $M\geq 2$

$\displaystyle\int_{-\infty}^{-NM}\left(-N-x_{1}\right)^{2H_{1}-3}dx_{1}$
$\displaystyle=\dfrac{1}{2-2H_{1}}\left(NM-N\right)_{+}^{2H_{1}-2}$
$\displaystyle=\dfrac{1}{2-2H_{1}}N^{2H_{1}-2}(M-1)^{2H_{1}-2}$
$\displaystyle\leq c\cdot M^{2H_{1}-2}$

while, using again the “Wiener isometry”,

$\displaystyle\int_{\mathbb{R}}$
$\displaystyle\left(\int_{-N}^{N}|\Psi(x)|\left|\int_{[0,x]}\left(s-x_{2}\right)_{+}^{H_{2}-3/2}\right|dx\right)^{2}dx_{2}$
$\displaystyle\leq$
$\displaystyle\,2N\left\lVert\Psi\right\rVert_{\infty}\sup_{x\in[-N,N]}\int_{\mathbb{R}}\left|\int_{[0,x]}\left(s-x_{2}\right)_{+}^{H_{2}-3/2}ds\right|^{2}dx_{2}$
$\displaystyle=$
$\displaystyle\,2N\left\lVert\Psi\right\rVert_{\infty}\sup_{x\in[-N,N]}\mathbb{E}\left[\left|B_{H_{2}}\left(x\right)-B_{H_{2}}\left(0\right)\right|^{2}\right]$
$\displaystyle\leq$
$\displaystyle\,2N\left\lVert\Psi\right\rVert_{\infty}\sup_{x\in[-N,N]}C_{H_{2}}\left(|x|\right)^{2H_{2}}\leq
c,$

where $B_{H_{2}}$ denotes the fractional Brownian motion with parameter
$H_{2}$. As a result, there exists a positive constant
$C^{*}_{\Psi,H_{1},H_{2}}$ such that, as we suppose $H_{1}\geq H_{2}$, one has

$\displaystyle\left\lVert\widecheck{c_{j,k}}^{M}\right\rVert_{L^{2}(\Omega)}\leq
C^{*}_{\Psi,H_{1},H_{2}}2^{-j(H_{1}+H_{2}-1)}M^{H_{1}-1}.$

∎

By Lemma 4.7, one can remark that as $M\rightarrow+\infty$,
$(\widetilde{\varepsilon_{j,k}}^{M})_{M}$ converges in $L^{2}(\Omega)$ to the
random variable

$\varepsilon_{j,k}:=\dfrac{c_{j,k}}{2^{-j(H_{1}+H_{2}-1)}}$

with, for all $M\in\mathbb{N}$,

$\varepsilon_{j,k}-\widetilde{\varepsilon_{j,k}}^{M}=\widecheck{\varepsilon_{j,k}}^{M}.$

By Theorem 3.15, there exists a constant $c_{1}>0$ such that, for all
$\lambda\in\Lambda$ and $y$ sufficiently large

$\displaystyle\mathbb{P}\left(\lvert\varepsilon_{j,k}\rvert\geq
y\right)\geq\exp{\left(-c_{1}y\right)}.$

Then, for all $M\in\mathbb{N}$, we have, for all such $\lambda$ and $y$

$\displaystyle\mathbb{P}\left(\lvert\widetilde{\varepsilon_{j,k}}^{M}\rvert\geq
y\right)$
$\displaystyle\geq\mathbb{P}\left(\\{\lvert\widetilde{\varepsilon_{j,k}}^{M}\rvert\geq
y\\}\cap\\{|\widecheck{\varepsilon_{j,k}}^{M}|\leq y\\}\right)$
$\displaystyle\geq\mathbb{P}\left(\\{\lvert\varepsilon_{j,k}\rvert-|\widecheck{\varepsilon_{j,k}}^{M}|\geq
y\\}\cap\\{|\widecheck{\varepsilon_{j,k}}^{M}|\leq y\\}\right)$
$\displaystyle\geq\mathbb{P}\left(\\{\lvert\varepsilon_{j,k}\rvert\geq
2y\\}\cap\\{|\widecheck{\varepsilon_{j,k}}^{M}|\leq y\\}\right)$
$\displaystyle\geq\mathbb{P}\left(\lvert\varepsilon_{j,k}\rvert\geq
2y\right)-\mathbb{P}\left((|\widecheck{\varepsilon_{j,k}}^{M}|>y\right).$

Using Lemma 4.7 and Theorem 3.14 one has

$\displaystyle\mathbb{P}\left(|\widecheck{\varepsilon_{j,k}}^{M}|>y\right)$
$\displaystyle\leq\mathbb{P}\left((|\widecheck{c_{j,k}}^{M}|>y\left\lVert\widecheck{c_{j,k}}^{M}\right\rVert_{L^{2}(\Omega)}(C^{*}_{\Psi,H_{1},H_{2}})^{-1}M^{1-\max\\{H_{1},H_{2}\\}}\right)$
$\displaystyle\leq\exp(-\overset{\star}{C}(C^{*}_{\Psi,H_{1},H_{2}})^{-1}M^{1-\max\\{H_{1},H_{2}\\}}y).$

Thus, if $M$ is large enough, one has, as $1-\max\\{H_{1},H_{2}\\}>0$,

$\exp(-\overset{\star}{C}(C^{*}_{\Psi,H_{1},H_{2}})^{-1}M^{1-\max\\{H_{1},H_{2}\\}}y)\leq\frac{1}{2}\exp{\left(-2c_{1}y\right)}$

which gives that, for all large enough $y$, one gets

$\mathbb{P}\left(|\widetilde{\varepsilon_{j,k}}^{M}|>y\right)\geq\exp{\left(-c_{2}y\right)}$
(94)

with $c_{2}:=2c_{1}$. In the sequel, we will implicitly always consider such
large enough $M$. ∎

In the following two subsections, Lemmata 4.9 and 4.12 follow the lines of
Lemmata 3.6 and 3.8 in [7] respectively, with some subtle modifications as the
authors in [7] deal with $\mathcal{N}(0,1)$ random variables while, here, we
focus on random variables in the second order Wiener chaos that depend on the
parameter $M$. For the sake of completeness and clarity, we write the proofs
in full details.

###  4.1 Ordinary Points

In this section our aim is to prove the following proposition.

######  Proposition 4.8.

There exists $\Omega^{*}_{1}\subset\Omega$ with probability 1 such that for
all $\omega\in\Omega^{*}_{1}$ and Lebesgue almost every $t\in(0,1)$ one has

$\displaystyle\limsup_{j\rightarrow+\infty}\dfrac{d_{j}(t,\omega)}{2^{-j(H_{1}+H_{2}-1)}\log
j}>0.$ (95)

To this end, as a first step, let us state the following lemma concerning the
random variable $\widetilde{\varepsilon_{\lambda}}^{M}$. If
$\lambda=\lambda_{j,k}$ is a dyadic interval and $m\in\mathbb{N}$,
$S_{\lambda,m}=S_{j,k,m}$ stands for the finite set of cardinality $2^{m}$
whose elements are the dyadic intervals of scale $j+m$ included in
$\lambda_{j,k}$, formally speaking

$\displaystyle
S_{j,k,m}:=\\{\lambda\in\Lambda_{j+m}\,:\,\lambda\subset\lambda_{j,k}\\}$

######  Lemma 4.9.

There is a deterministic constant $C>0$ such that the following holds: for all
$M\in\mathbb{N}$ and for all $t\in(0,1)$, there exists
$\Omega_{t,1}\subset\Omega$ with probability 1 such that for all
$\omega\in\Omega_{t,1}$ there are infinitely many $j\in\mathbb{N}$ such that

$\displaystyle\max_{\begin{matrix}\lambda^{\prime}\in
S_{\lambda,\lfloor\log_{2}(NM)\rfloor+2}\\\ \lambda\in
3\lambda_{j}(t)\end{matrix}}\left|\widetilde{\varepsilon_{\lambda^{\prime}}}^{M}(\omega)\right|\geq
C\log j.$

###### Proof.

Let us fix $t\in(0,1)$ and $j\in\mathbb{N}$. For any $\lambda\in
S_{j,k_{j}(t),m}$, there exists a unique decreasing finite sequence
$(I_{n})_{0\leq n\leq m}$ of decreasing dyadic intervals in the sense of
inclusion such that $I_{0}=\lambda_{j,k_{j}(t)}$, $I_{m}=\lambda$ and
$I_{n}\in S_{j,k_{j}(t),n}$. Then, define the sequence $(T_{n})_{1\leq n\leq
m}$ of unique dyadic intervals such that $I_{n-1}=I_{n}\cup T_{n}$. Note that
for all $1\leq n\leq m$, $T_{n}\in 3I_{n}$. Moreover, as $(I_{n})_{0\leq n\leq
m}$ is decreasing, $(T_{n})_{1\leq n\leq m}$ are pairwisely disjoint.
Furthermore, for every $n\in\\{1,...,m\\}$, there exist
$T^{\prime}_{n}=\lambda_{j_{n},k_{n}}\in S_{T_{n},\lfloor\log_{2}NM\rfloor+2}$
such that

$\displaystyle\left(\dfrac{k_{n}-NM}{2^{j_{n}}},\dfrac{k_{n}+N}{2^{j_{n}}}\right)\subset
T_{n}.$

As a consequence, the associated random variables
$\left(\widetilde{\varepsilon_{T^{\prime}_{n}}}^{M}\right)_{1\leq n\leq m}$
are independent as the dyadic intervals $(T^{\prime}_{n})_{1\leq n\leq m}$
satisfies condition $(C_{M})$ in Definition 4.5. Next, for a constant $C>0$ to
be chosen later, we set

$\displaystyle\mathcal{E}_{j,m}(t)=\left\\{\omega\in\Omega\,:\,\max_{1\leq
n\leq m}\left|\widetilde{\varepsilon_{T^{\prime}_{n}}}^{M}\right|\geq
C\log(2m)\right\\}.$

Note that, as the random variables
$\left(\widetilde{\varepsilon_{T^{\prime}_{n}}}^{M}\right)_{1\leq n\leq m}$
are independent,

$\displaystyle\mathbb{P}\left(\mathcal{E}_{j,m}(t)\right)=1-\prod_{n=1}^{m}\mathbb{P}\left(\left|\widetilde{\varepsilon_{T^{\prime}_{n}}}^{M}\right|<C\log(2m)\right)$

Recalling (94), and the fact that $\log(1-x)\leq-x$ if $x\in(0,1)$, one gets,
for $m$ is large enough,

$\displaystyle\mathbb{P}\left(\mathcal{E}_{j,m}(t)\right)\geq$
$\displaystyle\,1-(1-\exp(-Cc_{2}\log(2m))^{m}$ $\displaystyle=$
$\displaystyle\,1-\left(1-\left(\dfrac{1}{2m}\right)^{Cc_{2}}\right)^{m}$
$\displaystyle\geq$
$\displaystyle\,1-\exp\left(\dfrac{m}{(2m)^{Cc_{2}}}\right)$ $\displaystyle=$
$\displaystyle\,1-\exp\left(\dfrac{m^{1-Cc_{2}}}{2^{Cc_{2}}}\right).$

Finally, choosing $C$ such that $0<Cc_{2}<1$, one obtain that

$\displaystyle\sum_{p\in\mathbb{N}}\mathbb{P}\left(\mathcal{E}_{2^{p},2^{p}}(t)\right)=+\infty.$

Knowing that the events $\mathcal{E}_{2^{p},2^{p}}(t)$ are independent for all
$p\in\mathbb{N}$, one concludes using Borel-Cantelli Lemma that

$\displaystyle\mathbb{P}\left(\limsup_{m\rightarrow+\infty}\mathcal{E}_{2^{m},2^{m}}(t)\right)=1$

It follows that for a fixed $t\in\mathbb{R}$, almost surely, there are
infinitely many $j\in\mathbb{N}$ such that

$\displaystyle\max_{\begin{matrix}\lambda^{\prime}\in
S_{\lambda,\lfloor\log_{2}NM\rfloor+2}\\\ \lambda\in
3\lambda_{j}(t)\end{matrix}}\left|\widetilde{\varepsilon_{\lambda^{\prime}}}^{M}(\omega)\right|\geq
C\log j.$

∎

Concerning the “non-independent part” of the wavelet coefficients, one can
state the following Lemma.

######  Lemma 4.10.

There is a deterministic constant $C^{\prime}>0$ such that, for all
$M\in\mathbb{N}$ and for all $t\in(0,1)$, there exists
$\Omega_{t,2}\subset\Omega$ with probability 1 such that for all
$\omega\in\Omega_{t,2}$ there exists $J\in\mathbb{N}$ such that, for all
$j\geq J$,

$\displaystyle\max_{\begin{matrix}\lambda^{\prime}\in
S_{\lambda,\lfloor\log_{2}(NM)\rfloor+2}\\\ \lambda\in
3\lambda_{j}(t)\end{matrix}}\left|\widecheck{\varepsilon_{\lambda^{\prime}}}^{M}(\omega)\right|\leq
C^{\prime}M^{\max\\{H_{1},H_{2}\\}-1}\log j.$

###### Proof.

Let us fix $t\in(0,1)$. For any $C^{\prime}>0$, for all $j$ sufficiently large
and $\lambda\in 3\lambda_{j}(t)$, we have, by Theorem 3.14,

$\displaystyle\mathbb{P}$ $\displaystyle\left(\exists\lambda^{\prime}\in
S_{\lambda,\lfloor\log_{2}NM\rfloor+2}\,:\,\left|\widecheck{\varepsilon_{\lambda^{\prime}}}^{M}\right|\geq
C^{\prime}M^{\max\\{H_{1},H_{2}\\}-1}\log j\right)$
$\displaystyle\leq\sum_{\lambda^{\prime}\in
S_{\lambda,\lfloor\log_{2}NM\rfloor+2}}\mathbb{P}\left(\left|\widecheck{\varepsilon_{\lambda^{\prime}}}^{M})\right|\geq
C^{\prime}M^{\max\\{H_{1},H_{2}\\}-1}\log j\right)$
$\displaystyle\leq\sum_{\lambda^{\prime}\in
S_{\lambda,\lfloor\log_{2}NM\rfloor+2}}\mathbb{P}\left(\left|\widecheck{\varepsilon_{\lambda^{\prime}}}^{M}\right|\geq
C^{\prime}(C^{*}_{\Psi,H_{1},H_{2}})^{-1}\|\widecheck{\varepsilon_{\lambda^{\prime}}}^{M}\|_{L^{2}(\Omega)}\log
j\right)$ $\displaystyle\leq
4NM\exp(-\overset{\star}{C}C^{\prime}(C^{*}_{\Psi,H_{1},H_{2}})^{-1}\log j)$

Thus, for $C^{\prime}>C^{*}_{\Psi,H_{1},H_{2}}/\overset{\star}{C}$, the
conclusion follows by Borel-Cantelli Lemma. ∎

###### Proof of Proposition 4.8.

The constant $C$ and $C^{\prime}$ of Lemmata 4.9 and 4.10 being deterministic
and independent of $M$, on can choose $M$ large enough such that

$C-C^{\prime}M^{\max\\{H_{1},H_{2}\\}-1}>0.$

Let us fix $t\in(0,1)$ and consider $\omega\in\Omega_{t,1}\cap\Omega_{t,2}$,
where the events, of probability $1$, $\Omega_{t,1}$ and $\Omega_{t,2}$ are
given by the same Lemmata. For all $J\in\mathbb{N}$, by Lemma 4.9, there exist
$j\geq J$ and $\lambda^{\prime}(j)\subseteq 3\lambda_{j}(t)$ of scale
$j^{\prime}=j+\lfloor\log NM\rfloor+2$ such that

$\displaystyle\left|\widetilde{c_{\lambda^{\prime}(j)}}^{M}(\omega)\right|\geq
C2^{-j^{\prime}(H_{1}+H_{2}-1)}\log j.$

If $J$ is large enough, we also have, for all such $j\geq J$, by Lemma 4.10,

$\displaystyle\left|\widecheck{c_{\lambda^{\prime}(j)}}^{M}(\omega)\right|\leq
C^{\prime}M^{\max\\{H_{1},H_{2}\\}-1}2^{-j^{\prime}(H_{1}+H_{2}-1)}\log j.$

From this we deduce that

$\displaystyle d_{j}(t,\omega)\geq$
$\displaystyle\left|c_{\lambda^{\prime}(j)}(\omega)\right|$
$\displaystyle\geq$
$\displaystyle\left|\widetilde{c_{\lambda^{\prime}(j)}}^{M}(\omega)\right|-\left|\widecheck{c_{\lambda^{\prime}(j)}}^{M}(\omega)\right|$
$\displaystyle\geq$ $\displaystyle 2^{-j^{\prime}(H_{1}+H_{2}-1)}\log
j\left(C-C^{\prime}M^{\max\\{H_{1},H_{2}\\}-1}\right)$ $\displaystyle\geq$
$\displaystyle 2^{-j(H_{1}+H_{2}-1)}(4NM)^{1-H_{1}-H_{2}}\log
j\left(C-C^{\prime}M^{\max\\{H_{1},H_{2}\\}-1}\right)$

Therefore, (95) holds true for all $t\in(0,1)$ and
$\omega\in\Omega_{t,1}\cap\Omega_{t,2}$. The conclusion follows then from
Fubini Theorem. ∎

###  4.2 Rapid Points

In this section our aim is to prove the following proposition.

######  Proposition 4.11.

There exists $\Omega^{*}_{2}\subset\Omega$ with probability 1 such that, for
all $\omega\in\Omega^{*}_{2}$, there exist $t\in(0,1)$ such that

$\displaystyle\limsup_{j\rightarrow+\infty}\dfrac{d_{j}(t,\omega)}{2^{-j(H_{1}+H_{2}-1)}j}>0.$
(96)

As in the previous subsection, we start by working with the random variables
$\widetilde{\varepsilon_{\lambda}}^{M}$.

######  Lemma 4.12.

There exists a deterministic constant $C>0$ such that for all $M$ there is
$\Omega_{2}\subset\Omega$ with probability 1 such that for all
$\omega\in\Omega_{2}$ there exist $t\in(0,1)$ such that

$\displaystyle\limsup_{j\rightarrow+\infty}\dfrac{\left|\widetilde{\varepsilon_{\lambda_{j}(t)}}^{M}(\omega)\right|}{j}\geq
C.$ (97)

###### Proof.

Let us fix $a\in(0,1)$ and $C>0$ to be chosen later on. For every
$(j,l)\in\mathbb{N}\times\left\\{0,\dots,\lfloor
2^{j(1-a)}\rfloor-1\right\\}$, we set

$S_{j,l}^{M}=\left\\{l\lfloor 2^{aj}/(2NM)\rfloor,\dots,(l+1)\lfloor
2^{aj}/(2NM)\rfloor-1\right\\}$

and consider the event

$\displaystyle\mathcal{E}_{j,l}^{M}=\left\\{\omega\in\Omega\,:\,\max_{k\in
S_{j,l}^{M}}\left|\widetilde{\varepsilon_{j,2kNM}}^{M}(\omega)\right|\geq
Cj\right\\}$

Let $j_{0}$ be the smallest integer such that $\lfloor 2^{aj}/(2NM)\rfloor\geq
1$. If we assume that

$\displaystyle\Omega^{*}_{2}=\bigcup_{J\geq j_{0}}\bigcap_{j\geq
J}\bigcap_{l\in\left\\{0,\dots,\lfloor
2^{j(1-a)}\rfloor-1\right\\}}\mathcal{E}_{j,l}^{M}$ (98)

is an event of probability 1 and we consider $\omega\in\Omega^{*}_{2}$. For
every $j\geq j_{0}$, denote by

$\displaystyle
G_{j}^{M}(\omega):=\left(k\in\\{0,\dots,2^{j}-1\\}\,:\,\left|\widetilde{\varepsilon_{j,k}}^{M}(\omega)\right|\geq
Cj\right).$ (99)

Moreover, for every $n\geq j_{0}$, one considers

$\displaystyle O_{n}^{M}(\omega):=\bigcup_{j\geq
n}U_{j}^{M}(\omega),\quad\mbox{where }U_{j}^{M}(\omega):=\bigcup_{k\in
G_{j}^{M}(\omega)}\left(\frac{k}{2^{j}},\frac{k+1}{2^{j}}\right).$ (100)

If one proves that $O_{n}^{M}(\omega)$ is dense in $(0,1)$, then by Baire’s
theorem the set $\cap_{n\geq j_{0}}O_{n}^{M}(\omega)$ is non-empty and let t
be an element of this set. Then for every $n\geq j_{0}$, there is $j\geq n$
such that
$\left|\widetilde{\varepsilon_{\lambda_{j}(t)}}^{M}(\omega)\right|\geq Cj$,
and so desired statement (97) is true.
We still have to prove two points:

  1. 1.

$O_{n}^{M}(\omega)$ is dense in $(0,1)$.

  2. 2.

$\Omega_{2}^{*}$ is an event of probability 1.

Indeed, starting with statement 1, consider $t\in(0,1)$, $j\geq j_{0}$ and $k$
such that $\lambda_{j}(t)=\lambda_{j,k}$. Then, we have two cases:

  1. Case 1 : There is $l\in\\{0,\dots,\lfloor 2^{j(1-a)}\rfloor-1\\}$ such that

$\displaystyle k\in\left\\{l\lfloor 2^{aj}\rfloor,\dots,(l+1)\lfloor
2^{aj}\rfloor-1\right\\}$

Using (98) and (99), there is $k^{\prime}\in\\{l\lfloor
2^{aj}/(2NM)\rfloor,\dots,(l+1)\lfloor 2^{aj}/(2NM)\rfloor-1\\}$ such that
$2k^{\prime}NM\in G_{j}(\omega)$. Then, by (100),

$\displaystyle\left(\dfrac{2NMk^{\prime}}{2^{j}},\dfrac{2NMk^{\prime}+1}{2^{j}}\right)\subset
O_{n}^{M}(\omega).$

which is at is at most $2^{-j}\left(\lfloor 2^{aj}\rfloor+2NM\lfloor
2^{aj}/(2NM)\rfloor\right)$ from $t$. Finally, we get that $t$ is at a
distance at most $22^{j(a-1)}$ of $U_{j}^{M}(\omega)$.

  2. Case 2 : $k\in\\{\lfloor 2^{j(1-a)}\rfloor\lfloor 2^{ja}\rfloor,\dots,2^{j}-1\\}$. Again by (98) and (99), there is $k^{\prime}\in S_{j,l}^{M}$ such that $2k^{\prime}NM\in G_{j}^{M}(\omega)$, and similarly, we get that $t$ is at a distance at most $c2^{j(a-1)}$ of $U_{j}^{M}(\omega)$, for some constant $c>0$ depending only on $N$, $M$ and $a$.

Finally, in both cases $t$ is at a distance at most $c2^{j(a-1)}$, and so the
density follows.
Now for statement 2, in order to prove that $\Omega^{*}_{2}$ has a probability
1, it is enough to prove that

$\displaystyle\mathbb{P}\left(\mathcal{C}\left(\bigcap_{l\in\left\\{0,\dots,\lfloor
2^{j(1-a)}\rfloor-1\right\\}}\mathcal{E}_{j,l}^{M}\right)\right)$ (101)

is the general term of a convergent series, then the result follows by Borel-
Cantelli Lemma. Note that the variables
$\widetilde{\varepsilon_{j,2NMk}}^{M}(\omega)$, $k\in S_{j,l}^{M}$ and
$l\in\\{0,\ldots,\lfloor 2^{j(1-a)}\rfloor-1\\}$, are independent because for
every $k\neq k^{\prime}$, $|2NMk-2NMk^{\prime}|\geq 2NM$ and so
$\lambda_{j,2MNk}^{M}\cap\lambda_{j,2MNk^{\prime}}^{M}=\cmemptyset$.
Consequently, one has

$\displaystyle\mathbb{P}\left(\mathcal{C}\left(\bigcap_{l\in\left\\{0,\dots,\lfloor
2^{j(1-a)}\rfloor-1\right\\}}\mathcal{E}_{j,l}^{M}\right)\right)$
$\displaystyle=$ $\displaystyle
1-\mathbb{P}\left(\bigcap_{l\in\left\\{0,\dots,\lfloor
2^{j(1-a)}\rfloor-1\right\\}}\mathcal{C}\left(\mathcal{E}_{j,l}^{M}\right)\right)$
$\displaystyle=$ $\displaystyle 1-\prod_{l\in\left\\{0,\dots,\lfloor
2^{j(1-a)}\rfloor-1\right\\}}\left(1-\prod_{k\in
S_{j,l}^{M}}\mathbb{P}\left(\left|\widetilde{\varepsilon_{j,2NMk}}^{M}(\omega)\right|<Cj\right)\right)$
$\displaystyle=$ $\displaystyle
1-\left(1-\left(1-\mathbb{P}\left(|\varepsilon|\geq Cj\right)\right)^{\lfloor
2^{aj}/(2NM)\rfloor}\right)^{\lfloor 2^{j(1-a)}\rfloor}$ $\displaystyle\leq$
$\displaystyle 1-\exp{\left(2^{j(1-a)}\log(1-p_{j})\right)}$ (102)

where $\varepsilon$ is a random variable belonging to the Wiener chaos of
order 2 distributed according to the
$(\widetilde{\varepsilon_{\lambda}})_{\lambda\in\Lambda}$ and
$p_{j}=\left(1-\mathbb{P}\left(|\varepsilon|\geq Cj\right)\right)^{\lfloor
2^{aj}/(2NM)\rfloor}$. Remark that $p_{j}$ is a positive term that tends to
$0$ as $j\rightarrow+\infty$. Indeed, using the fact that $\log(1-x)\leq-x$ if
$x\in(0,1)$ together with (94), there exist $J\in\mathbb{N}$ such that for all
$j\geq J$,

$\displaystyle 0\leq p_{j}\leq$
$\displaystyle\left(1-\exp{(-C\,c_{2}\,j)}\right)^{\left\lfloor
2^{aj}/(2NM)\right\rfloor}$ $\displaystyle\leq$
$\displaystyle\exp{\left(-\left\lfloor\frac{2^{aj}}{2NM}\right\rfloor\exp{(-C\,c_{2}\,j)}\right)}$
$\displaystyle\leq$ $\displaystyle\exp{\left(-C^{\prime}\exp{(\log
2^{aj})}\exp{(-C\,c_{2}\,j)}\right)}$ $\displaystyle\leq$
$\displaystyle\exp{\left(-C^{\prime}\exp{j(a\log 2-C\,c_{2})}\right)}$ (103)

where $C^{\prime}$ depends only on $N$, $M$ and $a$ and $c_{2}$ is the
constant given in (94). It is enough to choose $C$ such that $a\log
2-C\,c_{2}>0$ to deduce that and so $p_{j}\rightarrow 0$ as
$j\rightarrow+\infty$. Similarly, one can get for all $j\geq J$

$\displaystyle 0\leq 2^{j(1-a)}p_{j}\leq\exp{\left(-C^{\prime}\exp{j(\log
2-C\,c_{2})}\right)}$

which indeed shows that $2^{j(1-a)}p_{j}$ tends to $0$ as
$j\rightarrow+\infty$. Now, using the fact that $\log(1-x)=-x+o(x)$ and
$\exp{(x)}=1+x+o(x)$ as $x\rightarrow 0$, together with (4.2) we obtain that
for all $\delta>0$

$\displaystyle\mathbb{P}\left(\mathcal{C}\left(\bigcap_{l\in\left\\{0,\dots,\lfloor
2^{j(1-a)}\rfloor-1\right\\}}\mathcal{E}_{j,l}^{M}\right)\right)\leq
2^{j(1-a)}\left(\delta(p_{j}+\delta p_{j})+p_{j}+\delta p_{j}\right)$

for $j$ large enough. Using the upper bound in (4.2), one can finally conclude
that (101) is indeed the general term of a convergent series. ∎

Concerning the random variable $\widecheck{\varepsilon_{\lambda}}^{M}$, one
can give an almost sure upper bound.

######  Lemma 4.13.

There exists a deterministic constant $C^{\prime}>0$ such that for all $M$
there is $\Omega_{2}^{\prime}\subset\Omega$ with probability 1 such that for
all $\omega\in\Omega_{2}^{\prime}$ there exist $J\in\mathbb{N}$ such that, for
all $j\geq J$, for all $\lambda\in\Lambda_{j}$, $\lambda\subseteq[0,1]$,

$\left|\widecheck{\varepsilon_{\lambda}}^{M}(\omega)\right|\leq
C^{\prime}M^{\max\\{H_{1},H_{2}\\}-1}j$

###### Proof.

If $C^{\prime}>0$, for all $j$ sufficiently large, we have, by Theorem 3.14

$\displaystyle\mathbb{P}$
$\displaystyle\left(\exists\lambda\in\Lambda_{j},\,\lambda\subseteq[0,1]\,:\,\left|\widecheck{\varepsilon_{\lambda}}^{M}(\omega)\right|\geq
C^{\prime}M^{\max\\{H_{1},H_{2}\\}-1}j\right)$
$\displaystyle\leq\sum_{\lambda\in\Lambda_{j}\,\lambda\subseteq[0,1]}\mathbb{P}\left(\left|\widecheck{\varepsilon_{\lambda}}^{M}(\omega)\right|\geq
C^{\prime}M^{\max\\{H_{1},H_{2}\\}-1}j\right)$ $\displaystyle\leq
2^{j}\exp(-\overset{\star}{C}C^{\prime}(C^{*}_{\Psi,H_{1},H_{2}})^{-1}j)$

and thus, if $C^{\prime}>\log(2)C^{*}_{\Psi,H_{1},H_{2}}/\overset{\star}{C}$,
the conclusion follows by Borel-Cantelli Lemma. ∎

###### Proof of Proposition 4.11.

Again, one can choose $M$ large enough such that

$C-C^{\prime}M^{\max\\{H_{1},H_{2}\\}-1}>0,$

where $C$ and $C^{\prime}$ are the constant given by Lemmata 4.12 and 4.13
respectively. Let us consider
$\omega\in\Omega^{*}_{2}:=\Omega_{2}\cap\Omega_{2}^{\prime}$ where the evnets,
of probability $1$, $\Omega_{2}$ and $\Omega_{2}^{\prime}$ are giving by the
same Lemmata. We use the same notations as in them. First there exist
$t\in(0,1)$ such that for all $J\in\mathbb{N}$ there exist $j\geq n$ such that

$\displaystyle\left|\widetilde{c_{\lambda_{j}(t)}}^{M}(\omega)\right|\geq
C\,j2^{-j(H_{1}+H_{2}-1)}.$ (104)

Moreover, if $J$ is large enough, for all such $j$ we also have

$\displaystyle\left|\widecheck{c_{\lambda_{j}(t)}}^{M}(\omega)\right|\leq
C^{\prime}\,M^{\max\\{H_{1},H_{2}\\}-1}2^{-j(H_{1}+H_{2}-1)}j.$ (105)

In this case, as in 4.8 we have that for all $J$ great enough, there is $j\geq
J$ such that

$\displaystyle d_{j}(t,\omega)\geq
2^{-j(H_{1}+H_{2}-1)}j\left(C-C^{\prime}M^{\max\\{H_{1},H_{2}\\}-1}\right)$