Keywords: Fractional Brownian motion, Rosenblatt process, Image set, Level set, Sojourn times, Wavelet series, Slow/Ordinary/Rapid points, Fractal dimensions, Macroscopic Hausdorff dimension, Potential theory for dimensions, Projection theorem. v

The topics of this thesis lie at the interference of probability theory with dimensional and harmonic analysis, accentuating the geometric properties of random paths of Gaussian and non-Gaussian stochastic processes. Such line of research has been rapidly growing in past years, paying off clear local and global properties for random paths associated to various stochastic processes such as Brownian and fractional Brownian motion. In this thesis, we start by studying the level sets associated to fractional Brownian motion using the macroscopic Hausdorff dimension. Then as a preliminary step, we establish some technical points regarding the distribution of the Rosenblatt process for the purpose of studying various geometric properties of its random paths. First, we obtain results concerning the Hausdorff (both classical and macroscopic), packing and intermediate dimensions, and the logarithmic and pixel densities of the image, level and sojourn time sets associated with sample paths of the Rosenblatt process. Second, we study the pointwise regularity of the generalized Rosenblatt and prove the existence of three kinds of local behavior: slow, ordinary and rapid points.

In the last chapter, we illustrate several methods to estimate the macroscopic Hausdorff dimension, which played a key role in our results. In particular, we build the potential theoretical methods. Then, relying on this, we show that the macroscopic Hausdorff dimension of the projection of a set E ⊂ R 2 onto almost all straight lines passing through the origin in R 2 depends only on E, that is, they are almost surely independent of the choice of straight line.

First and foremost I would like to deeply thank my thesis advisors, Ivan Nourdin and Stéphane Seuret, whose sincerity and encouragement I will never forget. They have been my source of inspiration as I hurdled through the path of this PhD degree. This thesis would not have been possible without them. Their guidance from the initial step in research enabled me to develop an understanding of the subject. I am thankful for the extraordinary experiences they arranged for me and for providing opportunities for me to grow professionally. It is an honor to learn from Prof. Nourdin and Prof. Seuret.

I express my deep gratitude to all the jury members, Ivan Nourdin, Stéphane Seuret, Giovanni Peccati, Stéphane Jaffard, Céline Esser and Yimin Xiao. Thank you for agreeing to read the manuscript and to participate in the defense of this thesis. It is a great honor for me to present my work in front of you.

My appreciation also extends to my colleagues in Luxembourg: thanks to my office mates, Valentin and Leonardo: sharing the office with you guys made my days more enjoyable. I thank Guenda, James, George, Massimo, Luca, Christian, Alexandre, Nicola, Robert, Mikolaj, Krzysztof, Tara, Xiaochuan, Arturo: on both a professional and a personal level, I am grateful for the time we spent together inside and outside the department, our lunch breaks, discussions, and ping-pong matches (even though I am a bad player). I would like also to thank my colleagues in Paris for the enjoyable time we spent during my frequent visits: in particular Elie, Edouard, and Guillaume: thank you guys for always encouraging and pushing me to speak French. I thank our secretaries, Marie and Katharina, together with Catherine from the doctoral office for maintaining the best possible work environment by taking care of the administration procedures. I want to thank my friends, I could not have completed this dissertation without your support guys: Ghina, Sara, Rancy, Elie, Joseph, Nadine, Nada, Adnan, Bob, Rim and Vita who provided stimulating discussions as well as happy distractions to rest my mind outside of my research.

Last but not least, I am grateful for my parents, Raif and Sanaa, whose constant love and support keep me motivated and confident. My accomplishments and success are because they believed in me. Deepest thanks to my siblings, Taher and Sara, who keep me grounded, remind me of what is important in life, and are always supportive of my adventures. Finally, I owe my deepest gratitude to Ryan, my love. I am forever thankful for the unconditional love and support throughout the entire thesis process and every day. I acknowledge support from the FNR grant APOGee (R-AGR-3585-10) at the University of Luxembourg, in the framework of which this thesis was elaborated. vii In this section, we provide an overview of the main theoretical tools that will be used in this manuscript. Our exposition is divided into 3 main building blocks:

Table of Contents

• In Section I.1.1, we introduce the class of self-similar stochastic processes with stationary increments (SSSI processes). Section I.1.1.1 presents the fractional Brownian motion and its local times, whereas Section I.1.1.2 is dedicated to define the Generalized Rosenblatt process which is investigated intensively in our work, and finally Section I.1.1.3 introduces the Hermite processes.

• In Section I.1.2, we define several fractal dimensions that are key tools in this manuscript. We mainly divide them onto two groups depending on the properties they reflect, local or global properties. Section I.1.2.2.2 is dedicated to the so-called macroscopic Hausdorff dimension, which plays a pivotal role in our work.

• In Section I.1.3, we expose basic wavelet tools and illustrate the so-called wavelet leaders methods.

The goal behind this expository part is to present the necessary material in a selfcontained way, hopefully allowing the reader to follow it easily without further referencing.

I.1.1 Self-similar processes with stationary increments

Here and throughout the thesis, every random object is defined on an appropriate probability space (Ω, F , P). The symbols 'E', 'V ar' and 'Cov' denote, respectively, the expectation, the variance and the covariance associated with P. A stochastic process (X t ) t≥0 is a R-valued random function on R. Two stochastic processes X and Y have the same distribution (noted X (d) = Y ) if they have the same finitedimensional distributions.

Self-similar processes are stochastic processes that are invariant in distribution under suitable scaling of time and space. Formally speaking, a stochastic process (X t ) t≥0 is said to be self-similar with exponent H > 0 if

(X ct ) t≥0 (d)
= (c H X t ) t≥0 for all c > 0, and has stationary increments if (X t+t 0 -X t 0 ) t≥0 (d) = (X t ) t≥0 for all t 0 ∈ R.

Self-similar processes with stationary increments (SSSI processes) appear as limits in various normalization procedures [START_REF] Lamperti | Semi-stable stochastic processes[END_REF][START_REF] Samorodnitsky | Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance: Stochastic Modeling[END_REF][START_REF] Taqqu | Convergence of integrated processes of arbitrary Hermite rank[END_REF]. In applications, they occur in various fields such as finance, hydrology, biomedicine and image processing. The simplest SSSI processes are the Brownian motion and, more generally, Lévy stable motions. A broad class of SSSI processes which belongs to the homogeneous Wiener chaos of an arbitrary order N ≥ 1 are Hermite processes of rank N . They generalize the fractional Brownian motion and the Rosenblatt process. In the following proposition, we discuss some properties of SSSI processes.

Proposition I.1.1. [START_REF] Taqqu | Fractional Brownian motion and long range dependence, in 'theory and applications of long-range dependence[END_REF] Fix H ∈ (0, 1] and let (X t ) t≥0 be an H-self-similar stochastic process with stationary increments. Then the following properties hold:

1. X 0 = 0 almost surely.

2. If H ̸ = 1, then E[X t ] = 0, for all t ∈ R.

3. If H = 1, then X t = tX 1 almost surely for t ∈ R. 2 1 ] < ∞, then the covariance function of the process X is given by

If E[X

E[X t X s ] = E[X 2 1 ] 2 t 2H + s 2H -|t -s| 2H .
SSSI processes, in particular the Hermite processes, are defined with the aid of a multiple stochastic integral called Wiener-Itô integral. One mentions that two classical books on Wiener chaoses, multiple Wiener integrals and related topics are [START_REF] Janson | Gaussian Hilbert Spaces[END_REF][START_REF] Nualart | The Malliavin calculus and related topics[END_REF]. First of all, let us define the multiple Wiener-Itô integrals.

Definition I.1.2. The multiple Wiener-Itô integral of order k ≥ 1 is defined for any f ∈ L 2 (R k ) as

I k (f ) = ′ R k f (x 1 , ..., x k )dB(x 1 )...dB(x k ),
where B is Brownian motion viewed as a random integrator, and ′ R k denotes integration over R k excluding the diagonals. Remark 1. The set of random variables I k (f ) forms the k-th Wiener chaos when f varies in L 2 (R k ). Moreover, I k (f ) has the following properties:

1. I k (.) is a linear mapping from L 2 (R k ) to L 2 (Ω).

2. If f σ (x 1 , ..., x k ) := f (x σ(1) , ..., x σ(k) ), where σ is a permutation, then I k (f σ ) = I k (f ).

As a result if we denote by f the symmetrization of f , namely

f (x 1 , ..., x k ) := 1 k! σ∈S d f σ (x 1 , ..., x k ), then I k (f ) = I k ( f ) for all f ∈ L 2 (R k ).
3. For f ∈ L 2 (R q ) and g ∈ L 2 (R p ), one has

E [I q (f )I p (g)] = 1 p! R k f (x)ĝ(x)dx, if q = p 0 if q ̸ = p .
In the next three sections, we define three SSSI process. We start with the fractional Brownian motion which is the simplest Gaussian Hermite process, then we move to the Rosenblatt proces which is the simplest non-Gaussian Hermite process, and finally we introduce the class of Hermite processes. Our aim is to provide definitions and properties of these processes allowing one to have all the essential tools for the coming chapters.

I.1.1.1 Fractional Brownian Motion

Brownian motion is a random phenomenon, of central theoretical importance. Nevertheless it often appears as too restrictive for applications. Brownian motion is the unique Gaussian process, which has stationary increments that are independent and of finite variance with mean 0. To obtain a less restrictive model, it is necessary to relax one or more of these conditions.

Fractional Brownian motion is a generalization of Brownian motion, which has stationary increments that are normally distributed but no longer independent. Fractional Brownian motion, which was first itroduced by Kolmogorov [START_REF] Kolmogorov | Wienersche spiralen und einige andere interessante kurven in hilbertscen raum, cr (doklady)[END_REF] and further developed by Mandelbrot and Van Ness [START_REF] Mandelbrot | Fractional brownian motions, fractional noises and applications[END_REF], is defined as follows.

Definition I. 1.3. Let H ∈ (0, 1]. A fractional Brownian motion (FBM) of Hurst index H is a centered continuous Gaussian process B H = B H t t>0 with covariance function

R H (t, s) := E[B H t B H s ] = 1 2 t 2H + s 2H -|t -s| 2H . (I.1.1)
Fractional Brownian motion exists for all H ∈ (0, 1]. Moreover, it admits a version with continuous paths, and for every t ≥ 0 and s > 0 the increment B H t+s -B H t has normal distribution with mean zero and variance s 2H , so that Proposition I.1.4. [START_REF] Nourdin | Selected aspects of fractional Brownian motion[END_REF] Let B H be a fractional Brownian motion of Hurst parameter H ∈ (0, 1]. Then B H has the following properties:

P(B H t+s -B H t ≤ x) = 1 s H √ 2π x -∞ exp -u 2 2s 2H du, x ∈ R.
(1) Self-similarity: The processes B H ct , t ≥ 0 and c H B H t , t ≥ 0 have the same distribution.

(2) Stationary increments: The distribution of the process B H t+s -B H s , t ≥ 0 does not depend on s ≥ 0.

(3) Time inversion: The processes B H t , t ≥ 0 and t 2H B H 1/t , t ≥ 0 have the same distribution.

(4) Brownian filtration: The natural filtration associated to a fractional Brownian motion is Brownian, i.e., there is a Brownian motion (B t ) t≥0 defined on the same probability space than B H such that its filtration satisfies σ B H s : s ≤ t ⊂ σ {B s : s ≤ t} , (I.1.2) for all t > 0.

Conversely, any continuous Gaussian process B H = (B H t ) t≥0 with B H 0 = 0, and V ar(B H 1 ) = 1, and such that (1) and (2) hold, is a fractional Brownian motion of index H. A fractional Brownian motion with Hurst index H admits Hölder continuous paths for all exponents less than H (see e.g. [START_REF] Nourdin | Selected aspects of fractional Brownian motion[END_REF]).

Proposition I. 1.5. Let B H be a fractional Brownian motion of Hurst parameter H ∈ (0, 1]. If 0 < δ < H and T > 0 then, with probability 1, there exists a random constant C T such that

|B H t+s -B H t | ≤ C T |s| δ
for all t, s ∈ [0, T ].

As we will see, the use of the local time will play a key role throughout Chapter II. Provided it exists, the local time x → L(x, t) of a process (X t ) t≥0 is, for each t, the density of the occupation measure µ t (A) = λ({s ∈ [0, t] : X s ∈ A}) associated with X, where λ stands for the Lebesgue measure; otherwise stated, one has L(., t) = dµt dλ . The case of Gaussian (and centered, say) processes has been widely studied in the literature. For instance, one of the main striking results in the Gaussian framework (see e.g. Dozzi [START_REF] Dozzi | Occupation density and sample path properties of N -parameter processes[END_REF]), in particular for fractional Brownian motion, is the following condition ensuring the existence of (L x t ) t∈[0,T ],x∈R in L 2 (Ω).

I := [0,T ] 2 ds dt R H (s, s)R H (t, t) -R H (s, t) 2
< +∞, (I. 1.3) where R H (s, t) = E B H s B H t ; moreover, in this case we have the Fourier type representation: L(x, t) = 1 2π R dy t 0 du e iy(B H u -x) . (I. 1.4) As B H is selfsimilar and satisfies (I.1.3), then it is immediate from (I. 1.4) that its local time at level x also has some selfsimilarity properties in time with index 1 -H but with a different level, as stated below.

Proposition I.1.6. Let c > 0. Assume B H is a fractional Brownian motion of Hurst index H ∈ (0, 1) and consider its local time (L(x, t)) t≥0,x∈R . One has

(L(x, ct)) t≥0,x∈R (d)
= c 1-H (L(c -H x, t)) t≥0,x∈R . (I. 1.5) Finally, the local time is Hölder continuous in both time and space. In particular Proposition I.1.7. [START_REF] Berman | Local nondeterminism and local times of Gaussian processes[END_REF] For every x ∈ R, almost surely, the local time L(x, t) is β-Hölder continuous in t for every β ∈ [0, 1 -H].

Proposition I.1.8. [START_REF] Geman | Occupation Densities[END_REF]Theorem 26.1] Assume B H is a fractional Brownian motion of Hurst index H ∈ (0, 1) and consider its local time (L(x, t)) x∈K , where K is a given compact interval in R. Then, for all β ∈ 0, 1

Generalized Rosenblatt Process

Like the fractional Brownian motion, the Rosenblatt process is a selfsimilar stochastic process with stationary increments. Both processes belong to the class of Hermite processes, fractional Brownian motion being of order 1 while Rosenblatt process is of order 2. However, unlike the fractional Brownian motion, the Rosenblatt process is not Gaussian.

Before giving a formal definition of the Rosenblatt process, let us recall some important notions related to the Hermite polynomials which are essential for our coming definitions. For m ≥ 0, the Hermite polynomial of degree m is given by

H m (x) := (-1) m e x 2 2
d m dx m e -x 2 2 .

Definition I.1.9. Given a function f ∈ L 2 (R), we say that f has Hermite rank equal to k if E[f (ξ)H m (ξ)] = 0 for m ≤ k -1 and E[f (ξ)H k (ξ)] ̸ = 0, where ξ ∼ N (0, 1).

The Rosenblatt process appears in the limit of Non-Central Limit Theorem of [START_REF] Breuer | Central limit theorems for non-linear functionals of Gaussian fields[END_REF]. Formally speaking, consider a stationary Gaussian sequence (ξ n ) n≥0 with mean zero and variance 1 such that, for all n ≥ 0, one has

E(ξ 0 ξ n ) = n H-1 L(n),
where H ∈ ( 1 2 , 1) and L is a slowly varying 1 function at infinity. Let f be a function such that E(f (ξ 0 )) = 0 and E(f (ξ 0 ) 2 ) < ∞. Suppose that f has Hermite rank equal to 2. Then the Non-Central Limit Theorem of [START_REF] Breuer | Central limit theorems for non-linear functionals of Gaussian fields[END_REF] asserts that 1 n H ⌊nt⌋ j=1 f (ξ j ) converges as n → ∞ in the sense of finite dimensional distributions to the process

R H t = c H ′ R 2 t 0 (s -x 1 ) H 0 -3 2 + (s -x 2 ) H 0 -3 2 +
ds dB(x 1 )dB(x 2 ), (I. 1.7) where

x + = max(x, 0) and

H 0 = H + 1 2 .
The above integral is Wiener-Itô stochastic integrals with respect to Brownian motion (B(t)) t∈R (see Definition I.1.2), and the constant c H is positive and satisfies E((R H 1 ) 2 ) = 1. The process (R H t ) t≥0 is called the Rosenblatt process (it was actually been named in this way by Taqqu in [START_REF] Taqqu | The Rosenblatt process[END_REF]) and it is non-Gaussian, H-selfsimilar, with stationary increments. In addition it has the same second order properties as fractional Brownian motion, namely,

E[(R H t ) 2 ] = t 2H , E[R H t R H s ] = 1 2 (t 2H + s 2H -|t -s| 2H
). (I.1.8) 1 A positive measurable function L is said to be slowly varying if lim t→+∞ L(xt) L(t) = 1 for all x > 0.

We call (I.1.7) the time-domain representation. It is known that Rosenblatt process admits other representations in terms of Wiener-Itô integrals, among which we note the spectraldomain representation (see [START_REF] Taqqu | The Rosenblatt process[END_REF] and [START_REF] Dobrushin | Non-central limit theorems for nonlinear functionals of Gaussian fields[END_REF]):

R H • (d) = C(H) R 2
e i(x+y)• -1 i(x + y) Z G (dx)Z G (dy), (I. 1.9) where the double Wiener-Ito integral is taken over x ̸ = ±y and Z G (dx) is a complex-valued random white noise with control measure G satisfying G(tA) = t 1-H G(A) for all t ∈ R and G(dx) = |x| -H dx. The constant C(H) in (I.1.9) is such that E((R H 1 ) 2 ) = 1. Remark 2. Note that in the notation of [START_REF] Taqqu | The Rosenblatt process[END_REF], Z G (dx) = |x| -H/2 d B(x), with (B(t)) t∈R the Brownian motion and d B(x) is viewed as the complex-valued Fourier transform of dB(x). For more details, see [START_REF] Taqqu | Convergence of integrated processes of arbitrary Hermite rank[END_REF].

In Chapter III, our main interest consists in studying the geometric properties of the random paths of the Rosenblatt process. In this analysis, the local time of the Rosenblatt process plays a pivotal role. Its existence was shown in [START_REF] Shevchenko | Properties of trajectories of the multifractional Rosenblatt process[END_REF] together with this L 2 representation:

L(x, t) = 1 2π R t 0 e iξ(x-R H s ) dsdξ. (I.1.10)
As mentioned before, since R H is selfsimilar of index H, its local time at level x also has some self-similarity properties in time with index 1 -H, but with a different level. More precisely, one has, for every c > 0:

(L(x, ct)) t≥0,x∈R (d)
= c 1-H (L(c -H x, t)) t≥0,x∈R .

(I. 1.11) In Chapter IV we study the generalized Rosenblatt process which is a generalization of the Rosenblatt process introduced in [START_REF] Maejima | Selfsimilar processes with stationnary increments in the second Wiener chaos[END_REF], and is defined as follows:

Definition I.1.10. Given two parameters H 1 , H 2 ∈ ( 1 2 , 1) such that H 1 + H 2 > 3 2 , the generalized Rosenblatt process {R H 1 ,H 2 (t, •)} t∈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, •) = ′ R 2 K H 1 ,H 2 (t, x 1 , x 2 ) dB(x 1 )dB(x 2 ), (I.1.12)
where ′ R 2 denotes integration over R 2 excluding the diagonal. The kernel function in (I.1.12) is expressed, for all (t, x 1 , x 2 ) on R + × R 2 , by

K H 1 ,H 2 (t, x 1 , x 2 ) = 1 Γ H 1 -1 2 Γ H 2 -1 2 t 0 (s -x 1 ) H 1 -3 2 + (s -x 2 ) H 2 -3 2 + ds,
where Γ stands for the usual Gamma Euler function, and where for (x, α) ∈ R 2

x α + =

x α if x ≥ 0 0 otherwise.

Note that the (standard) Rosenblatt process defined in (I. 1.7) is the process {R H,H (t, •)} t∈R + for H ∈ (3/4, 1). The generalized Rosenblatt process {R H 1 ,H 2 (t, •)} t∈R + is non-Gaussian, belongs to the second Wiener chaos, has stationary increments, and is (H 1 + H 2 -1)-selfsimilar.

I.1.1.3 Hermite processes

Hermite processes are SSSI processes that naturally arise as limits of normalized sums of long-range dependent random variables [START_REF] Dobrushin | Non-central limit theorems for nonlinear functionals of Gaussian fields[END_REF]. Since the seminal works of Taqqu [START_REF] Taqqu | Weak convergence to fractional Brownian motion and to the Rosenblatt process[END_REF][START_REF] Taqqu | Convergence of integrated processes of arbitrary Hermite rank[END_REF], the class of Hermite processes has attracted considerable interest in probability and statistics. A Hermite process can be defined by any of its equivalent representations. Here by equivalent representations, we mean that the represented processes share the same finite-dimensional distributions. The most well known representation is the time domain representation in terms of multiple Wiener-Itô integrals.

Definition I. 1.11. Fix an integer N ≥ 1 and a real number H ∈ (1 -1/(2N ), 1). The Hermite process of rank N and parameter H is defined through the multiple Wiener integral with respect to Brownian motion (B(t)) t∈R :

Z N H (t) = c N,H ′ R N t 0 N p=1 (s -x p )
H-3/2 + ds dB(x 1 ), ..., dB(x N ), (I. 1.13) where x + = max(x, 0), and c N,H is some positive constant that makes Var(Z N H (1)) = 1.

Remark 3. The Hermite process Z N H has stationary increments and is self-similar with Hurst index H. When the rank N = 1, we recover the classical (Gaussian) fractional Brownian motion. When N ≥ 2, the law of Z N H is non-Gaussian, and if N = 2, the process is also known as the Rosenblatt process.

Another important representation is the spectral domain representation which is given by

Z N H (t) = C N,H " R N e it(x 1 +...+x N ) -1 i(x 1 + ... + x N ) N p=1
|x p | 1/2-H d B(x 1 ), ..., d B(x N ), (I. 1.14) where B is a complex-valued Hermitian Gaussian random measure (see [START_REF] Pipiras | Long-range dependence and self-similarity[END_REF]Definition B.1.3]) with Lebesgue control measure, the double prime " at the top of the integral sign indicates the exclusion of the hyperdiagonals x i = ±x j , i ̸ = j, in the N -tuple stochastic integral, and C N,H is a constant such that V ar(Z N H (1)) = 1. Remark 4. All Hermite processes with Hurst index H, regardless of the order, share the same covariance structure as a standard fractional Brownian motion, that is,

E[Z N H (t)Z N H (s)] = 1 2 (t 2H + s 2H -|t -s| 2H ).

I.1.2 Fractal Dimensions

From an early age, we learned that straight lines and curves have dimension 1, planes and surfaces have dimension 2, solids such as a ball have dimension 3, and so on. More properly, we say that a set is n-dimensional if we need n independent variables to describe a neighborhood of any point. However, one can map a real line into a plane bijectively and continuously. In other words, a one-dimensional curve can cover a two-dimensional plane completely, which is known by space-filling curve. Fractal geometry generalizes this notion of dimension to a wider class of sets by defining non-integer dimensions. Roughly speaking, these fractal dimensions measures how much space is occupied near each point of a set. In this section we give an overall summary of all fractal dimensions used in this manuscript. We split this section into three subsections:

1. Dimensions reflecting microscopic properties: We introduce Hausdorff, box, packing, and intermediate dimensions, following [START_REF] Falconer | Fractal geometry[END_REF][START_REF] Falconer | Intermediate dimensions[END_REF][START_REF] Burrell | Projection theorems for intermediate dimensions[END_REF][START_REF] Falconer | Packing dimensions of projections and dimension profiles[END_REF][START_REF] Howroyd | Box and packing dimensions of projections and dimension profiles[END_REF].

2. Dimensions reflecting macroscopic properties: We introduce another group of dimensions which are logarithmic density, pixel density, and macroscopic Hausdorff dimension, following [START_REF] Khoshnevisan | Intermittency and multifractality: a case study via parabolic stochastic PDEs[END_REF][START_REF] Khoshnevisan | On the macroscopic fractal geometry of some random sets[END_REF][START_REF] Barlow | Defining fractal subsets of Z d[END_REF][START_REF] Barlow | Fractional dimension of sets in discrete spaces[END_REF].

3. Overview of the types of fractal dimensions: We compare all dimensions mentioned above and give a few relations between them to give the reader an intuition.

Our main bibliographic sources serving as guiding inspiration for this section is the book by Falconer [START_REF] Falconer | Fractal geometry[END_REF]. In this section, we let (R d , ∥∥ 2 ) be the d-dimension Euclidean space equipped with its usual L 2 -norm.

I.1.2.1 Dimensions reflecting microscopic properties

I.1.2.1.1 Box dimensions
What is the relation between an object length (area or volume) and its diameter? This question leads us to think about dimensions from different perspectives. To clarify our idea, let us consider some examples. When we want to cover a unit square with little squares of side δ, we obviously need 1/δ 2 little squares. Whereas, if we want to cover a unit cube, we will need exactly 1/δ 3 of little cubes of diameter δ. We note that the exponents we got here are the dimensions of the objects that we are covering, which is not a coincidence. This was the main idea behind the box dimension, also known as Minkowski-Bouligand dimension, that determines the fractal dimension of a set F ⊂ R d using box-counting analysis. Formally speaking, given a non-empty set If s = dim B (F ), then (I.1.17) roughly states that N δ (F ) ∼ δ -s for δ small enough, or more precisely one has

F ⊂ R d , we call {U i } an exact δ-cover of F if {U i } is a countable
lim δ→0 N δ (F )δ s = ∞ if s < dim B (F ) 0 if s > dim B (F )
Motivated by the above limit, we want to mention another equivalent definition for the lower and upper box dimensions, which is more convenient to use.

Definition I.1.12. For a given set F ⊂ R d , the lower box dimension is given by:

dim B (F ) := inf s ≥ 0 : ∀ε > 0, ∃ cover {U i } ∞ i=1 of F, s.t. |U i | = |U j | ∀i, j and ∞ i=1 |U i | s ≤ ε . (I.1.18)
Similarly, we define the upper box dimension:

dim B (F ) := inf s ≥ 0 : ∀ε > 0, ∃δ > 0, ∀ cover {U i } ∞ i=1 of F, s t. |U i | ≤ δ, |U i | = |U j | ∀i, j and ∞ i=1 |U i | s ≤ ε . (I.1.19)
Releasing the constraints on the covers of F , we get another fractal dimension, called Hausdorff dimension, that we introduce in the next section.

I.1.2.1.2 Hausdorff dimensions

The Hausdorff dimension is one of the oldest fractal dimensions. It can be defined for any set, and its definition is based on a measure which gives it some advantages on other dimensions. For a given non-empty set F ⊂ R d , we call {U i } an δ-cover of F if {U i } is a countable or finite collection of sets with diameter at most δ covering F , i.e. F ⊂ ∪ i U i . To define the Hausdorff dimension we start by defining the Hausdorff measure. For F ⊂ R d , s ≥ 0, and δ > 0, define

H s δ (F ) = inf ∞ i=1 |U i | s : {U i } ∞ i=1 is a δ-cover of F , (I.1.20)
where the infimum is taken over all possible covers with diameter at most δ. Moreover, as δ decreases, the set of possible covers decreases too, and so δ → H s δ (F ) is an non-decreasing function. On the other hand, for a given δ < 1, s → H s δ (F ) is non-increasing, and for s < t, one has

H t δ (F ) ≤ δ t-s H s δ (F
(F ) = inf{s ≥ 0 : H s (F ) = 0} = sup{s ≥ 0 : H s (F ) = +∞} (I.1.23)
with sup ∅ = 0 by convention. Then, one has

H s (F ) = ∞ if 0 ≤ s < dim H (F ) 0 if s > dim H (F ) .
As illustrated in

H (F ) := inf s ≥ 0 : ∀ε > 0, ∃ cover {U i } ∞ i=1 of F, s.t. ∞ i=1 |U i | s ≤ ε . (I.1.24)
If we compare the Hausdorff dimension with the box dimension (see Definition I.1.12), we see that the box dimension is more restrictive with the covering of the set, i.e. all covers should have same diameter.

At the end of this section, we introduce two well-known techniques for calculating the Hausdorff dimension. Firstly we start by the potential theoretical methods which are mainly based on integral analysis. To this end we define, for s ≥ 0, the s-potential at a point x ∈ R d resulting from a mass distribution2 µ on R d :

ϕ s (x) = dµ(y) ∥x -y∥ s ,
and the s-energy of µ:

I s (µ) = ϕ s (x)dµ(x) = dµ(x)dµ(y) ∥x -y∥ s .
The following theorem states the potential theoretical methods which are usually used to bound the Hausdorff dimension from below.

Theorem I.1.14. [40, Theorem 4.13] Let F be a subset of R d .

1. If there exists a mass distribution µ on F such that

I s (µ) < ∞, then H s (F ) = ∞ and dim H (F ) ≥ s.
2. If F is a Borel set with 0 < H s (F ) ≤ ∞ then, for all 0 < t < s, there exists a mass distribution µ of F with I t (µ) < ∞.

The second technique, called the mass distribution principle, is based on a given measure µ, and estimating the µ-mass of small sets in order to bound the Hausdorff dimension from below too.

Theorem I. 1.15. [40,Section 4] Let µ be a mass distribution on F and suppose that for some s > 0, there is a number c > 0 and ε > 0 such that

µ(U ) ≤ c|U | s for all sets |U | ≤ ε. Then H s (E) ≥ µ(F )/c and dim H (F ) ≥ s.

I.1.2.1.3 Packing dimensions

With the Hausdorff dimension we are able to outpace most dimensions based on the fact that it is defined in terms of measures. In fact this is not the case for the box dimension (see definition I.1.12), though one can construct a measure based dimension, the packing dimension, which is in some sense "dual" to the Hausdorff dimension. To this end, for F ⊂ R d and s > 0, let us recall the definition of the s-dimensional packing measure of

F P s (F ) := inf n P s 0 (F n ) : F ⊆ n F n ,
where for F ⊂ R d ,

P s 0 (F ) := lim ε→0 sup i (2r i ) s : B(x i , r i ) are disjoint , x i ∈ F, r i < ε . Definition I.1.16. Given F ⊂ R d and a Borel measure µ on R d , the packing dimension of F is dim P (F ) := inf{s > 0 : P s (F ) = 0}, (I.1.25)
and the packing dimension of µ is defined by

dim P (µ) := inf{dim P (F ) : µ(F ) > 0 and F ⊂ R d is a Borel set}.
Next, we recall the concept of packing dimension profiles first conceived by Falconer and Howroyd in [START_REF] Falconer | Packing dimensions of projections and dimension profiles[END_REF] and [START_REF] Howroyd | Box and packing dimensions of projections and dimension profiles[END_REF]. For finite Borel measures µ on R d and for any s > 0, let

F µ s (x, r) = R ψ s x -y r dµ(y)
be the potential with respect to the kernel ψ s (x) = min {1, ∥x∥ -s }, ∀x ∈ R d .

Definition I.1.17. Let µ be a Borel measure on R d . The packing dimension profile of µ is defined as

dim P,s (µ) = sup β ≥ 0 : lim inf r→0 F µ s (x, r) r β = 0 for µ -a.e. x ∈ R d .
Now for any Borel set F ⊂ R d , we define M + c (F ) to be the family of finite Borel measures with compact support in F . Then an equivalent definition of the packing dimension can be established.

Proposition I.1.18. Given F ⊂ R d , the packing dimension of F is equal to dim P (F ) = sup dim P (µ) : µ ∈ M + c (F ) .
Motivated by this, Falconer and Howroyd [START_REF] Falconer | Packing dimensions of projections and dimension profiles[END_REF] define s-dimensional packing dimension profile of F ⊂ R d by dim P,s (F ) = sup dim P,s (µ) :

µ ∈ M + c (F ) .
It is easy to see that 0 ≤ dim P,s (F ) ≤ s, and dim P,s (F ) = dim P (F ) for any s ≥ d.

I.1.2.1.4 Intermediate dimensions

By comparing the covering restrictions between the box and the Hausdorff dimensions, Falconer et al [START_REF] Falconer | Intermediate dimensions[END_REF] introduced a new continuum of dimensions intermediate between the box and the Hausdorff dimension, named intermediate dimensions. As we will see later in details, for a bounded and non-empty set F ⊂ R d , θ ∈ (0, 1] and s ∈ [0, d], they defined

H s r,θ (F ) = inf i |U i | s : {U i } i is a cover of F such that r ≤ |U i | ≤ r θ for all i . (I.1.26)
In particular, for θ = 0, H s r,0 (F ) is the s-dimensional Hausdorff measure of f . The following lemma enables us to define the intermediate dimensions.

Lemma I.1.19. [22, Lemma 2.1] Let θ ∈ (0, 1) and F ⊂ R d . For each 0 < r < 1 and all 0 ≤ t ≤ s ≤ d, -(s -t) ≤ log(H s r,θ (F )) -log r - log(H t r,θ (F )) -log r ≤ -θ(s -t)
In particular, there is a unique s ∈ [0, d] such that lim inf Similarly, the upper θ-intermediate dimension of E is defined by

dim θ (F ) = the unique s ∈ [0, d] such that lim sup r→0 log H s r,θ (F ) -log r = 0 . (I.1.28) When dim θ (F ) = dim θ (F ), we refer to the θ-intermediate dimension dim θ (F ) = dim θ (F ) = dim θ (F ).
Thus, the classical Hausdorff (I.1.24) and box dimensions (I.1.18), (I. 1.19) can be viewed as the extremes of a continuum of dimensions with increasing restrictions on the relative sizes of covering sets. Indeed, for every bounded F ⊂ R,

dim 0 F = dim 0 F = dim H (F ) , dim 1 F = dim B (F ) and dim 1 F = dim B (F ) .
Moreover, the intermediate dimensions can be defined in terms of capacities with respect to an appropriate kernel denoted by ϕ s,m r,θ (see [START_REF] Burrell | Projection theorems for intermediate dimensions[END_REF]). For each collection of parameters θ ∈ (0, 1], m ∈ {1, ..., d}, 0 ≤ s ≤ m and 0 < r < 1, let ϕ s,m r,θ : R d → R be the function

ϕ s,m r,θ (x) :=      1 0 ≤ |x| < r, r |x| s r ≤ |x| < r θ , r θ(m-s)+s |x| m r θ ≤ |x|. (I.1.29)
Using this kernel we define the capacity of a compact set

F ⊂ R d as C s,m r,θ (F ) := inf µ∈M(F ) ϕ s,m r,θ (x -y)dµ(x)dµ(y) -1 , (I.1.30)
where M(F ) is the set of probability measures supported in F . The following lemma, which is similar to Lemma I.

0 ≤ t ≤ s ≤ d, -(s -t) ≤ log(C s,m r,θ (F )) -log r -s - log(C t,m r,θ (F )) -log r -t ≤ -θ(s -t)
In particular, there is a unique s ∈ [0, d] such that lim inf The intermediate dimension profiles are increasing in m and for

F ⊂ R d , dim θ,d (F ) = dim θ (F ) and dim θ,d (F ) = dim θ (F ) .

I.1.2.2 Dimensions reflecting macroscopic properties

In this section, we would like to note that all our definitions are independent of the choice of the norm on R d . But for the coherence of the definitions, we continue to work with the d-dimension Euclidean space (R d , ∥∥ 2 ). For x ∈ R d and r > 0, B(x, r) denotes the Euclidean ball with center x and radius r. For F ⊂ R d , the diameter of a set F is denoted by |E|.

I.1.2.2.1 Logarithmic and pixel densities

In the scope of measuring macroscopic properties of a given set F , we recall the definitions of logarithmic and pixel densities. But, first, we have to introduce some notions. For all x = (x 1 , ..., x d ) ∈ R d , define

Q(x) = [x 1 , x 1 + 1) × ... × [x d , x d + 1).
One defines the pixelization of a set

F ⊂ R d as pix(F ) = {x ∈ Z d : F ∩ Q(x) ̸ = ∅}. (I.1.33) It is clear that pix(F ) = F if F ⊂ Z d , and pix(R d ) = Z d .
Definition I.1.22. (see [START_REF] Khoshnevisan | Intermittency and multifractality: a case study via parabolic stochastic PDEs[END_REF][START_REF] Khoshnevisan | On the macroscopic fractal geometry of some random sets[END_REF])

For F ⊂ R d , the pixel density of F is Den pix (F ) := lim sup n→∞ log 2 #pix(F ∩ B(0, 2 n )) n ,
where # denotes cardinality. The logarithmic density of F is given by

Den log (F ) := lim sup n→∞ log 2 Leb(F ∩ B(0, 2 n )) n ,
where 'Leb' is the d-dimensional Lebesgue measure.

Note that for any F ⊂ R d , both Den pix (F ) and Den log (F ) range between 0 and d.

I.1.2.2.2 Macroscopic Hausdorff dimensions

The macroscopic Hausdorff dimension Dim H (F ) of a set F ⊂ R d was introduced by Barlow and Taylor [START_REF] Barlow | Defining fractal subsets of Z d[END_REF][START_REF] Barlow | Fractional dimension of sets in discrete spaces[END_REF] to define the notion of fractals in a discrete setup. It is a discrete analog of Hausdorff dimension, and the word macroscopic comes from the fact that this dimension ignores the local structure of the sets. In this section we aim to define this macroscopic Hausdorff dimension. To this end, define for all integer n ∈ N, the n-th shell of R d by S 0 = B(0, 1) and

S n := B(0, 2 n ) \ B(0, 2 n-1 ) for all n ≥ 1. (I.1.34)
Both standard Hausdorff dimension and macroscopic Hausdorff dimension describe how a set F can be efficiently covered by balls. Nevertheless, the macroscopic Hausdorff dimension is concerned only with large scale behaviors, and so Barlow and Taylor proposed to cover the intersections F ∩S n by balls with diameters at least 1. In this capacity, let us introduce, for F ⊆ R d , the set of covers of F restricted to S n defined by

I n (F ) = {B(x i , r i )} m i=1 : m ∈ N, x i ∈ S n , r i ≥ 1, F ∩ S n ⊂ m i=1 B(x i , r i ) .
Finally, for s ≥ 0 and n ∈ N, set

ν s n (F ) = inf m i=1 r i 2 n s : {B i = B(x i , r i )} m i=1 ∈ I n (F ) . (I.1.35)
Observe that ν s n is sub-additive, i.e. ν s n (A ∪ B) ≤ ν s n (A) + ν s n (B) for every sets A and B, but is not a measure (because of the constraints on r i ). Now we define the Barlow-Taylor macroscopic Hausdorff dimension.

Definition I.1.23. For every s ≥ 0 and

F ⊂ R d , define ν s (F ) = n≥1 ν s n (F ).
The macroscopic Hausdorff dimension of F ⊂ R d is defined by

Dim H (F ) = inf {s ≥ 0 : ν s (F ) < +∞} . (I.1.36) One easily checks that Dim H (F ) ∈ [0, d] for all F ⊂ R d , Dim H (F ) = 0 when F is bounded,

and an alternative definition for Dim

H (F ) is Dim H (F ) = sup {s ≥ 0 : ν s (F ) = +∞} ,
where sup ∅ = 0 by convention.

Remark 5. Both Dim H (F ) and Den pix (F ) (resp. Den log (E)) give an intuition about the macroscopic geometry of F . The main difference is that Dim H (F ) not only counts the number of points of F ∩ S n as Den pix (F ) (resp. measures F ∩ S n as Den log (F )) but also takes into account the geometry of the set F , in particular by considering the most efficient covering of F ∩ S n . Thus, as an intuition, the value of ν s n (F ) is larger when the points F ∩ S n are scattered all over S n , while it is smaller when these points are all located in the same region. For instance, for 0 < α < 1, define the two sets A α and B α by for all n ≥ 1,

A α ∩ S n = 2 n-1 + k 2 n-1 2 nα : k ∈ {0, ..., 2 nα -1} ; B α ∩ S n = 2 n-1 + k 2 nα : k ∈ {0, ..., 2 nα -1} .
Even though both sets have same cardinality, we have Dim H A α = α whereas Dim H B α = 0.

I.1.2.3 Overview of the types of fractal dimensions

Throughout this thesis, we use various fractal dimensions. In this section we aim to compare between the dimensions mentioned so far to give the reader some intuition.

(•) Hausdorff Covering (0, δ] [0, d] δ → 0 dim B (•) Box Upper -Covering Lower -Packing δ [0, d] δ → 0 dim P (•) Packing Packing (0, δ] [0, d] δ → 0 dim θ (•) Intermediate ∈ (δ 1/θ , δ) [0, d] δ → 0 Den log (•) Logarithmic density Balls [1, 2 n ] [0, d] n → ∞ Den pix (•) Pixel density Balls [1, 2 n ] [0, d] n → ∞ Dim H (•) Macroscopic Hausdorff Collections of balls in B(0, 2 n )/B(0, 2 n-1 ) [1, 2 n ] [0, d] n → ∞
R d Z d dim H (•) Classical Hausdorff 0 [0, d] d 0 dim B (•) Box 0 [0, d] d 0 dim P (•) Packing 0 [0, d] d 0 dim θ (•) Intermediate 0 [0, d] d 0 Den log (•) Logarithmic density [0, d] 0 d d Den pix (•) Pixel density [0, d] 0 d d Dim H (•) Macroscopic Hausdorff [0, d] 0 d d
Table I.2: A summary of the fractal dimensions of discrete and bounded sets.

Remark 6. We also mention a few relations between the dimensions mentioned so far to give the reader some intuition:

dim H (F ) ≤ dim B (F ) ≤ dim B (F ) ; dim H (F ) ≤ dim θ (F ) ≤ dim θ (F ) ≤ dim B (F ) ; dim P (F ) ≤ dim B (F ) ; Den log (F ) ≤ Den pix (F ) ; Dim H (F ) ≤ Den pix (F )

I.1.3 Wavelets tools and wavelet leaders method

Another field of study we were interested in is studying precisely the path behavior, and in particular regularity, of stochastic processes. To this aim, wavelet analysis allowed to obtain series expansions for many stochastic processes which made it a key tool in studying point-wise properties.

There seems to be no agreement in the literature on one unique definition of a wavelet. Nevertheless, the following conditions are commonly used.

Definition I.1.24. We say ψ :

R → R is a wavelet if ψ ∈ L 1 (R) ∩ L 2 (R) satisfying the so-called admissibility condition R | ψ(ξ)| |ξ| dξ < ∞, (I.1.37)
where ψ is the Fourier transform of ψ.

Remark 7. An immediate but important consequence of the admissibility condition is that ψ has (at least) 1 vanishing moment, i.e. R ψ(x)dx = 0, which makes ψ orthogonal to polynomials of degree 0. In many situations, it is preferable to use wavelets that are orthogonal to all low-order polynomials. Therefore, it is generally required that ψ has M (M ∈ N) vanishing moments, i.e. for each m ∈ N such that m < M , the function

x → x m ψ(x) belongs to L 1 (R) and R x m ψ(x)dx = 0.
The regularity properties of a function can be studied by decomposing it in an orthonormal wavelet basis of the space L 2 (R).

Proposition I.1.25. Under some general assumptions ( [START_REF] Daubechies | Ten lectures on wavelets[END_REF][START_REF] Meyer | Wavelets and operators[END_REF][START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF]), it is possible to build a wavelet ψ such that {ψ(

2 j • -k) : j ∈ Z, k ∈ Z} forms an orthogonal basis of L 2 (R). Therefore, any function f ∈ L 2 (R) can be decomposed as f = j∈Z k∈Z c j,k ψ(2 j • -k), where c j,k = 2 j R f (x)ψ(2 j x -k) dx.
Remark 8. The coefficients c j,k are called the wavelet coefficients of f . For a given scale j (j ∈ N) and position k (k ∈ Z) the wavelet coefficients c j,k are usually associated with the dyadic cube λ j,k of R defined as

λ j,k := k 2 j , k + 1 2 j .
The notation Λ j will stand for the set of dyadic intervals λ of R with side length 2 -j . The unique dyadic interval from Λ j containing the point t ∈ R will be denoted λ j (t). The set of dyadic intervals is Λ := ∪ j∈N Λ j . Two dyadic intervals λ and λ ′ are adjacent if there exist j ∈ N such that λ, λ ′ ∈ Λ j and dist(λ, λ ′ ) = 0.

These dyadic cubes allow an easy geometric visualization of the concepts of wavelet leaders related to a point and to a cube defined in the following. Definition I.1.26. The wavelet leader of t 0 ∈ R at the scale j is defined as

d j (t 0 ) = max λ∈3λ j (t 0 ) sup λ ′ ⊆λ |c ′ λ |, (I.1.38)
where λ j (t 0 ) is the unique dyadic cube at the scale j containing t 0 , and 3λ j (t 0 ) is the set of dyadic intervals adjacent to λ j (t 0 ).

The wavelet leaders (d j (t 0 )) j∈N of t 0 are key quantities to study the pointwise regularity of t 0 , as we will see in Chapter III when studying the pointwise regularity of the generalized Rosenblatt process R H 1 ,H 2 .

I.2 Main contributions and structure of the thesis

In this section, we take a closer look at the contributions of this thesis. We give some of our results in a simplified and informal way summing-up all our main findings. Nevertheless, complete formulations and technical details can be find in referred respective chapters. Chapters II-IV deal with describing the geometric properties of sample paths, which played a significant role in modern stochastic analysis, and have been investigated using various methods. Two of the most relevant tools utilized are multifractal analysis and harmonic analysis. These are the approaches that we are going to develop in the majority of this thesis. In Chapter V we investigate the macroscopic Hausdorff dimension which played a pivotal role in our research. In particular, we develop a potential theoretical method to estimate this dimension. Then, we apply this method to obtain projection theorems that link between the macroscopic Hausdorff dimension of a set in R 2 , and its projections on almost every straight line passing through the origin.

This line of research started by studying the regularity and irregularity of the real valued Brownian motion. Paley, Wiener and Zygmund [START_REF] Paley | Notes on random functions[END_REF] have shown that its local Hölder regularity cannot be larger than 1/2. Followed by investigating the behavior of a Brownian motion on a given point, Khinchin [START_REF] Khinchin | On a theorem of probability calculation[END_REF] introduced a new notion of ordinary points by proving that the law of iterated logarithm holds almost surely. Later, Oray and Taylor [START_REF] Orey | How often on a Brownian path does the law of iterated logarithm fail[END_REF] proved that there exist exceptional points, called rapid points, where the law of the iterated logarithm fails. Furthermore, Kahane [START_REF] Kahane | of Cambridge Studies in Advanced Mathematics[END_REF] obtained the existence of a third category of points, presenting a slower oscillation. These points are called slow points.

Another natural question is studying the graph of a Brownian motion using various fractal dimensions, such as box, packing and Hausdorff dimension, which lead to some global and local geometric properties (see [START_REF] Mörters | Brownian motion[END_REF][START_REF] Taylor | The Hausdorff α-dimensional measure of Brownian paths in n-space[END_REF][START_REF] Taylor | The α-dimensional measure of the graph and set of zeros of a Brownian path[END_REF]). A further approach for understanding the features of a random path was through assessing the proportion of time spent by a Brownian motion in a given region, which is known by sojourn times. Many authors studied sojourn times associated to Brownian motion (see [START_REF] Ciesielski | First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path[END_REF][START_REF] Ray | Sojourn times and the exact Hausdorff measure of the sample path for planar Brownian motion[END_REF][START_REF] Uchiyama | The proportion of Brownian sojourn outside a moving boundary[END_REF][START_REF] Seuret | On sojourn of Brownian motion inside moving boundaries[END_REF]). As a consequence of these remarkable efforts, the Brownian case is well understood, and many authors have tried to extend these results to more general stochastic processes such as fractional Brownian motion and Rosenblatt process.

Fractional Brownian motion is a generalization of the Brownian motion. Mathematicians were motivated to extend the Brownian motion properties mentioned above, and to understand how much those findings rely on the specific features of Brownian motion, such as the (strong and weak) Markov properties. In this capacity, the law of iterated logarithm and the study of the set of fast points has naturally been studied and extended for more general classes of Gaussian processes, such as the fractional Brownian motion, see e.g. [START_REF] Marcus | Hölder conditions for Gaussian processes with stationary increments[END_REF][START_REF] Orey | Growth rate of certain Gaussian processes[END_REF][START_REF] Bingham | Variants on the law of the iterated logarithm[END_REF][START_REF] Marcus | Moduli of continuity of local times of strongly symmetric Markov processes via Gaussian processes[END_REF][START_REF] Monrad | Small values of Gaussian processes and functional laws of the iterated logarithm[END_REF][START_REF] Khoshnevisan | Fast sets and points for fractional Brownian motion[END_REF]. In 1999, Meyer, Sellan and Taqqu introduced their famous decomposition of the fractional Brownian motion using the Lemarié-Meyer wavelet [START_REF] Meyer | Wavelets, generalized white noise and fractional integration: the synthesis of fractional Brownian motion[END_REF], which can be used to generalize the notion of ordinary, rapid, and slow points for Gaussian wavelets series [START_REF] Esser | Slow, ordinary and rapid points for Gaussian Wavelets Series and application to Fractional Brownian Motions[END_REF], in particular for fractional Brownian motion.

One the other hand, fractal analysis played a major role in studying the random path of the fractional Brownian motion. Various random sets such as graph, level sets and sojourn times associated to fractional Brownian motion were assessed using packing dimension [START_REF] Xiao | Packing dimension of the image of fractional Brownian motion[END_REF], Hausdorff dimension [START_REF] Kahane | of Cambridge Studies in Advanced Mathematics[END_REF][START_REF] Falconer | Fractal geometry[END_REF], and macroscopic Hausdorff dimension [START_REF] Nourdin | Sojourn time dimensions of fractional Brownian motion[END_REF]. In Chapter II, we assess the level sets associated to the fractional Brownian motion using the macroscopic Hausdorff dimension. Our results recover Seuret-Yang's results [START_REF] Seuret | On sojourn of Brownian motion inside moving boundaries[END_REF] for Brownian motion and can be considered as an addendum to Nourdin-Peccati-Seuret's work [START_REF] Nourdin | Sojourn time dimensions of fractional Brownian motion[END_REF].

In this thesis, we mainly focus on generalizing all the results mentioned above to the Rosenblatt process case. Our investigations are essentially motivated by the fact that, unlike the fractional Brownian motion, the Rosenblatt process is not Gaussian. A natural question is how much this property impact random paths? In Chapter III, we study the fractal properties of the random sets and measures determined by the sample paths of a Rosenblatt process, and in Chapter IV we assess its pointwise regularity where the existence of three types of points is proved: slow, ordinary and rapid.

The last chapter of this thesis develops various methods for estimating the macroscopic Hausdorff dimension. Recalling the fact that the macroscopic Hausdorff dimension is a discrete analog of the Hausdorff dimension, we developed similar estimating methods used for the Hausdorff dimension. The two usual methods are the mass distribution principle and the potential theoretic method. The potential theoretic method is based on an integral analysis, and it is a practical tool with various applications. As an application of the new potential theoretic method, we obtain a Marstrand-like projection theorem, describing the dimension of almost all projections on lines of sets in R d .

Here below we give a global outline of each of the chapters.

Chapter II: A uniform result for the dimension of fractional Brownian motion level sets

This chapter is concerned with estimating the size of level sets of the fractional Brownian motion (B H t ) t≥0 , which are defined for any x ∈ R as

L B (x) = {t ≥ 0 : B H t = x}.
Due to self-similarity property of B H , L B (x) may look like a fractal, so in order to describe it quantitatively one can use a type of fractal dimension. In this aim, the macroscopic Hausdorff dimension Dim H has proven to be relevant in the present chapter because it gives an intuition about the geometry of the set, precisely whether it is scattered or not.

Our work add supplementary results to [START_REF] Nourdin | Sojourn time dimensions of fractional Brownian motion[END_REF]. In particular, in [START_REF] Nourdin | Sojourn time dimensions of fractional Brownian motion[END_REF] they proved

∀x ∈ R, P(Dim H L B (x) = 1 -H) = 1.
Our aim is to extend this result uniformly for all x which is a non-trivial mission. Formally, we proved:

Theorem I.2.1 (L. Daw (2021)).

P(∀x ∈ R : Dim H L B (x) = 1 -H) = 1. (I.2.1)
We note that Theorem I.2.1 also recovers Seuret-Yang's result [105, Theorem 2] (Brownian motion), using a more natural approach in our opinion, where the local time of the fractional Brownian motion plays a crucial role.

Chapter III: Fractal dimensions of the Rosenblatt process

In this chapter we focus on the fractal properties of the random sets and measures determined by the sample paths of the Rosenblatt process Z, i.e., we study the function Z t = Z t (ω), for a fixed ω ∈ Ω. Some (random) sets of interest are then:

Image set: Z(E) := {Z(t) : t ∈ E} ; (I.2.2) Graph set: Gr Z (E) := {(t, Z(t)) ∈ E × R : t ∈ E} ; (I.2.3) Level set: L Z (x) := {t ∈ R + : Z(t) = x} , x ∈ R; (I.2.4) Sojourn set: E Z (γ) := {t ∈ R + : |Z(t)| ≤ t γ } , γ > 0; (I.2.5) Inverse image: Z -1 (E ′ ) := {t ∈ R + : Z(t) ∈ E ′ } , (I.2.6)
where E ⊂ R + and E ′ ⊂ R are Borel sets. For instance, by self-similarity of Z, these sets may look like a fractal. As a result in this chapter, we measure the above sets using the Hausdorff (both classical and macroscopic), packing and intermediate dimensions, and the logarithmic and pixel densities (see Section I.1.2 for exact definitions). Our results can be collected in three theorems. First, we assess the image sets Z(E), for all E ⊂ R + , using intermediate dimension.

Theorem I.

< ε < 1, ∀x ∈ R, P (dim θ (L Z (x) ∩ [ε, 1]) = 1 -H) = 1 (I.2.9) ∀x ∈ R, P(dim P (L Z (x) ∩ [ε, 1]) = 1 -H) = 1 (I.2.10) P(∀x ∈ R : Dim H (L Z (x)) = 1 -H) = 1. (I.2.11)
Finally, we establish a result for the sojourn times E Z (γ).

Theorem I.2.4 (L. Daw, G. Kerchev (2021)). For E ⊂ R, let Den pix (E) and Den log (E) denote the pixel and logarithmic densities of E. Then, for all γ ∈ [0, H],

Den pix (E Z (γ)) = Den log (E Z (γ)) = γ + 1 -H, a.s. (I.2.12) Dim H (E Z (γ)) = 1 -H a.s. (I.2.13)
Many of the results listed above rely on Hölder regularity conditions for the sample paths, and more precisely, for the local time of the process. The existence of local time of Z was first established in [START_REF] Shevchenko | Properties of trajectories of the multifractional Rosenblatt process[END_REF]. Hölder regularity was then recovered in the recent paper [START_REF] Kerchev | Local times and sample path properties of the Rosenblatt process[END_REF]. For our analysis, as a preliminary step we also establish the time inversion property of the Rosenblatt process: Proposition I.2.5 (L. Daw, G. Kerchev (2021)). The inverse time process

t → Zt := t 2H Z 1/t , (I.2.14)
is also a Rosenblatt process.

In addition to that, a few properties of the density for the joint process (Z t 1 , Z t 2 ) are needed. Using techniques from [START_REF] Kerchev | Local times and sample path properties of the Rosenblatt process[END_REF] we were able to prove the following: Proposition I.2.6 (L. Daw, G. Kerchev (2021)). (i) The probability density function f : R → R + of Z 1 is continuous and f (x) > 0 for x ≥ 0.

(ii) For every t 1 , . . . , t n ≥ 0, the vector (Z t 1 , . . . , Z tn ) has a continuous density.

Chapter IV: Wavelet methods to study the pointwise regularity of the generalized Rosenblatt process

In this chapter we prove that the generalized Rosenblatt process (R H 1 ,H 2 (t)) t≥0 presents three kinds of local behaviors: slow, ordinary and rapid points. On this purpose, fine bounds on the increments of this process are needed, both from above and below. For the upper bounds, we take advantage of the wavelet-type representation of the generalized Rosenblatt process established in [START_REF] Ayache | Wavelet-type expansion of the generalized Rosenblatt process and its rate of convergence[END_REF] by the means of the Meyer's wavelet. For the lower bounds we use the compactly supported Daubechies wavelets basis, and our main tools are the wavelet leaders. These representations are our key tools to prove the following Theorem I.2.7 which is the main result of this chapter.

Theorem I.2.7 (L. Daw, L. Loosveldt (2022)). For all

H 1 , H 2 ∈ ( 1 2 , 1) such that H 1 +H 2 > 3 2
, there exists an event Ω H 1 ,H 2 of probability 1 satisfying the following assertions for all ω ∈ Ω H 1 ,H 2 and every non-empty interval I of R.

For almost every t ∈ I,

0 < lim sup s→t |R H 1 ,H 2 (t, ω) -R H 1 ,H 2 (s, ω)| |t -s| H 1 +H 2 -1 log log |t -s| -1 < +∞. (I.2.15)
Such points are called ordinary points.

There exists a dense set of points t ∈ I such that

0 < lim sup s→t |R H 1 ,H 2 (t, ω) -R H 1 ,H 2 (s, ω)| |t -s| H 1 +H 2 -1 log |t -s| -1 < +∞. (I.2.16)
Such points are called rapid points.

There exists a dense set of points t ∈ I such that

lim sup s→t |R H 1 ,H 2 (t, ω) -R H 1 ,H 2 (s, ω)| |t -s| H 1 +H 2 -1 < +∞. (I.2.17)
Such points are called slow points.

In [START_REF] Esser | Slow, ordinary and rapid points for Gaussian Wavelets Series and application to Fractional Brownian Motions[END_REF], Esser and Loosveldt proved the existence of slow, ordinary and rapid points for Gaussian wavelet series, in particular for the fractional Brownian motion. Theorem I.2.7 shows in particular that slow, ordinary and rapid points are not specific to Gaussian processes.

Chapter V: Potential methods and projection theorems for macroscopic Hausdorff dimension

In this chapter we build various methods for estimating the macroscopic Hausdorff dimension (see Section I.1.2.2.2 for formal definitions), which is a discrete analog of the standard Hausdorff dimension. A natural approach is to extend the estimating methods used for the standard Hausdorff dimension. In general, when assessing the standard Hausdorff dimension of a given set E ⊂ R d the challenging part is finding a lower bound. To this end, a famous approach is the potential theoretical method, which is based on integral analysis. Our aim in this chapter is to establish the potential theoretical method for the macroscopic Hausdorff dimension which requires careful analysis. Let us first introduce the macroscopic s-energy of a measure. Definition I.2.8. Let s ≥ 0, and let µ be a finite mass distribution on R d . The macroscopic (µ, s)-potential at a point x is defined as

ϕ s µ (x) := R d dµ(y) ∥x -y∥ s 2 ∨ 1 . (I.2.18)
The macroscopic s-energy of µ is

I s (µ) := R d ϕ s µ (x)dµ(x) = (R d ) 2 dµ(x)dµ(y) ∥x -y∥ s 2 ∨ 1 . (I.2.19)
This result is quite comparable to the standard Hausdorff dimension (see Section I.1.2.1.2), except that in the integrals (I.2.18) and (I.2.19), the quantity ∥x -y∥ s 2 ∨ 1 is simply ∥x -y∥ s 2 . This modification is justified by the fact that Dim H is not concerned with local behavior, so we are not interested in small interactions ∥x -y∥ 2 < 1. The following theorem illustrates the potential theoretical methods for the macroscopic Hasudorff dimension, and is one of the main results in this chapter.

Theorem I.2.9 (L. Daw, S. Seuret (2022)). Let E be a subset of R d .

1. If there exists a Radon measure µ on R d such that µ(E) = +∞ and if n≥0 2 ns I s (µ |Sn ) < +∞, then ν s (E) = +∞ and Dim H (E) ≥ s.

2. If ν s (E) = +∞, then for all 0 < ε < s there exists a Radon measure

µ ε on R d such that µ ε (E) = +∞ and n≥0 2 n(s-ε) I s-ε (µ ε |Sn ) < +∞.
Although the potential theoretic methods are very comparable to the ones established for the standard Hausdorff dimension [START_REF] Falconer | Fractal geometry[END_REF]Theorem 4.13], for the macroscopic Hausdorff dimension we consider the measure µ which is defined on R d , and we focus on the restriction of µ on every annulus S n . For this reason, we deal with sums over n.

A key ingredient in proving Theorem I.2.9 is the existence of macroscopic s-sets which can be defined as follows.

Definition I.2.10. Let s ≥ 0. A set E ⊂ R d is called a macroscopic s-set when Dim H (E) = s and ν s (E) < +∞.

We prove the existence of macroscopic s-sets.

Theorem I.2.11 (L. Daw, S. Seuret (2022)). Let E ⊂ R d be such that ν s (E) = +∞. Then there exists a macroscopic s-set E such that E ⊂ E. This extraction theorem is a key ingredient at various places in our proofs, in particular for the projection theorems which are an application of the potential theoretic methods we demonstrate in Theorem I.2.9.

Theorem I.2.12 (L. Daw, S. Seuret (2022)). Let E ⊂ R 2 be a Borel set. Define L θ as the straight line passing through 0 with angle θ, and proj θ E as the orthogonal projection of

E onto L θ . (a) If Dim H (E) < 1, then Dim H (proj θ E) = Dim H (E) for Lebesgue almost every θ ∈ [0, π]. (b) If Dim H (E) ≥ 1, then Dim H (proj θ E) = 1 for Lebesgue almost every θ ∈ [0, π].
It is natural to seek projection results for fractal dimensions, hence it is quite satisfactory to obtain the projection theorems for the macroscopic Hausdorff dimension. We expect that Theorem I.2.12 can be extended in higher dimensional spaces, and Theorem I.2.9 is useful in this situation.

Chapter II A Uniform Result for the Dimension of Fractional Brownian Motion Level Sets

The content of this chapter is a copy of the paper entitled "A Uniform Result for the Dimension of Fractional Brownian Motion Level Sets", and published in "Statistics and Probability Letters".

II.1 Introduction

Let B = {B t : t ≥ 0} be a fractional Brownian motion of index H ∈ (0, 1), that is, a centered, real-valued Gaussian process with covariance function

R(s, t) = E (B s B t ) = 1 2 |s| 2H + |t| 2H -|s -t| 2H , s, t ≥ 0. (II.1.1) Since E (B s -B t ) 2 = |s -t| 2H
, it is an immediate consequence of the Kolmogorov-Centsov continuity theorem that B admits a continuous modification. Throughout this note, we will always assume that B is continuous. It is also immediate (see, e.g., [START_REF] Nourdin | Selected aspects of fractional Brownian motion[END_REF]) that B is a self-similar process of exponent H, that is, for any a > 0,

{B at : t ≥ 0} d = a H B t : t ≥ 0 ,
where X d = Y means that two processes X and Y have the same distribution. Moreover, B has stationary increments, that is, for every s ≥ 0 ,

{B t+s -B s : t ≥ 0} d = {B t : t ≥ 0} .
This article is concerned with estimating the size of the level sets of B, which are defined for any x ∈ R as

L x = {t ≥ 0 : B t = x} . (II.1.2)
This line of research started with the seminal work of Taylor [START_REF] Taylor | The α-dimensional measure of the graph and set of zeros of a Brownian path[END_REF], who was the first to study the Hausdorff dimensions of the level sets in the case of a standard Brownian motion. His results were extended later on by Perkins [START_REF] Perkins | The exact Hausdorff measure of the level sets of Brownian motion[END_REF] who showed that, with probability one, the level sets L x have a Hausdorff dimension 1 2 for all x ∈ R. Hence, the local structure of the level sets in the Brownian case is well understood.

Another method to describe the geometric properties of the sample paths of a given process is in terms of its sojourn times. Here, the goal is to study the dimension of the amount of time spent by the stochastic process inside a moving boundary, that is, of the form

E(ϕ) := {t ≥ 0 : |B t | ≤ ϕ(t)} ,
where ϕ : R + → R is an appropriate function.

Strongly related to our note, we mention the recent work of Nourdin, Peccati and Seuret [START_REF] Nourdin | Sojourn time dimensions of fractional Brownian motion[END_REF], in which a specific large scale dimension is computed for the sojourn times

E γ := {t ≥ 0 : |B t | ≤ t γ } , 0 < γ < H, (II.1.3)
of the fractional Brownian motion B. Note that this choice for ϕ is completely natural here because, on the one hand, the fractional Brownian motion is selfsimilar (hence the choice of a power function for ϕ) and, on the other hand, it satisfies a law of iterated logarithm as t → ∞ (hence the range (0, H) for γ). Actually, [START_REF] Nourdin | Sojourn time dimensions of fractional Brownian motion[END_REF] extended to the fractional Brownian motion the results given by Seuret and Yang [START_REF] Seuret | On sojourn of Brownian motion inside moving boundaries[END_REF] in the framework of the standard Brownian case.

In general, defining a notion of fractal dimension for a subset of R d involves taking into consideration the microscopic (i.e. local) properties of this set. However, many models in statistical physics are based on the Euclidean lattice Z d ; in this case, it may look more natural to rely on the macroscopic (i.e. global) properties of the set to define a notion of dimension. This is what Barlow and Taylor proposed in [START_REF] Barlow | Fractional dimension of sets in discrete spaces[END_REF][START_REF] Barlow | Defining fractal subsets of Z d[END_REF]. Their dimension, called macroscopic Hausdorff dimension, has proven to be relevant in many contexts. This is the one that was used in [START_REF] Nourdin | Sojourn time dimensions of fractional Brownian motion[END_REF][START_REF] Seuret | On sojourn of Brownian motion inside moving boundaries[END_REF], and also the one we will use in the present note, because it can give a good intuition about the geometry of the set into consideration, precisely whether it is scattered or not. Precise definitions will be given in Section II.2.1. At this stage, we only mention that we denote this macroscopic Hausdorff dimension by Dim H .

Our note can be considered as an addendum to [START_REF] Nourdin | Sojourn time dimensions of fractional Brownian motion[END_REF]. Let L x be the level sets associated with a fractional Brownian motion. In [START_REF] Nourdin | Sojourn time dimensions of fractional Brownian motion[END_REF], the following is shown.

Theorem II.1.1. Fix x ∈ R. Then P(Dim H L x = 1 -H) = 1.
Our aim is to extend Theorem II.1.1 from "∀x, P(. . .) = 1" to "P(∀x : . . .) = 1". To this end, new and non-trivial arguments are required. We will prove the following.

Theorem II.1.2.

P(∀x ∈ R : Dim H L x = 1 -H) = 1.
(II. 1.4) We note that our Theorem II.1.2 also recovers Seuret-Yang's result [105, Theorem 2] (Brownian motion), and provides a proof that we find more natural.

Throughout the note, every random object is defined on a common probability space (Ω, A, P), and E denotes the expectation with respect to P.

II.2 Preliminaries

This section gathers the different tools that will be needed in order to prove Theorem II.1.2.

II.2.1 Macroscopic Hausdorff Dimension

Following the notations of [START_REF] Khoshnevisan | Intermittency and multifractality: a case study via parabolic stochastic PDEs[END_REF][START_REF] Khoshnevisan | On the macroscopic fractal geometry of some random sets[END_REF], we consider the intervals S -1 = [0, 1/2) and S n = [2 n-1 , 2 n ) for n ∈ N. For E ⊂ R + , we define the set of proper covers of E restricted to S n by

I n (E) = {I i } m i=1 : I i = [x i , y i ] with x i , y i ∈ N, y i > x i , I i ⊂ S n and E ∩ S n ⊂ m i=1 I i .
For any set E ⊂ R + , ρ ≥ 0 and n ≥ -1, we define

ν n ρ (E) = inf m i=1 diam(I i ) 2 n ρ : {I i } m i=1 ∈ I n (E) , (II.2.1) where diam([a, b]) = b -a.
The key point in the definition of ν n ρ (E) is that the sets I i are non-trivial intervals with integer boundaries; in particular, the infimum is reached.

Definition II.2.1. Let E ⊂ R + . The macroscopic Hausdorff dimension of E is defined by Dim H E = inf ρ > 0 : n≥-1 ν n ρ (E) < +∞ . (II.2.2)
We observe that Dim H E always belongs to [0, 1], whatever E ⊂ R + . Indeed, consider the family -ρ) . Thus, ν n 1+ε (E) ≤ 2 -nε for all ε > 0, implying in turn that Dim H E ≤ 1 + ε for all ε > 0. As a result, we have that Dim H E ∈ [0, 1]. In (II.2.1), the covers are chosen to have length larger than 1. This shows that the macroscopic Hausdorff dimension does not rely on the local structure of the underlying set.

I i = [2 n-1 + i -1, 2 n-1 + i], 1 ≤ i ≤ 2 n-1 , which belongs to I n (E) and satisfies m i=1 diam(I i ) 2 n ρ ≤ 1 2 2 n(1
The dimension of a set is unchanged when one removes any bounded subset, since the series in (II.2.2) converges if and only if its tail series converges. Consequently, the dimension of any bounded set E is zero. But the converse is not true, for example Dim H ({2 n , n ≥ 1}) = 0.

The macroscopic Hausdorff dimension not only counts the number of covers of a set but also it gives an intuition about the geometry of the set. Precisely, the more the points of the set are spread-out, the larger its dimension. For instance for 0 < α < 1, define the two sets A α and B α by for all n ≥ 1,

A α ∩ S n = 2 n-1 + k 2 n-1 2 nα : k ∈ {0, ..., 2 nα -1} ; B α ∩ S n = 2 n-1 + k 2 nα : k ∈ {0, ..., 2 nα -1} .

Even though both sets have same cardinality but Dim

H A α = α whereas Dim H B α = 0.
These features make the macroscopic Hausdorff dimension an interesting quantity describing the large scale geometry of a set; in particular, it appears to be well suited for the study of the level sets L x .

As we will see in our upcoming analysis, it might be sometimes wise to slightly modify the way Dim H E is defined, to get a definition that is more amenable to analysis. For this reason, let us introduce, for any E ⊂ R + , ρ > 0, ξ ≥ 0, and n ≥ -1, the quantity

ν n ρ,ξ (E) = inf m i=1 diam(I i ) 2 n ρ log 2 diam(I i ) 2 n ξ : {I i } m i=1 ∈ I n (E) . (II.2.3)
The difference between ν n ρ (E) and ν n ρ,ξ (E) is that we introduce a logarithmic factor in the latter. This modification has actually no impact on the definition of Dim H E, as stated by the following lemma.

Lemma II.2.2. Let ξ ≥ 0. For every set E ⊂ R + , Dim H E = inf ρ > 0 : n≥-1 ν n ρ,ξ (E) < +∞ . (II.2.4) Proof. Define dξ = inf ρ > 0 : n≥-1 ν n,ξ ρ (E) < +∞ . For n ≥ -1, consider {I i } m i=1 ∈ I n (E). As I i ⊂ S n , one has diam(I i ) ≤ 2 n-1 , implying in turn that log 2 diam(I i ) 2 n ξ ≥ 1.
Thus, ν n ρ,ξ (E) ≥ ν n ρ (E) and then Dim H E ≤ dξ . If Dim H E = 1, the conclusion is straightforward. So, let us assume that Dim H E < 1 and let us fix ε > 0 small enough and ρ < 1 such that ρ > Dim H E + ε. Since the function x → x ε |log 2 x| ξ is continuous on (0, 1] and tends to zero as x tends to zero, it follows that there exists c > 0 such that

|log 2 x| ξ ≤ cx -ε , ∀x ∈ (0, 1]
We deduce that, for all

{I i } m i=1 ∈ I n (E), m i=1 diam(I i ) 2 n ρ log 2 diam(I i ) 2 n ξ ≤ c m i=1 diam(I i ) 2 n ρ-ε
By taking the infimum over all {I i } m i=1 ∈ I n (E) and recalling the definitions (II.2.1) and (II.2.3), one deduces that

ν n ρ,ξ (E) ≤ cν n ρ-ε (E), implying in turn d ξ ≤ ρ -ε. Letting ρ tend to Dim H E + ε yields the result.

II.2.2 Local Time of Fractional Brownian Motion

As we will see, the use of the local time will play a key role throughout the proof of Theorem II.1.2.

Provided it exists, the local time x → L x (t) of a given process (X t ) t≥0 is, for each t, the density of the occupation measure µ t (A) = λ({s ∈ [0, t] : X s ∈ A}) associated with X, where λ stands for the Lebesgue measure; otherwise stated, one has

L(t) = dµt dλ .
In what follows, we shall also freely use the notation

L x ([a, b]) to indicate the quantity L x (b)-L x (a).
The case where X is Gaussian (and centered, say) has been widely studied in the literature. For instance, we can refer to the survey by Dozzi [START_REF] Dozzi | Occupation density and sample path properties of N -parameter processes[END_REF]. One of the main striking results in the Gaussian framework is the following easy-to-check condition that ensures that (L x (t)) t∈[0,T ],x∈R exists in L 2 (Ω) :

I := [0,T ] 2 ds dt R(s, s)R(t, t) -R(s, t) 2 < +∞, (II.2.5)
where R(s, t) = E (X s X t ); morever, in this case we have the Fourier type representation:

L x (t) = 1 2π R dy t 0 du e iy(Bu-x) . (II.2.6)
If X is Gaussian, selfsimilar of index H and satisfies (II.2.5), then it is immediate from (II.2.6) that its local time at level x also have some selfsimilarity properties in time with index 1 -H, but with a different level as stated below. More precisely, one has, for every c > 0:

(L x (ct)) t≥0,x∈R d = c 1-H (L c -H x (t)) t≥0,x∈R . (II.2.7)
When X stands for the fractional Brownian motion B of Hurst index H ∈ (0, 1), it is immediate that (II.2.5) and (II.2.7) are satisfied. But we can go further. A consequence of Berman's work [START_REF] Berman | Local nondeterminism and local times of Gaussian processes[END_REF] is that the local time associated to B is β-Hölder continuous in t for every β ≤ 1 -H and uniformly in x. On their side, German and Horowitz (see [START_REF] Geman | Occupation Densities[END_REF]Theorem 26.1]) proved that, for all fixed t, the local time (L x (t)) x∈R admits the Hölder regularity in space stated in the following lemma.

Lemma II.2.3 (Spatial Hölder continuity of local time). Assume X is a fractional Brownian motion of Hurst index H ∈ (0, 1) and consider its local time (L x (t)) x∈K , where K is a given compact interval in R. Then, for all β ∈ 0, 1 2 1 H -1 and for all t ≥ 0,

P sup x,y∈K |L x (t) -L y (t)| |x -y| β < ∞ = 1. (II.2.8)
As we will see, Lemma II.2.3 will be one of our main key tools in order to prove Lemma II.3.3 (which is one of the steps leading to the proof of Theorem II.1.2).

II.2.3 Filtration of Fractional Brownian Motion

A last crucial property of the fractional Brownian B that we will use in order to to prove Theorem II.1.2, is that the natural filtration associated with B is Brownian. We mean by this that there exists a standard Brownian motion (W u ) u≥0 defined on the same probability space than B such that its filtration satisfies, for all t > 0,

σ{B u : u ≤ t} ⊂ σ{W u : u ≤ t}.
(II.2.9) Property (II.2.9) is an immediate consequence of the Volterra representation of B (see, e.g., [START_REF] Baudoin | Equivalence of Volterra processes[END_REF]). It will be exploited together with the Blumenthal's 0 -1 law, in the end of the proof of Proposition II.3.1.

II.3 Proof of Theorem II.1.2 II.3.1 Upper bound for Dim H L x

By a theorem in [START_REF] Nourdin | Sojourn time dimensions of fractional Brownian motion[END_REF], for every γ ∈ (0, H), a.s.

Dim H E γ = 1 -H.
On the other hand, observe that for a fixed γ > 0 and x ∈ R, the level set L x is ultimately included in E γ . Indeed,

L x ∩ [|x| 1/γ , +∞) ⊂ E γ .
We have recalled in Section II.2.1 that the macroscopic Haussdorff dimension is insensitive to the suppression of any bounded subset. As a result, a.s. for every x ∈ R,

Dim H L x = Dim H L x ∩ [|x| 1/γ , +∞) ≤ Dim H E γ = 1 -H.

II.3.2 Lower bound for Dim H L x

Recall S n from Section II.2.1, and let us introduce the random variables

Z x n = L x (S n ) 2 n(1-H) and F x N = N n=1 Z x n . (II.3.1)
The random variables (Z x n ) n≥-1 are positive, so (F x N ) N ≥1 is non-decreasing. We denote by

F x ∞ its limit, i.e. F x ∞ = ∞ n=-1 Z x n ∈ [0, +∞]. Using (II.2.7), we have for all n ≥ 0 Z x n d = Z 2 -nH x 0 . (II.3.2)
We note that similar random variables H) were introduced in [87, Section 5.3]. However, the fact that we are dealing with other space variables compared to [START_REF] Nourdin | Sojourn time dimensions of fractional Brownian motion[END_REF] induce several differences in our proofs. Although its statement is exactly the same than [87, Lemma 5], the meaning and the context of our proof are different. This is why we provide all the details, for the convenience of the reader. Our aim now is to link the random variable Z x n to the microscopic Hausdorff dimension. To this end, let us introduce the random variables

Y x n = L 2 n x (S n ) 2 n(1-
A n := sup 0≤t≤2 n sup 0≤h≤2 n-1 sup y∈R L y ([t, t + h]) h 1-H (n -log 2 h) H , (II.3.3)
where log 2 stands for the binary logarithm (base 2). By (II.2.7), we have

A n = sup 0≤t≤1 sup 0≤h≤1/2 sup y∈R L y ([2 n t, 2 n (t + h)]) (2 n h) 1-H (-log 2 h) H (II.3.4) d = sup 0≤t≤1 sup 0≤h≤1/2 sup y∈R L y ([t, t + h]) h 1-H (-log 2 h) H .
First, let us prove that A n is finite almost surely. We start by making use of a result of Xiao [128, Theorem 1.2] that describes the scaling behavior of local times of Gaussian processes with stationary increments; in particular, this applies to the fractional Brownian motion and we have, with probability one:

M := lim r↓0 sup 0≤t≤1 sup 0≤h≤r sup y∈R L y ([t, t + h]) h 1-H (-log 2 h) H < ∞.
By the very definition of a limit, we deduce the existence of a (random) real number 0 < r < 1/2 such that, almost surely,

sup 0≤t≤1 sup 0≤h≤r sup y∈R L y ([t, t + h]) h 1-H (-log 2 h) H ≤ 2M. (II.3.5) Now for r ≤ h ≤ 1/2, we have h 1-H (-log 2 h) H ≥ r 1-H and L y ([t, t + h]) ≤ L y ([0, 3/2]) for all 0 ≤ t ≤
sup y∈R L y ([t, t + h]) h 1-H (-log 2 h) H ≤ r H-1 sup 0≤t≤1 sup r≤h≤1/2 sup y∈R L y ([t, t + h]) (II.3.6) ≤r H-1 sup y∈R L y ([0, 3/2]) < ∞.
Finally, by summing up (II.3.5) and (II.3.6), one has

P sup 0≤t≤1 sup 0≤h≤1/2 sup y∈R L y ([t, t + h]) h 1-H (-log 2 h) H < ∞ = 1. Now for K > 0 define the event Ω K := sup 0≤t≤1 sup 0≤h≤1/2 sup y∈R L y ([t, t + h]) h 1-H (-log 2 h) H ≤ K . (II.3.7)
Fix x ∈ R and consider the level set L x defined by (II.1.2). By recalling Definition II.2.3, we have: if

(I i = [s i , t i ]) m i=1 ∈ I n (L x ) is a cover minimizing ν n 1-H,H (L x ) then, ν n 1-H,H (L x ) = m i=1 |t i -s i | 2 n 1-H log 2 |t i -s i | 2 n H . (II.3.8)
Using (II.3.4) and a scaling argument with t =

s i 2 n , h = t i -s i 2 n
, and y = 2 -nH x, we deduce that

|t i -s i | 2 n 1-H log 2 |t i -s i | 2 n H ≥ K -1 L x (I i ) 2 n(1-H) on Ω K .
Back to (II.3.8), we have

ν n 1-H,H (L x ) ≥ K -1 m i=1 L x (I i ) 2 n(1-H) ≥ K -1 L x (S n ) 2 n(1-H) = K -1 Z x n , on Ω K , (II.3.9)
where the last inequality holds because the local time L x • increases only on the set I i (whose union covers L x S n ). Finally, one gets

Ω K ⊂ ∀x ∈ R, ∀n ≥ -1 : ν n 1-H,H (L x ) ≥ K -1 Z x n .
Using (II.3.9) for the first inclusion and Lemma II.2.2 for the second one, we can write

Ω K ∩ {∀x ∈ R, F x ∞ = +∞} ⊂{∀x ∈ R, n≥-1 ν n 1-H,H (L x ) = +∞} (II.3.10) ⊂{∀x ∈ R, Dim H L x ≥ 1 -H}.
But by definition of Ω K we have

P(Ω K ) ----→ K→+∞ P sup 0≤t≤1 sup 0≤h≤1/2 sup y∈R L y ([t, t + h]) h 1-H (-log 2 h) H < ∞ = 1. (II.3.11)
As a consequence, in order to conclude the proof of Theorem II.1.2, it remains to check that

P(∀x ∈ R, F x ∞ = +∞) = 1.
Then, using II.3.10, by letting K ↑ ∞ an a.s. uniform lower bound of Dim H L x is attained. The object of the next proposition is prove that F x ∞ = +∞ almost surely for all x ∈ R.

Proposition II.3.1. We have

P(∀x ∈ R, F x ∞ = +∞) = 1 (II.3.12)
Note that the following stronger statement of Proposition II.3.1 was shown in [START_REF] Nourdin | Sojourn time dimensions of fractional Brownian motion[END_REF]: for all x ∈ R, P(F x ∞ = +∞) = 1. Our main contribution in the present note is precisely to prove the strongest version stated in Proposition II.3.1.

II.3.3 Proof of Proposition II.3.1

For every a > 0, define

Z a n = inf x∈[-a,a] Z x n and F a ∞ = n≥1 Z a n . (II.3.13)
Recalling (II.2.7), we get for all n ≥ 0

Z a n = inf x∈[-a,a] Z x n d = inf x∈[-a,a] Z 2 -nH x 0 = inf x∈[-2 -nH a,2 -nH a] Z x 0 = Z 2 -nH a 0 . (II.3.14)
In the three forthcoming lemmas, the following three facts are established:

(i) the existence of ε > 0 such that P(Z 0 0 > 4ε) > 0 (Lemma II.3.2), (ii) the existence of a > 0 such that P(Z 0 0 > 4ε) ≤ 2P( Z a 0 > 0) (Lemma II.3.3), (iii) that P F b ∞ = ∞ ≥ P Z a 0 > 0 for all b > 0 (Lemma II.3.4).
Combining the results obtained in (i) to (iii), we deduce that

P F b ∞ = ∞ > 0 for all b > 0. (II.3.15) Set B u = u 2H B 1/u , u > 0.
By the time inversion property of the fractional Brownian motion, B is a fractional Brownian motion of Hurst index H as well. We can write

L x (S n ) = 1 2π R dy e -iyx 2 n 2 n-1 due iyu 2H B 1/u .
As a result, we get that

x → L x (S n ) is σ B u : u ≤ 2 -(n-1) -measurable, implying in turn that σ Z b n : n ≥ M ⊂ σ B u : u ≤ 2 -(M -1) (II.3.16)
for every M ≥ 1. Consequently, 1) .

F b ∞ = ∞ ∈ M ≥1 σ B u : u ≤ 2 -(M -
Using (II.2.9), there exists a standard Brownian motion (W u ) u≥0 defined on the same probability space such that

F b ∞ = ∞ ∈ M ≥1 σ W u : u ≤ 2 -(M -1) . (II.3.17)
By the Blumenthal's 0-1 law, the probability P F b ∞ = ∞ is either 0 or 1. But by (II.3.15), this probability is strictly positive; hence we conclude that

P F b ∞ = ∞ = 1 for all b > 0. (II.3.18)
For every b > 0, one has

P (∀x ∈ [-b, b] : F x ∞ = ∞) = P inf x∈[-b,b] F x ∞ = ∞ = P inf x∈[-b,b] N ≥1 Z x N = ∞ ≥ P N ≥1 inf x∈[-b,b] Z x N = ∞ = P F b ∞ = ∞ = 1.
We finally conclude that

P (∀x ∈ R, F x ∞ = ∞) = lim b→∞ P (∀x ∈ [-b, b], F x ∞ = ∞) = 1,
which is the desired conclusion of Proposition II.3.1.

To conclude, it remains to state and prove the three lemmas mentioned in points (i) to (iii).

Lemma II.3.2. For all ε > 0 small enough such that P(Z 0 0 > 4ε) > 0.

Proof.

Using that L 0 [ 1 2 , 1] = 1 2π R dy 1 1 2
du e iyBu , we have

E L 0 1 2 , 1 = 1 2π 1 1 2 u -H du R e -z 2 2 dz = 1 √ 2π 1 1 2 u -H du > 0.
As a result, P (Z 0 0 > 0) = P L 0 1 2 , 1 > 0 > 0, and the desired conclusion follows. Lemma II.3.3. For every ε > 0 small enough, there exists a real number a > 0 such that

0 < P(Z 0 0 > 4ε) ≤ 2P( Z a 0 > 0). Proof. Let β < 1 2 1 H -1 , K = [-1, 1] and J = [ 1 2 , 1]. Set c = c(ω) := sup x∈K\{0} L 0 (J)(ω) -L x (J)(ω) |x| β . By Lemma II.2.3, we have that P(c < ∞) = 1. Set η ε = η ε (ω) := min ε c(ω) 1/β , 1 . As [-η ε , η ε ] ⊂ [-1, 1], one has ∀|x| ≤ η ε (ω), L 0 (1)(ω) -L x (1)(ω) -L 0 1 2 (ω) -L x 1 2 (ω) ≤ ε. (II.3.19)
By triangle inequality,

L x (1) -L x 1 2 ≥ L 0 (1) -L 0 1 2 -(L 0 (1) -L x (1)) -L 0 1 2 -L x 1 2 .
(II.3.20)

Using (II. 3.19) and (II.3.20), we have

Z 0 0 = L 0 (1) -L 0 ( 1 2 ) > 4ε ⊂ ∀|x| ≤ η ε (ω), |L x (1) -L x ( 1 2 )| ≥ 3ε . (II.3.21) But ∀|x| ≤ η ε (ω), L x (1) -L x 1 2 ≥ 3ε) = inf x∈[-ηε,ηε] L x (1) -L x 1 2 ≥ 3ε . Re-
calling the definition of Z ηε 0 , we deduce that

P Z ηε 0 > 0 ≥ P Z ηε 0 > 3ε ≥ P Z 0 0 > 4ε > 0. (II.3.22)
Now for all a > 0, we have

Z ηε 0 > 0 ⊂ Z a 0 > 0 ∪ {η ε ≤ a} . (II.3.23)
Since c < ∞ a.s., one has that P (c ≥ M ) → 0 as M → ∞. We can then choose a > 0 small enough such that

P (η ε ≤ a) = P c ≥ ε 2aβ ≤ 1 2 P Z 0 0 > 4ε . (II.3.24)
Using (II.3.22), (II.3.23) and (II.3.24) we deduce that

P Z 0 0 > 4ε ≤ P Z ηε 0 > 0 ≤ P Z a 0 > 0 + P (η ε ≤ a) ≤ P Z a 0 > 0 + 1 2 P Z 0 0 > 4ε .
Finally, by Lemma II.3.2

0 < P Z 0 0 > 4ε ≤ 2P Z a 0 > 0 ,
which is the desired conclusion.

Lemma II.3.4. For any a, b > 0, we have

P F b ∞ = ∞ ≥ P Z a 0 > 0 . Proof. Fix γ > 0 and a, b > 0, consider the event A γ,b = F b ∞ ≤ γ . By Fubini's theorem, γ ≥ E 1 A γ,b F b ∞ = n≥-1 E 1 A γ,b Z b n = n≥-1 ∞ 0 P A γ,b ∩ { Z b n > u} du.
Using P (A ∩ B) ≥ (P(A) -P(B c )) + where B c denotes the complement of B, and recalling (II.3.14), we deduce that

γ ≥ n≥0 ∞ 0 P (A γ,b ) -P Z b n ≤ u + du = n≥0 ∞ 0 P (A γ,b ) -P Z 2 -nH b 0 ≤ u + du.
There exists M ≥ 1 such that 2 -nH b ≤ a for all n ≥ M . Then, for all n ≥ M ,

P Z 2 -nH b 0 ≤ u ≤ P Z a 0 ≤ u and γ ≥ n≥M ∞ 0 P (A γ,b ) -P Z a 0 ≤ u + du.
Since the summand does not depend on n and the series is bounded by γ and thus finite, one has necessarily

∞ 0 P (A γ,b ) -P Z a 0 ≤ u + du = 0. P F b ∞ ≤ γ = P (A γ,b ) ≤ P Z a 0 ≤ u . (II.3.25)
We know that P( Z a 0 ≤ u) is increasing as a function of u. Hence, (II.3.25) is actually true for every u ≥ 0 and γ ≥ 0. Hence

P F b ∞ > n ≥ P Z a 0 > 1 n for all n ∈ N. One conclude that P F b ∞ = ∞ ≥ P Z a 0 > 0 .
Chapter III

Fractal dimensions of the Rosenblatt process

The content of this chapter is a copy of the paper entitled "Fractal dimensions of the Rosenblatt process", written with "George Kerchev", and submitted to "Stochastic Processes and their Applications".

III.1 Introduction

The Rosenblatt process Z = (Z t ) t≥0 is a stochastic process that is a limit of normalized sums of long-range dependent random variables. It belongs to the class of Hermite processes and is the simplest member that is non-Gaussian. It has continuous but nowhere differentiable paths and is selfsimilar of order H ∈ (1/2, 1) with stationary increments.

The process Z, due to its self-similarity, can find applications across a multitude of fields like internet traffic [START_REF] Chaurasia | Performance of synthetic Rosenblatt process under multicore architecture[END_REF], hydrology, and turbulence [START_REF] Sakthivel | Retarded stochastic differential equations with infinite delay driven by Rosenblatt process[END_REF][START_REF] Lakhel | Existence, uniqueness and stability of impulsive stochastic neutral functional differential equations driven by Rosenblatt process with varyingtime delays[END_REF]. We refer the reader to [START_REF] Embrechts | Selfsimilar processes[END_REF] and [START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF] for a detailed review of the properties associated with self-similarity. In particular, the Rosenblatt process is used in finance [START_REF] Torres | Donsker type theorem for the Rosenblatt process and a binary market model[END_REF][START_REF] Stoyanov | Pricing derivatives in Hermite markets[END_REF][START_REF] Fauth | Multifractal random walk driven by a Hermite process[END_REF] and statistical inference [START_REF] Lévy-Leduc | Asymptotic properties of U -processes under long-range dependence[END_REF][START_REF] Dehling | Non-parametric change-point tests for long-range dependent data[END_REF][START_REF] Nourdin | Statistical inference for Vasicek-type model driven by Hermite processes[END_REF].

From a mathematical standpoint the process has received a lot of interest since its inception in [START_REF] Rosenblatt | A central limit theorem and a strong mixing condition[END_REF]. Its distribution is not known in explicit form but was studied first in [START_REF] Albin | A note on Rosenblatt distributions[END_REF] and more recently in [START_REF] Maejima | On the distribution of the Rosenblatt process[END_REF] and [START_REF] Veillette | Properties and numerical evaluation of the Rosenblatt distribution[END_REF]. There are three integral representations: in terms of time, the spectrum and on finite intervals, see [START_REF] Taqqu | The Rosenblatt process[END_REF]. There is also a wavelet representation [START_REF] Pipiras | Wavelet-type expansion of the Rosenblatt process[END_REF] (see also the recent article [START_REF] Ayache | Wavelet-type expansion of the generalized Rosenblatt process and its rate of convergence[END_REF] for the wavelet representation of the generalized Rosenblatt process and its rate of convergence). From a statistical point of view, the value of the Hurst index H is important for practical applications and various estimators exist, see [START_REF] Bardet | A wavelet analysis of the Rosenblatt process: chaos expansion and estimation of the self-similarity parameter[END_REF]121].

In the present paper, we focus on the fractal properties of the random sets and measures determined by the sample paths of Z, i.e., if the underlying probability space is (Ω, F , P), we study the function Z(t) = Z t (ω), for a fixed ω ∈ Ω. Some (random) sets of interest are then: Fractal dimensions give you an intuition about the geometry of a set. Having identified some interesting random sets and possible ways to measure them, we note that such studies can be traced to the pioneering work of Lévy [START_REF] Lévy | La mesure de Hausdorff de la courbe du mouvement Brownien[END_REF] and Taylor [START_REF] Taylor | The Hausdorff α-dimensional measure of Brownian paths in n-space[END_REF][START_REF] Taylor | The α-dimensional measure of the graph and set of zeros of a Brownian path[END_REF][START_REF] Taylor | Sample path properties of a transient stable process[END_REF] on the sample path properties of the Brownian motion. We refer the reader to [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF] and [START_REF] Xiao | Random fractals and Markov processes[END_REF] for surveys of such results for Lévy and Markov processes respectively.

Image set: Z(E) := {Z(t) : t ∈ E} ; (III.1.1) Graph set: Gr Z (E) := {(t, Z(t)) ∈ E × R : t ∈ E} ; (III.1.2) Level set: L Z (x) := {t ∈ R + : Z(t) = x} , x ∈ R; (III.1.3) Sojourn set: E Z (γ) := {t ∈ R + : |Z(t)| ≤ t γ } , γ > 0; (III.1.4) Inverse image: Z -1 (E ) := {t ∈ R + : Z(t) ∈ E } , ( III
An important class of such dimensions reflects local properties of the set. One important example is the classical Hausdorff dimension, which can be defined as follows using the Hausdorff content, see [START_REF] Falconer | Fractal geometry[END_REF]Section 3.2]. For E ⊂ R,

dim H (E) := inf s ≥ 0 : ∀ε > 0, ∃ cover {U i } ∞ i=1 of E, s.t. ∞ i=1 |U i | s ≤ ε , (III.1.6)
where |F | denotes the diameter of the set F . Moreover by imposing further restrictions on the sets in the cover {U i } one can recover the definitions of box dimension. In particular, for E ⊂ R, the lower box dimension is given by:

dim B (E) := inf s ≥ 0 : ∀ε > 0, ∃ cover {U i } ∞ i=1 of E, s.t. |U i | = |U j | ∀i, j and ∞ i=1 |U i | s ≤ ε . (III.1.7)
Similarly, we define the upper box dimension:

dim B (E) := inf s ≥ 0 : ∀ε > 0, ∃δ > 0, ∀ cover {U i } ∞ i=1 of E, s t. |U i | ≤ δ, |U i | = |U j | ∀i, j and ∞ i=1 |U i | s ≤ ε . (III.1.8)
The box dimension dim B (E) is then given by the common value (if it exists) of dim B (E) and dim B (E). Next for θ ∈ [0, 1], the θ-intermediate dimensions dim θ (E) is a dimension that interpolate between the Hausdorff and box dimensions by increasing restriction on the relative sizes of covering sets as θ increases (δ

1/θ ≤ |U i | ≤ δ for all i).
In particular, one defines dim θ (E) and dim θ (E) similarly to dim B (E) and dim B (E). Then dim θ (E) is the common value if it exists of dim θ (E) and dim θ (E).

One need not consider only covers for the set E. For example, dim B (E) can be defined alternatively using coverings by small balls of equal radius (corresponding to dim B (E)) or using packings by disjoint balls of equal radius that are as dense as possible (corresponding to dim B (E)), see [START_REF] Falconer | Fractal geometry[END_REF]Section 3.4]. If the radii are allowed to differ the covering procedure corresponds to the classical Hausdorff dimension while the packing one is associated to the packing dimension dim P (E). In linear programming the packing and covering problems are dual of each other and thus the packing dimension can be considered as the dual analogue to the classical Hausdorff dimension. The precise definitions are delayed to Section III.3.

Other definitions of the packing and intermediate dimensions are possible by employing methods from potential theory. Thus, dim P (E), dim θ (E) and dim θ (E) can be expressed via capacities with respect to certain kernels, see [START_REF] Falconer | Packing dimensions of projections and dimension profiles[END_REF][START_REF] Burrell | Projection theorems for intermediate dimensions[END_REF]. This gives rise to packing and intermediate dimension profiles -dim P,α (E), dim θ,α (E) and dim θ,α (E) respectively. See III.3.2 for the precise definitions.

All the dimensions in the discussion above pertain to local properties of the set. It is often the case, for instance in statistical physics, that one needs to quantify global properties of an infinite set. The simplest way of assessing the size of such a set is given by its (Lebesgue) density at infinity. In particular, we utilize the logarithmic density Den log (E) and the pixel density Den pix (E) (the latter corresponding to the "pixelated" image). Alternatively, one can use the macroscopic Hausdorff dimension Dim H (E) introduced in [START_REF] Barlow | Fractional dimension of sets in discrete spaces[END_REF][START_REF] Barlow | Defining fractal subsets of Z d[END_REF] for the study of the macroscopic properties of random walks. More recent applications can be found in the study of high peaks of solutions of the stochastic heat equation [START_REF] Khoshnevisan | Intermittency and multifractality: a case study via parabolic stochastic PDEs[END_REF][START_REF] Khoshnevisan | On the macroscopic fractal geometry of some random sets[END_REF]. Definitions of these concepts are provided in Section III.4. A brief summary of all dimensions discussed can be seen in Table III.1.

We also mention a few relations between the dimensions mentioned so far to give the reader some intuition:

dim H (E) ≤ dim B (E) ≤ dim B (E) ; dim H (E) ≤ dim θ (E) ≤ dim θ (E) ≤ dim B (E) ; dim P (E) ≤ dim B (E) ; Den log (E) ≤ Den pix (E) . Dimension Name Cover Size Values Limit dim H (•) Classical Hausdorff Covering (0, δ] [0, 1] δ → 0 dim B (•) Box Upper -Covering Lower -Packing δ [0, 1] δ → 0 dim P (•) Packing Packing (0, δ] [0, 1] δ → 0 dim θ (•) Intermediate ∈ (δ 1/θ , δ) [0, 1] δ → 0 Den log (•) Logarithmic density Interval [1, 2 n ] [0, 1] n → ∞ Den pix (•) Pixel density Interval* [1, 2 n ] [0, 1] n → ∞ Dim H (•) Macroscopic Hausdorff Collections of sets in [2 n-1 , 2 n ) (0, δ] [0, 1] n → ∞
Table III.1: Overview of the types of fractal dimensions. For the pixel density the cover consists of the integer points in the interval at distance less than 1 from E.

Before we list our main results, we outline what is known regarding fractal properties of sample paths of a Hermite process of rank 1, i.e., the fractional Brownian motion. The fractional Brownian motion X = (X t ) t≥0 , like Z, is a selfsimilar stochastic process with stationary increments. Both processes, X and Z, share the same covariance structure and are governed by a parameter H (called Hurst parameter in both cases). Unlike the Rosenblatt process, the process X is Gaussian and H ∈ (0, 1). See Table III.2 for an overview of some fractal properties of sets associated with the sample paths of the fractional Brownian motion. For completeness we mention also some results regarding the graph and the inverse sets. If X : R N → R d is a fractional Brownian sheet, it has been proved in [START_REF] Adler | Hausdorff dimension and Gaussian fields[END_REF] that, almost surely, dim H Gr X ([0, 1] N ) = min {N/H, N + (1 -H)d}. The box dimension of the graph of the fractional Brownian sheet over a non degenerate cube Q of R N was determined in [START_REF] Kamont | On the fractional anisotropic Wiener field[END_REF].

X(E) L X (x) E X (γ) dim P (•) 1 H dim P,H (E) [129] 1 dim θ (•) 1 H dim θ,H (E) [21] 1 dim H (•) min 1, 1 H dim H (E) [54] 1 -H [38] 1 Dim H (•) 1 -H [28] 1 -H [87] Den pix (•) γ + 1 -H [87] Den log (•) γ + 1 -H [87]
Moreover, with probability 1, dim B (Gr X (Q)) = N + 1 -H. Regarding the inverse set, the following holds: for E a closed subset of R d , dim H (X -1 (E)) = N -Hd + dim H (E) (see [START_REF] Monrad | Local nondeterminism and Hausdorff dimension[END_REF]). We believe that analogous results can be established for the Rosenblatt process, but the sets in question are not the subject of the current paper.

Many of the results listed above rely on Hölder regularity conditions for the sample paths, and more precisely, for the local time of the process. Such properties have been established for stationary Gaussian processes, like the fractional Brownian motion, by Berman in [START_REF] Berman | Local times and sample function properties of stationary Gaussian processes[END_REF]. His analytic approach, which is based on properties of the Fourier transform of the underlying process, has been adapted to the Rosenblatt setting in [START_REF] Shevchenko | Properties of trajectories of the multifractional Rosenblatt process[END_REF] where existence of the local time of Z was first established. Hölder regularity was then recovered in the recent paper [START_REF] Kerchev | Local times and sample path properties of the Rosenblatt process[END_REF]. These new results now allow to generalize some of the results in Table III.2 for the Rosenblatt case. See Table III. [START_REF] Arneodo | multifractales et turbulences : de l'ADN aux croissances cristallines[END_REF]. All results in Table III.3 but the ones for the dimensions of the image of the process Z(E) are new. Our findings are collected in the following three propositions. First, for the image set we extend the results of [START_REF] Shieh | Hausdorff and packing dimensions of the images of random fields[END_REF] to the intermediate dimensions setting, as in [START_REF] Burrell | Dimensions of fractional brownian images[END_REF]: Then, we study the proportion of time spent by a stochastic process in a given region. We describe the size of the level sets L Z (x) in terms of intermediate dimensions and macroscopic Hausdorff dimension. The following holds: Theorem III.1.2. For E ⊂ R and θ ∈ [0, 1], let dim θ (E) and Dim H (E) denote the θ-intermediate and macroscopic Hausdorff dimensions of E. Then, for any x ∈ R and 0

Z(E) L Z (x) E Z (γ) dim P (•) 1 H dim P,H (E) [107] 1 -H 1 dim θ (•) 1 H dim θ,H (E) 1 -H 1 dim H (•) min 1, 1 H dim H (E) [107] 1 -H 1 Dim H (•) 1 -H 1 -H Den pix (•) γ + 1 -H Den log (•) γ + 1 -H Table III.3:
Theorem III.1.1. Let θ ∈ (0, 1] and E ⊂ R + be compact. Then almost surely: dim θ (Z(E)) = 1 H dim θ,H (E) , (III.1.9) and dim θ (Z(E)) = 1 H dim θ,H (
< ε < 1, ∀x ∈ R, P (dim θ (L Z (x) ∩ [ε, 1]) = 1 -H) = 1. (III.1.11)
And,

∀x ∈ R, P(dim P (L Z (x) ∩ [ε, 1]) = 1 -H) = 1. (III.1.12)
Moreover,

P(∀x ∈ R : Dim H (L Z (x)) = 1 -H) = 1. (III.1.13)
We believe that the same uniform result holds for classical Hausdorff dimension but we only prove the pointwise one. Finally, we establish the results for the sojourn times E Z (γ):

Theorem III.1.3. For E ⊂ R, let Den pix (E) and Den log (E) denote the pixel and logarithmic densities of E. Then, for all γ ∈ [0, H],

Den pix (E Z (γ)) = Den log (E Z (γ)) = γ + 1 -H, a.s. (III.1.14)
Moreover,

Dim H (E Z (γ)) = 1 -H a.s. (III.1.15)
To fill the missing entries in Table III.3 one needs new techniques. In particular, the macroscopic Hausdorff dimension and the two densities of the image set Z(E) should depend on the fractional properties of E (in particular should be 0 if E is bounded). However, intuition regarding this relation is missing. Regarding, the level set L Z (x), the approach for the macroscopic Hausdorff dimension does not translate since the key result (Lemma III.4.2) is an artifact of the definition of Dim H .

The authors believe that many of the results above can be extended to some generalizations of the Rosenblatt process, for instance, when the time and space sets are N and d dimensional, or when the Hurst index is a function of time, as in [START_REF] Shevchenko | Properties of trajectories of the multifractional Rosenblatt process[END_REF]. To ease the presentation only the case N = d = 1 and H ∈ (1/2, 1) -fixed is considered. However, establishing the results for Hermite processes of rank above 2 requires new techniques and is beyond the scope of the current paper. In particular, the Berman analytic approach relies on a "good" representation for the Fourier transform of the process and this is not known for Hermite processes of higher rank.

The structure of the paper is as follows. The three main results listed above are established in Sections III.3-III.5. Some necessary technical properties of the Rosenblatt process are reviewed and proved in Section III.2.

III.2 Properties of the Rosenblatt process

The Rosenblatt process is formally defined, for t ≥ 0 and H ∈ (1/2, 1), as

Z H t := lim n→∞ σ n H ⌊nt⌋ k=1 (Y 2 k -1), (III.2.1)
where (Y k ) k≥0 is a Gaussian sequence of mean zero, unit variance and covariance

E[Y 0 Y k ] = (1+k 2 ) -(1-H)/2
. The series converges in terms of finite dimensional distributions but also as weak convergence of probability measures (see [START_REF] Taqqu | The Rosenblatt process[END_REF] for further details). The parameter σ is an arbitrary constant and is taken such that the limit Z H t has unit variance. Letting t = 1, one recovers the example, constructed by Rosenblatt in [START_REF] Rosenblatt | A central limit theorem and a strong mixing condition[END_REF], to highlight the limitations of a cental limit theorem for stronly mixing sequences also stated in [START_REF] Rosenblatt | A central limit theorem and a strong mixing condition[END_REF].

The Rosenblatt process can also be defined in terms of Wiener-Itô stochastic integrals.

Following [START_REF] Taqqu | The Rosenblatt process[END_REF], (Z H t ) t≥0 is defined as the double Wiener integral with respect to a standard Brownian motion {B(x)} x∈R :

Z H t = ′ R 2 K H (t, x 1 , x 2 ) dB(x 1 )dB(x 2 ), (III.2.2)
where ′ R 2 denotes integration over R 2 excluding the diagonal and the kernel function K H given, for all (t, x 1 , x 2 ) on R + × R 2 , by

K H (t, x 1 , x 2 ) = c(H) t 0 (s -x 1 ) H-3 2 + (s -x 2 ) H-3 2 + ds, with x + = max(x, 0). The constant c(H) is a positive normalizing constant and it is chosen such that E((Z H 1 ) 2 ) = 1. More precisely, c(H) 2 = (H -1/2)(4H -3) β(H -1/2, 2 -2H)
This definition is also known as the time representation of the Rosenblatt process. Another closely related representation is the spectral representation of Z (see [START_REF] Taqqu | The Rosenblatt process[END_REF] and [START_REF] Dobrushin | Non-central limit theorems for nonlinear functionals of Gaussian fields[END_REF]):

Z H t = C(H) R 2 e i(x+y)t -1 i(x + y) Z G (dx)Z G (dy), (III.2.3)
where the double Wiener-Ito integral is taken over x ̸ = ±y and Z G (dx) is a complex-valued random white noise with control measure G satisfying G(tA

) = t 1-H G(A) for all t ∈ R and G(dx) = |x| -H dx. The constant C(H) in (III.2.3) is such that E[Z 2 t ] = t 2H and E[Z t Z s ] = 1 2 t 2H + s 2H -|t -s| 2H ,
for all s, t ≥ 0.

Remark 9. Note that in the notation of [START_REF] Taqqu | The Rosenblatt process[END_REF], Z G (dx) = |x| -H/2 d B(x), with (B(t)) t∈R the Brownian motion and d B(x) is viewed as the complex-valued Fourier transform of dB(x).

For more details, see [START_REF] Taqqu | Convergence of integrated processes of arbitrary Hermite rank[END_REF].

It is known (see [START_REF] Tudor | Analysis of the Rosenblatt process[END_REF]) that the Rosenblatt process has the following properties:

(1) self-similarity: Z is H-self-similar; that is, the processes {Z ct , t ≥ 0} and c H Z t , t ≥ 0 have the same distribution.

(2) stationary increments: Z has stationary increments; that is, the distribution of the process {Z t+s -Z s , t ≥ 0} does not depend on s ≥ 0.

(3) continuity: the trajectories of the Rosenblatt process Z are δ-Hölder continuous for every δ < H.

We will mention one more property that will be needed in our proofs, and is a consequence of the finite time interval representation [113,

= ∞ n=1 λ n (t 1 )(ε 2 n -1), . . . , ∞ n=1 λ n (t d )(ε 2 n -1) , (III.2.5)
where (ε n ) n≥1 are i.i.d N (0, 1) random variables and (λ n (t)) n≥1 are the (real) eigenvalues of a self-adjoint Hilbert-Schmidt operator associated with the process Z (see [START_REF] Dobrushin | Non-central limit theorems for nonlinear functionals of Gaussian fields[END_REF]).

For our analysis a few properties of the density for the joint process (Z t 1 , Z t 2 ) are needed. Using techniques from [START_REF] Kerchev | Local times and sample path properties of the Rosenblatt process[END_REF] we can establish the following:

Proposition III.2.1. (i) The probability density function f : R → R + of Z 1 is continuous and f (x) > 0 for x ≥ 0.
(ii) For every t 1 , . . . , t n ≥ 0, the vector (Z t 1 , . . . , Z tn ) has a continuous density.

Proof of Proposition III.2.1. (i) The density f of Z 1 is continuous (see [START_REF] Veillette | Properties and numerical evaluation of the Rosenblatt distribution[END_REF]Corollary 4.3]) and unimodal (see [START_REF] Maejima | On the distribution of the Rosenblatt process[END_REF]). Therefore f (0) > 0 since E[Z 1 ] = 0. To see that f (y) > 0 for all y > 0, recall [123, Corrolary 4.5]: for α > 0,

lim u→∞ P(Z 1 > u + α) P(Z 1 > u) = c H ,
for a deterministic constant c H > 0. In particular, this shows that for every y ∈ R + , there is x > y, such that f (x) > 0. Combined with the fact that f is continuous and unimodal, this implies that f (x) > 0 for every x ∈ R + .

(ii) If the characteristic function μ(z) of a probability measure µ in R d is integrable, then µ has a continuous density g(x) that tends to 0 as |x| → ∞ (see [

L 2 G := R |f (x)| 2 G(x)dx. For t ∈ R n + , ξ ∈ R n , let A t,ξ : L 2 G (R) → L 2 G (R)
be the operator given by

(A t,ξ f )(x) = R n j=1 ξ j e it j (x-y) -1 i(x -y) f (y)|y| -H/2 dy.
Let (λ k (t, ξ)) k≥1 be the set of eigenvalues of A t,ξ . Then,

E exp i n j=1 ξ j Z t j = k≥1 1 (1 + 4λ k (t, ξ)) 1/4 . Moreover, if t 0 = 0 < t 1 < • • • < t n ≤ 1, for every k ≥ 1, λ k (t, ξ) ≥ C(H)( max 1≤j≤n |ξ j -ξ j-1 ||t j -t j-1 | H ) 2 λ4 k , (III.2.6)
where λk ∼ k -H/2 (independent of t and ξ), ξ 0 = 0 and C(H) > 0 is a constant that only depends on H. Now, we follow a similar procedure to the one employed for the proof of [57, Proposition 1

.3]. Let f 0 : R n + × R n → R + be given by f 0 (t, y) := t H 1 |y 1 | ∨ t H 2 |y 2 | ∨ • • • ∨ t H n |y n |. (III.2.7) Further, let ξ ′ = (ξ 1 -ξ 0 , ξ 2 -ξ 1 , . . . , ξ n -ξ n-1 ) and t ′ = (t 1 -t 0 , t 2 -t 1 , . . . , t n -t n-1 ). Then R d E exp i n j=1 ξ j Z t j dξ = R d k≥1 (1 + 4λ k (ξ, t)) -1/4 dξ ≤ R d k≥1 1 + 4C(H)( max 1≤j≤n |ξ j -ξ j-1 ||t j -t j-1 | H ) 2 λ4 k -1/4 dξ = R d k≥1 1 + 4C(H)f 2 0 (t ′ , ξ ′ ) λ4 k -1/4 dξ ′ . (III.2.8) Let G(s) := k≥1 (1 + 4s 2 λ4 k ) -1/4 .
We can now switch to polar coordinates in (III.2.8) via

|ξ ′ | = r ′ , ξ ′ /r ′ = w ′ : R d k≥1 1 + 4C(H)f 2 0 (t ′ , ξ ′ ) λ4 k -1/4 dξ ′ ≤C |w ′ |=1 ∞ 0 (r ′ ) n-1 G( C(H)r ′ f 0 (t ′ , w ′ ))dr ′ H n-1 (dw ′ ) =C ∞ 0 R n-1 G(R)dR |w ′ |=1 (f 0 (t ′ , w ′ )) -n H n-1 (dw ′ ) ,
where H n-1 (dw ′ ) is the (n -1)-dimensional Hausdorff measure on the unit sphere, C > 0 is a constant that depends on H and the last equality follows with the change of variables

R = C(H)r ′ f (t ′ , w ′ ).
Next, recall [57, Lemma 2.3] that G(s) is finite and positive for any s > 0 and moreover there are constants c 1 , c 2 > 0 such that for all β ≥ 1,

∞ 0 s β-1 G(s)ds ≤ c 2 Hc -βH 1 Γ(βH),
where Γ is the Gamma function.

Finally, since t 1 , . . . , t n > 0 are fixed, by the definition (III.2.7) of f 0 (t ′ , w ′ ),

|w ′ |=1 (f 0 (t ′ , w ′ )) -n H n-1 (dw ′ ) ≤C(t ′ ) |w ′ |=1 (|w 1 | ∨ • • • ∨ |w n |) -n H n-1 (dw ′ ) ≤C(t ′ ) |w ′ |=1 (n -1/2 ) -n H n-1 (dw ′ ) < ∞,
where C(t ′ ) := inf{t ′Hn 1 , . . . , t ′Hn n } is a positive constant. Therefore, the characteristic function of (Z t 1 , . . . , Z tn ) is integrable and thus the joint distribution has a continuous density.

Next we establish a time inversion property for the Rosenblatt process: Proposition III.2.3. The inverse time process

t → Zt := t 2H Z 1/t , (III.2.9)
is also a Rosenblatt process.

Proof. First, using the spectral representation of a Rosenblatt process (III.2.3),

t 2H Z 1/t (d) = C(H)t 2H R 2 e i(x+y)/t -1 i(x + y) Z G (dx)Z G (dy) = C(H)t 2H R 2 e i(x ′ +y ′ )t -1 i(x ′ + y ′ )t 2 Z G (t 2 dx ′ )Z G (t 2 dy ′ ),
with the change of variables x = x ′ t 2 and y = y ′ t 2 . Now recall the change of variables formula for the Itô integral [32, Proposition 4.2]:

Proposition III.2.4. Let G and G ′ be two non-atomic spectral measures such that G is absolutely continuous with respect to G ′ , and let g(x) be a complex valued function such that

g(x) =g(-x), |g 2 (x)| = d(G(x)) d(G ′ (x)) .
Let f : R 2 → C be a measurable function such that:

1. f (-x 1 , -x 2 ) = f (x 1 , x 2 ), and
2. ||f || 2 = |f (x 1 , x 2 )| 2 G(dx 1 )G(dx n ) < ∞. Then, for f ′ (x 1 , x 2 ) = f (x 1 , x 2 )g(x 1 )g(x 2 ), f (x 1 , x 2 )Z G (dx 1 )Z G (dx 2 ) (d) = f ′ (x 1 , x 2 )Z G ′ (dx 1 )Z G ′ (dx 2 ). Let G t 2 (A) := G(At 2 ) = t 2(1-H) G(A)
for every measurable A. We apply Proposition III.2.4 with G and G t 2 , i.e., with |g(x)| 2 = t 2(1-H) a constant depending on t. Then,

C(H)t 2H R 2 e i(x ′ +y ′ )t -1 i(x ′ + y ′ )t 2 Z G (t 2 dx ′ )Z G (t 2 dy ′ ) = C(H)t 2H R 2 e i(x ′ +y ′ )t -1 i(x ′ + y ′ )t 2 Z G t 2 (dx ′ )Z G t 2 (dy ′ ) (d) = C(H)t 2H R 2 e i(x ′ +y ′ )t -1 i(x ′ + y ′ )t 2 t 2(1-H) Z G (dx ′ )Z G (dy ′ ) = C(H) R 2 e i(x ′ +y ′ )t -1 i(x ′ + y ′ ) Z G (dx ′ )Z G (dy ′ ),
and we recover the spectral representation of Z t as desired.

Remark 10. For the fractional Brownian motion B H t of Hurst index H ∈ (1/2, 1), the same fact is established using that the process is Gaussian and by comparing covariance functions. However, this property can also be recovered using the approach above. Indeed, we have the following spectral representation:

B H t (d) = C(H) R e iλt -1 iλ
1 |λ| H-1/2 d B(λ).
The same change of variables yields the desired conclusion.

We also recall a result [57, Proposition 4.2] regarding oscillations: Proposition III.2.5. Let (Z t ) t≥0 be the Rosenblatt process. Then for any s > 0 and h ∈ (0, s),

P sup t∈[s-h,s+h] |Z t -Z s | ≥ u ≤ C exp - u c 1 h H ,
where c 1 and C are constants that depend only on H.

We need the following properties of the local time of the Rosenblatt process. Its existence was shown in [START_REF] Shevchenko | Properties of trajectories of the multifractional Rosenblatt process[END_REF] and one has the representation:

L(x, t) = 1 2π R t 0 e iξ(x-Zs) dsdξ. (III.2.10)
As we mentioned in the beginning of this section, Z is selfsimilar of index H, then its local time at level x also has some selfsimilarity properties in time with index 1 -H, but with a different level as stated below. More precisely, one has, for every c > 0:

(L(x, ct)) t≥0,x∈R

(d) = c 1-H (L(c -H x, t)) t≥0,x∈R . (III.2.11)
Indeed, for every c > 0, t ≥ 0 and x ∈ R, one has There exist positive constants C 1 and C 2 such that, almost surely, for any s ∈ I,

L(x, ct) = 1 2π R ct 0 e iξ(x-Zs) dsdξ = c 1 2π R t 0 e iξ(x-Zcs) dsdξ (d) = c 1 2π R t 0 e iξ(x-c H Zs) dsdξ = c 1-H 1 2π R t 0 e iξ(c -H x-Zs) dsdξ = L(c -H x, t).
lim sup r→0 L * ([s -r, s + r]) r 1-H (log log r -1 ) 2H ≤ C 1 ,
(III.2.12) and lim sup

r→0 sup s∈I L * ([s -r, s + r]) r 1-H (log r -1 ) 2H ≤ C 2 . (III.2.13)
Furthermore, we can establish the following property which is key in the study of the classical Hausdorff dimension of the level sets.

Proposition III.2.7. For β ∈ 0, 1 2 1 H -1 , P sup x∈[-1,1]\{0} L 0, 1 2 , 1 -L x, 1 2 , 1 |x| β < ∞ = 1.
Proof. The result relies on a celebrated lemma due to Garsia, Rodemich and Rumsey [START_REF] Garsia | A real variable lemma and the continuity of paths of some Gaussian processes[END_REF], as well as on the moment estimates for the local time in [START_REF] Kerchev | Local times and sample path properties of the Rosenblatt process[END_REF]. First, let us recall the lemma from [START_REF] Garsia | A real variable lemma and the continuity of paths of some Gaussian processes[END_REF]:

Lemma III.2.8. Let Ψ(u) be a non-negative even function on (-∞, ∞) and p(u) be a non-negative even function on [-1, 1]. Assume both p(u) and Ψ(u) are non decreasing for u ≥ 0. Let f be continuous on [0, 1] and suppose that

1 0 1 0 Ψ f (u) -f (v) p(u -v) dudv ≤ B < ∞.
Then, for all x, y ∈ [0, 1],

|f (x) -f (y)| ≤ 8 |x-y| 0 Ψ -1 4B u 2 dp(u).
where Ψ -1 denotes the generalized inverse of Ψ.

Let Ψ(u) = |u| p and p(u) = |u| α+1/p where α ≥ 1/p and p ≥ 1. Then for any continuous f and x ∈ [0, 1],

|f (x) -f (y)| p ≤ C α,p |x -y| αp-1 [0,1] 2 |f (r) -f (v)| p |r -v| -αp-1 drdv.
Here the constant C α,p is given by C α,p = 4 • 8 p α + p -1 p α -p -1 -p . Thus, for fixed α and large enough p, we have C α,p ≤ C(α) p , where C(α) > 0 is a constant that depends on the chosen α. We apply this to f (x) = L 2x -1, 1 2 , 1 :

sup x∈[-1,1]\{0} L x, 1 2 , 1 -L 0, 1 2 , 1 p x 2 αp-1 ≤ C ′p [0,1] 2 L r, 1 2 , 1 -L v, 1 2 , 1 p |r -v| -αp-1 drdv.
Using the moment bounds for the occupation density established in [57, Theorem 3.1], one has

E sup x∈[-1,1]\{0} L x, 1 2 , 1 -L 0, 1 2 , 1 p x 2 αp-1 ≤ C ′p [0,1] 2 c(γ, H) p p p2H(1+γ) 2 γp |r -v| γp 2 (1-H-γH) |r -v| -αp-1 drdv,
where γ ∈ 0, 1-H 2H and c(γ, H) > 0 is a constant depending only on γ and H. Let α = γ/2 and p > 4/γ. Then,

E sup x∈[-1,1]\{0} L x, 1 2 , 1 -L 0, 1 2 , 1 p x 2 αp-1 ≤ C(γ, H, p),
where C(γ, H, p) > 0 is a constant that depends on γ, H and p.

Fatou's lemma implies that

P sup x∈[-1,1]/{0} L 0, 1 2 , 1 -L x, 1 2 , 1 |x| β < ∞ = 1,
as desired.

Finally, the local time is Hölder continuous in both time and space [57, Corollary 3.2]. In particular: Proposition III.2.9. For every x ∈ R,almost surely, the local time L(x, t) is Hölder continuous in t of order α for every α ∈ [0, 1 -H).

III.3 Image sets

The present section is dedicated to the study of intermediate dimensions and profiles. To make a comparison, we recall the more popular packing dimensions and profiles.

III.3.1 Packing dimensions

First, recall the definition of the packing dimension. For any α > 0, the α-dimensional packing measure of E ⊂ R N is

P s (E) := inf n P s 0 (E n ) : E ⊆ n E n ,
where for E ⊂ R,

P s 0 (E) := lim ε→0 sup i (2r i ) s : B(x i , r i ) are disjoint , x i ∈ E, r i < ε .
The packing dimension of E is dim P (E) := inf{s > 0 :

P s (E) = 0} (III.3.1)
and the packing dimension of a Borel measure µ on R N is defined by dim P (µ) := inf{dim P (E) : µ(E) > 0 and E ⊂ R N is a Borel set}.

Next, we recall the concept of packing dimension profiles first conceived by Falconer and Howroyd in [START_REF] Falconer | Packing dimensions of projections and dimension profiles[END_REF] and [START_REF] Howroyd | Box and packing dimensions of projections and dimension profiles[END_REF]. For finite Borel measures µ on R N and for any s > 0, let

F µ s (x, r) = R ψ s x -y r dµ(y),
be the potential with respect to the kernal ψ s (x) = min {1, ∥x∥ -s },∀x ∈ R N . The packing dimension profile of µ is defined as follows

dim P,s (µ) = sup β ≥ 0 : lim inf r→0 F µ s (x, r) r β = 0 for µ -a.e. x ∈ R N .
Now for any Borel set E ⊂ R N , we define M + c (E) to be the family of finite Borel measures on E with compact support in E. Then dim P (E) = sup dim P (µ) : µ ∈ M + c (E) . Motivated by this, Falconer and Howroyd [START_REF] Falconer | Packing dimensions of projections and dimension profiles[END_REF] define s-dimensional packing dimension profile of E ⊂ R N by dim P,s (E) = sup dim P,s (µ) : µ ∈ M + c (E) . It is easy to see that 0 ≤ dim P,s (E) ≤ s and for any s ≥ N , dim P,s (E) = dim P (E).

III.3.2 Intermediate dimensions

For a bounded and non-empty set E ⊂ R N , θ ∈ (0, 1] and s ∈ [0, N ], define

H s r,θ (E) = inf i |U i | s : {U i } i is a cover of E such that r ≤ |U i | ≤ r θ for all i . (III.3.2)
In particular, for θ = 0, H s r,0 (E) is the s-dimensional Hausdorff measure of E. Now, the intermediate dimensions are defined as in [START_REF] Falconer | Fractal geometry[END_REF]:

Definition III.3.1. Let E ⊂ R N be bounded. For 0 ≤ θ ≤ 1, the lower θ-intermediate dimension is dim θ (E) = the unique s ∈ [0, N ] such that lim inf r→0 log H s r,θ (E) -log r = 0. (III.3.3)
Similarly, the upper θ-intermediate dimension of E is defined by

dim θ (E) = the unique s ∈ [0, N ] such that lim sup r→0 log H s r,θ (E) -log r = 0. (III.3.4) When dim θ (E) = dim θ (E), we refer to the θ-intermediate dimension dim θ (E) = dim θ (E) = dim θ (E).
Thus, the classical Hausdorff (III.1.6) and box dimensions (III.1.7), (III.1.8) can be viewed as the extremes of a continuum of dimensions with increasing restrictions on the relative sizes of covering sets. Indeed, for every bounded E ⊂ R,

dim 0 E = dim 0 E = dim H (E) , dim 1 E = dim B (E) and dim 1 E = dim B (E) .
Moreover, the intermediate dimensions can be defined in terms of capacities with respect to an appropriate kernel denoted by ϕ s,m r,θ (see [START_REF] Burrell | Projection theorems for intermediate dimensions[END_REF]). For each collection of parameters θ ∈ (0, 1], 0 < m ≤ 1, 0 ≤ s ≤ m and 0 < r < 1, let ϕ s,m r,θ : R N → R be the function The intermediate dimension profiles are increasing in m and for

ϕ s,m r,θ (x) :=      1 0 ≤ |x| < r,
E ⊂ R N , dim θ,N (E) = dim θ (E) and dim θ,N (E) = dim θ (E) .
We note that originally the definitions of capacities and profiles above were established for E ⊂ R N and integers m ∈ (0, N ]. However, the recent result [21, Lemma 2.1], allows one to work with the version stated above. In fact, our first main result Theorem III.1.1 is an extension of a similar result in [START_REF] Burrell | Dimensions of fractional brownian images[END_REF] obtained for the index-α fractional Brownian motion. We proceed with the proof of Theorem III.1.1

III.3.3 Proof of Theorem III.1.1

Let θ ∈ (0, 1]. We first state two results due to Burrell [START_REF] Burrell | Dimensions of fractional brownian images[END_REF]. The first one establishes an upper bound for the intermediate dimensions of Hölder images using dimension profiles:

Lemma III.3.2. [21, Theorem 3.1] Let E ⊂ R be a compact, θ ∈ (0, 1], m ∈ {1, ..., n} and f : E → R. If there exist c > 0 and 0 < α ≤ 1 such that |f (x) -f (y)| ≤ c|x -y| α , for all x, y ∈ E, then dim θ (f (E)) ≤ 1 α dim θ,α (E) and dim θ (f (E)) ≤ 1 α dim θ,α (E) .
The second result gives a lower bound for the intermediate dimensions of image of a compact set E under measurable functions satisfying certain properties:

Lemma III.3.3. [21, Theorem 3.3] Let E ⊂ R be a compact, θ ∈ (0, 1], γ > 1 and s ∈ [0, 1). If f : Ω × E → R is a random function such that for each ω ∈ Ω, f (ω, .
) is a continuous measurable function and there exists c > 0 satisfying

P ({ω ∈ Ω : |f (ω, x) -f (ω, y)| ≤ r}) ≤ cϕ 1/γ,1/γ r γ ,θ (x -y), for all x, y ∈ E and r > 0, then dim θ (f (ω, E)) ≥ γdim θ,α (E) and dim θ (f (ω, E)) ≥ γdim θ,α (E) ,
for almost all ω ∈ Ω. Now let 0 < ε < H < 1. The Rosenblatt process Z has Hölder continuous paths in time of order H -ε, see [120, Propostion 3.5], and so there exists, almost surely, M > 0 such that

|Z s -Z t | ≤ M |s -t| H-ε ,
for all s, t ∈ E. In addition by Proposition III.2.1(i), the density function f of Z 1 is continuous and f (0) > 0. Then for all s, t ∈ E and r > 0, one has

P (|Z s -Z t | ≤ r) = P |Z 1 | ≤ r |s -t| H ≤ 4f (0) r |s -t| H = 4f (0) ϕ H,H r 1/H ,θ (s -t).
Now since the profiles are monotonically increasing, by Lemmas III.3.2 and III.3.3, one has almost surely

1 H dim θ,H (E) ≤ dim θ (Z(E)) ≤ 1 H -ε dim θ,H-ε (E) ≤ 1 H -ε dim θ,H (E) , and
1 H dim θ,H (E) ≤ dim θ (Z(E)) ≤ 1 H -ε dim θ,H-ε (E) ≤ 1 H -ε dim θ,H (E) .
Letting ε → 0 establishes the result.

III.4 Level sets

The present section is devoted to the proof of Theorem III.1.2. First, we establish (III.1.11) and (III.1.12) -the result regarding the θ-intermediate dimensions and the packing dimension. Recall that the definition of dim θ (E) and dim P (E) for E ⊂ R are given in definition III.3.1 and III.3.1 respectively. Note that the techniques employed in this section apply for the fractional Brownian motion case. As mentioned earlier, [9, Theorem 5] establishes dim [START_REF] Falconer | Fractal geometry[END_REF]. Thus from the defintion of the θ-intermediate dimensions (see

B (L X (x) ∩ [ε, 1]) ≤ 1 -H and dim H (L X (x) ∩ [ε, 1]) = 1 -H was shown in
III.3.2) dim θ (L X (x) ∩ [ε, 1]) = 1 -H, as well.
Relevant results about the local time can be found in [START_REF] Xiao | Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields[END_REF], which allows us to establish dim P (L X (x)) = 1 -H.

Proof of (III.1.11) and (III.1.12). Let θ ∈ [0, 1]. Recall that for any set For n ≥ 1 we cover [ε, 1] by ⌈n 1/H ⌉ subintervals R n,ℓ of length n -1/H , with ℓ ∈ {1, 2, . . . , ⌈n 1/H ⌉}. Let 0 < δ < 1 be fixed and τ n,ℓ be the left endpoint of the interval R n,ℓ . We first bound the probability P(x ∈ Z(R n,ℓ )):

E ⊆ R, one has dim H (E) ≤ dim θ (E) ≤ dim θ (E) ≤ dim B (E) , and dim H (E) ≤ dim P (E) ≤ dim B (E) . It is enough to show that dim B (L Z (x) ∩ [ε, 1]) ≤ 1 -H and dim H (L Z (x) ∩ [ε, 1]) ≥ 1 -
P(x ∈ Z(R n,ℓ )) ≤ P( sup t∈R n,ℓ |Z t -Z τ n,ℓ | ≤ n -(1-δ) , x ∈ Z(R n,ℓ )) + P( sup t∈R n,ℓ |Z t -Z τ n,ℓ | ≥ n -(1-δ) ) ≤ P(|Z τ n,ℓ -x| ≤ n -(1-δ) ) + C 1 exp(-c 1 n -(1-δ) /n -1 ) ≤ C 2 n -(1-δ) + C 1 exp(-c 1 n δ ) = O(n -(1-δ) ), (III.4.1)
where we have used Proposition III.2.5, and the fact that the density of Z t is continuous.

We can cover the set

L Z (x) ∩ [ε, 1] by a sequence of intervals R ′ n,ℓ with R ′ n,ℓ = R n,ℓ if x ∈ Z(R n,ℓ ) and R ′
n,ℓ = ∅, otherwise, for ℓ ∈ {1, 2, . . . , ⌈n 1/H ⌉}. We need to show that

E   ⌈n 1/H ⌉ ℓ=1 |R ′ n,ℓ | η   < ∞, (III.4.2)
for η = 1 -H(1 -δ) and arbitrary δ > 0. In turn this would imply by Fatou's lemma that dim B (L Z (x) ∩ [ε, 1]) ≤ η almost surely. Then, letting δ → 0 yields the upper bound on the upper Box dimension.

We establish (III.4.2):

E   ⌈n 1/H ⌉ ℓ=1 |R ′ n,ℓ | η   ≤ E   ⌈n 1/H ⌉ ℓ=1 n -1/H η 1 x∈Z(R n,ℓ )   ≤ cn 1/H-1/H(1-H(1-δ))-(1-δ) = c,
where the last inequality follows from the bound (III.4.1) on P(x ∈ Z(R n,ℓ )).

For the lower bound we first recall a relation between the Hölder regularity and the Hausdorff dimension.

Proposition III.4.1 (Theorem 27 in [START_REF] Dozzi | Occupation density and sample path properties of N -parameter processes[END_REF]). Let [u, v] ⊂ R be a finite interval and f : [u, v] → R be a continuous function with occupation density denoted by L. Suppose that L satisfies a Hölder condition of order γ ∈ (0, 1) (in the set variable). Then dim

H f -1 [u,v] (x) ≥ γ for all x ∈ R such that L(x, [u, v]) ̸ = 0.
A Hölder regularity condition for the local time of the Rosenblatt process was recently obtained in [START_REF] Kerchev | Local times and sample path properties of the Rosenblatt process[END_REF]. In particular, see Proposition III.2.6, for a finite closed interval I ⊂ (0, ∞), there exists a constant C > 0 such that almost surely, lim sup

r→0 sup s∈I sup x∈R L(x, [s -r, s + r]) r 1-H | log r| 2H ≤ C.
Therefore, the occupation density of the Rosenblatt process satisfies a Hölder condition in the set variable of order γ for all γ < 1 -H, and thus dim

H (L Z (x) ∩ [ε, 1]) ≥ 1 -H.
Before we establish the second part (III.1.13) of Theorem III.1.2 we recall some definitions and properties regarding the macroscopic Hausdorff dimension. Of special interest is a relation between Dim H (E Z (γ)) and Dim H (L Z (x)) which eases the proofs of both (III.1.13) and (III.1.15).

III.4.1 Macroscopic Hausdorff dimension

To set up the notation as in [START_REF] Khoshnevisan | On the macroscopic fractal geometry of some random sets[END_REF][START_REF] Khoshnevisan | Intermittency and multifractality: a case study via parabolic stochastic PDEs[END_REF], consider the intervals S -1 = [0, 1/2) and S n = [2 n-1 , 2 n ) for n ≥ 0. For E ⊂ R + , we define the set of proper covers of E restricted to S n by

I n (E) = {I i } m i=1 : I i = [x i , y i ] with x i , y i ∈ N, y i > x i , I i ⊂ S n and E ∩ S n ⊂ m i=1 I i .
.

For any set E ⊂ R + , ρ ≥ 0 and n ≥ -1, define ν n ρ (E) := inf m i=1 diamI i 2 n ρ : {I i } m i=1 ∈ I n (E) , where diam[a, b] = b -a.
The macroscopic Hausdorff dimension of E ⊂ R + is defined as:

Dim H (E) := inf ρ ≥ 0 : n≥0 ν n ρ (E) < ∞ .
Next we establish a relation between (III.1.13) of Theorem III.

L Z (x) ∩ t ≥ |x| 1 γ ⊂ E Z (γ).
The macroscopic Hausdorff dimension is left unchanged after the removal of any bounded subset. Then, almost surely, for every x ∈ R,

Dim H (L Z (x)) = Dim H L Z (x) ∩ t ≥ |x| 1 γ ≤ Dim H (E Z (γ)) . (III.4.3)
Therefore, to prove (III.1.13) and (III.1.15) it suffices to show that the following two statements hold almost surely:

For any x ∈ R, Dim H (L Z (x)) ≥ 1 -H, (III.4.4) Dim H (E Z (γ)) ≤ 1 -H. (III.4.5)
The proof of (III.4.4) follows in the next subsection while (III.4.5) is established in Section III.5.3.

III.4.2 Lower bound for Dim H (L Z (x))

In this section we aim to find a lower bound for Dim H (L Z (x)). We first establish a result regarding macroscopic Hausdorff dimension in general.

Lemma III.4.2. Let E ⊂ R + and suppose that there exist M > 0 and s ∈ [0, 1] such that there exists a family of finite measures {µ n } n≥-1 on S n such that for all intervals

I ⊂ S n , we have µ n (I) ≤ M (diamI) s . If Dim H (E) = t for some 0 ≤ t < s, then n≥-1 µ n (E ∩ S n ) 2 ns < +∞.
Proof. As t < s and using the definition of macroscopic Hausdorff dimension we have

ν s (A) < +∞. Let {I i } m i=1 ∈ I n (E), then µ n (E ∩ S n ) ≤ m i=1 µ n (I i ) ≤ m i=1 M diamI i s = 2 ns M m i=1 diamI i 2 n s . Then µ n (E ∩ S n ) 2 ns ≤ M ν s n (E ∩ S n ) and so n≥-1 µ n (A ∩ S n ) 2 ns < +∞.
By Proposition III.2.9, the local time is Hölder continuous in t of order α for every α ∈ [0, 1 -H). Now we will be using this property and the preceding lemma in order to get a lower bound for Dim H (L Z (x)). To this end, fix α ∈ [0, 1 -H) and introduce the following random variables

Y x n = L (x, S n ) 2 nα and F x N = N n=1 Y x n . (III.4.6)
The random variables (Y x n ) n≥-1 are positive, so (F x N ) N ≥1 is non-decreasing. We denote by

F x ∞ its limit, i.e. F x ∞ = ∞ n=-1 Y x n ∈ [0, +∞].
As a direct consequence of Lemma III.4.2, there is a connection between Dim H (L x ) and the r.v. Y x n . Indeed, for n ≥ -1 consider the sequence of measures µ n (I) := L(x, I), for all I ⊂ S n , By Proposition III.2.9, there exists M > 0 such that for all n ≥ -1 a.s.

µ n (I) ≤ M diamI α , for all I ⊂ S n .
Now by Lemma III.4.2, a.s. for every

x ∈ R, Dim H (L x ) ≥ α if n≥-1 µ n (L x ∩ S n ) 2 nα = F x ∞ = +∞.
As a consequence, we see that Dim H (L x ) ≥ α for all x ∈ R such that F x ∞ = +∞. Moreover in order to conclude the proof of Theorem III.1.2, it is enough to prove that for all α ∈ [0, 1 -H), a.s. for all x ∈ R, Dim H (L x ) ≥ α . Letting α ↑ 1 -H gives that a.s. for all x ∈ R, Dim H (L x ) ≥ 1-H. Finally it remains to check that P(∀x ∈ R, F x ∞ = +∞) = 1, for all α ∈ [0, 1 -H). This is the object of the next proposition.

Proposition III.4.3. Let α ∈ [0, 1 -H) and Y x n = L(x, S n ) 2 nα , for n ≥ -1, and
F x ∞ = n≥-1 Y x n .
Then,

P(∀x ∈ R, F x ∞ = +∞) = 1. (III.4.7)
Proof. We follow the technique in [START_REF] Daw | A uniform result for the dimension of fractional Brownian motion level sets[END_REF]. For every a > 0, let

Y a n = inf x∈[-a,a] Y x n , for n ≥ 1, and F a ∞ = n≥1 Y a n .
Using the self-similarity property of the local time (III.2.11), for all n ≥ 0,

Y a n = inf x∈[-a,a] Y x n (d) = inf x∈[-a,a] Y 2 -nH x 0 = inf x∈[-2 -nH a,2 -nH a] Y x 0 = Y 2 -nH a 0 .
The proof now relies on the following technical result:

Lemma III.4.4. For any b > 0, one has

P( F b ∞ = ∞) > 0.
Proof of Lemma III.4.4. We first show that there exists ε > 0 such that

P(Y 0 0 > ε) > 0. Recall that Y 0 0 = L(0, [1/2, 1]
) and is non-negative. Thus, it is enough to show that E[L(0, [1/2, 1])] > 0. Using the following representation of the local time, see [START_REF] Samorodnitsky | Stochastic processes and long range dependence[END_REF]Chapter 10], one gets

L(0, [1/2, 1]) = lim ε→0 1 2ε 1 1/2 1 -ε,ε (Z t )dt.
Then using self-similarity of Z and then Proposition III.2.1(i) with some constant c 1 > 0, one gets

E[L(0, [1/2, 1])] = lim ε→0 1 2ε 1 1/2 P(Z t ∈ [-ε, ε])dt = lim ε→0 1 2ε 1 1/2 P(Z 1 ∈ [-εt -H , εt -H ])dt ≥ lim ε→0 1 2ε 1 1/2 2c 1 εt -H dt = c 1 1 -H (1 -(1/2) 1-H ).
Therefore, E[L(0, [1/2, 1])] > 0 and thus P(Z 0 0 > ε) > 0 for some ε > 0. The rest of the proof is based on the following two facts:

1. For every ε > 0 small enough, there exists a ∈ R + , such that:

0 < P(Y 0 0 > ε) ≤ 2P( Y a 0 > 0).
2. For any a, b > 0, we have

P( F b ∞ = ∞) ≥ P( Y a 0 ).
The statements above correspond to Lemmas II.3.3 and II.3.4 in Chapter II and the proofs are identical as long as the following holds:

P sup x∈[-1,1]/{0} L 0, 1 2 , 1 -L x, 1 2 , 1 |x| β < ∞ = 1, where β ∈ 0, 1 2 1
H -1 . In [START_REF] Daw | A uniform result for the dimension of fractional Brownian motion level sets[END_REF], this property corresponds to Lemma 5 which is originally due to Geman in Horowitz [START_REF] Geman | Occupation densities[END_REF]Theorem 26.1]. For the Rosenblatt case, the above is established in Proposition III.2.7.

Using the result of Lemma III.4.4 we can establish that P( F b ∞ = ∞) = 1 if we can apply Blumental's 0-1 law. This is possible since

F b ∞ = ∞ ∈ M ≥1 σ W u : u < 2 -(M -1) , (III.4.8)
where (W t ) t≥0 is the standard Brownian motion. Indeed, the time inverted process Z t = t 2H Z 1/t , t > 0 is distributed as the Rosenblatt process (see Proposition III.2.3). Then, using the representation (III.2.10), the local time

L x (S n ) is σ Z u : u ≤ 2 -(n-1) -measurable. Moreover, σ Y b n : n ≥ M ⊂ σ Z u : u ≤ 2 -(M -1) ,
for M ≥ 1 and thus 1) .

F b ∞ = ∞ ∈ M ≥1 σ Z u : u < 2 -(M -
(III.4.9)

At this point by (III.2.4) (with Z instead of Z), there exists standard Brownian motion (W t ) t≥0 such that σ Zu : u ≤ t ⊂ σ {W u : u ≤ t}. This fact combined with (III.4.9) establishes (III.4.8). Then using (III.4.8) and the fact that P( F b ∞ = ∞) > 0 for all b > 0, one can apply Blumental's 0-1 law and thus gets that P( F b ∞ = ∞) = 1, for all b > 0. Finally, for every b > 0,

P(∀x ∈ [-b, b] : F x ∞ = ∞) = P inf x∈[-b,b] F x ∞ = ∞ = P inf x∈[-b,b] n≥1 Y x n = ∞ ≥ P n≥1 inf x∈[-b,b] Y x n = ∞ = P( F b ∞ = ∞) = 1.
Therefore,

P(∀x ∈ R : F x ∞ = ∞) = lim b→∞ P(∀x ∈ [-b, b], F x ∞ = ∞) = 1,
and (III.4.7) is established.

Next, we establish (III.1.12)-the result regarding Packing dimension. Recall that dim P (L Z (x)) ≤ dim B (L Z (x)) = 1 -H. It is enough to show that dim P (L Z (x)) ≥ 1 -H, which is the aim of the next section.

III.5 Sojourn times

This section is dedicated to the proof of Theorem III.1.3. We first establish (III. 1.14). Recall the definitions of logarithmic and pixel densities. For E ⊂ R + , the logarithmic density of E is given by Den log (E) := lim sup

n→∞ log 2 Leb(E ∩ [1, 2 n ]) n ,
where 'Leb' is the one-dimensional Lebesgue measure.

Let pix(E) := {n ∈ N : dist(n, E) ≤ 1}. Then, the pixel density of E is

Den pix (E) := lim sup n→∞ log 2 #pix(E ∩ [1, 2 n ]) n .
The two quantities are closely related, see [START_REF] Khoshnevisan | On the macroscopic fractal geometry of some random sets[END_REF]:

Den log (E) ≤ Den pix (E) . (III.5.1)
We want to show that for γ ∈ [0, H),

Den pix (E Z (γ)) = Den log (E Z (γ)) = γ + 1 -H, almost surely. Our strategy is then to establish that Den pix (E Z (γ)) ≤ γ + 1 -H and Den log (E Z (γ)) ≥ γ + 1 -H, almost surely.

III.5.1 Upper bound for Den pix (E Z (γ))

Our goal is to obtain an upper bound for #pix(E Z (γ)) ∩ [1, 2 n ] that holds with probability 1 for all large n. We first study the expectation

E[pix(E Z (γ) ∩ [1, 2 n ])] = 2 n m=1 P (∃s ∈ [m -1, m + 1], |Z s | ≤ s γ ) = 2 n m=1 P ∃s ∈ 1 - 1 m , 1 + 1 m , |Z s | ≤ s γ m γ-H = 2 n m=1 P ∃s ∈ 1 - 1 m , 1 , |Z s | ≤ s γ m γ-H + P ∃s ∈ 1, 1 + 1 m , |Z s | ≤ s γ m γ-H ≤ 2 n m=1 (A - 1/m + A + 1/m ), (III.5.2)
where

A - ε :=P(∃s ∈ [1 -ε, 1], |Z s | ≤ ε H-γ ), A + ε :=P(∃s ∈ [1, 1 + ε], |Z s | ≤ 2ε H-γ ).
Lemma III.5.1. There is a universal constant c > 0, such that, for every ε small enough,

max(A - ε , A + ε ) ≤ cε H-γ . (III.5.3) Proof. Consider A - ε first. We have A - ε ≤ P(∃s ∈ [1 -ε, 1], |Z s | ≤ ε H-γ , |Z 1 | ≤ 2ε H-γ ) + P(∃s ∈ [1 -ε, 1], |Z s | ≤ ε H-γ , |Z 1 | ≥ 2ε H-γ ) ≤ P(|Z 1 | ≤ 2ε H-γ ) + P(∃s ∈ [1 -ε, 1], |Z s -Z 1 | ≥ ε H-γ ). (III.5.4)
To bound the first term on the right-hand side above, we use Proposition III.2.1(i), i.e, the density function f of Z 1 is continuous and f (0) > 0. Then one can show, for instance, that for ε > 0 small enough,

P(|Z 1 | ≤ 2ε H-γ ) ≤ 4f (0)ε H-γ .
(III.5.5)

We are left to study the term P(∃s

∈ [1 -ε, 1], |Z s -Z 1 | ≥ ε H-γ ). Write P(∃s ∈ [1 -ε, 1], |Z s -Z 1 | ≥ ε H-γ ) ≤ P( sup s∈[1-ε,1+ε] |Z s -Z 1 | ≥ ε H-γ ) ≤ C exp -c 1 ε -γ , (III.5.6)
where the last inequality follows from Proposition III.2.5 and C, c 1 > 0 are constants depending only on H. Note that exp(-c 1 ε -γ ) = O(ε δ ), for any δ > 0 if any ε is small enough.

Finally, for ε small enough, combining (III.5.6) and (III.5.5) in (III.5.4) yields the bound of (III.5.3) for A - ε . Same arguments as above can be applied to A + ε to get an equivalent bound and establish (III.5.3).

Next, applying Lemma III.5.1 in (III.5.2) yields, for some absolute constant C > 0,

E[#pix(E Z (γ) ∩ [1, 2 n ])] ≤ 2C 2 n m=1 m γ-H = O 2 n(γ+1-H) . Choose ρ > γ + 1 -H. Then, n≥1 P (#pix(E Z (γ) ∩ [1, 2 n ]) > 2 nρ ) ≤ C n≥1 2 n(1+γ-h) 2 nρ < ∞.
By the Borel-Cantelli lemma, with probability one,

#pix(E Z (γ) ∩ [1, 2 n ]) ≤ 2 nρ ,
for every large enough n. Hence, Den pix (E Z (γ)) ≤ ρ.

Letting ρ ↓ γ + 1 -H yields Den pix (E Z (γ)) ≤ γ + 1 -H.

III.5.2 Lower bound for Den

log (E Z (γ)) Introduce S γ ([t 1 , t 2 ]) = Leb({t 1 ≤ s ≤ t 2 : |Z s | ≤ s γ }), for all 0 ≤ t 1 ≤ t 2 .
We will prove that for infinitely many integers n, S γ ([0, 2 n ]) ≥ c 2 2 n(γ+1-H) , for any c ∈ (0, 1). This implies that Den log (E Z (γ)) ≥ γ +1-H almost surely. Then using (III.5.1), we also obtain Den pix (E Z (γ)) ≤ γ + 1 -H and the proof of (III.1.14) is completed.

First we show that for any c ∈ (0, 1), there is a constant c ′ > 0 such that

P(S γ ([0, 2 n ]) ≥ c2 n(1+γ-H) ) ≥ c ′ .
(III.5.7)

By Paley-Zygmund inequality, for any c ∈ (0, 1), we have

P(S γ ([0, 2 n ]) ≥ c2 n(1+γ-H) ) ≥ (1 -c) E[S γ ([0, 2 n ])] 2 E[S γ ([0, 2 n ]) 2 ]
.

(III.5.8)

The numerator can be rewritten as:

E[S γ ([0, t])] = t 0 P(|Z s | ≤ s γ )ds = t 0 P(|Z 1 | ≤ s γ-H )ds.
Now, we establish a lower bound for P(|Z 1 | ≤ s γ-H ). Apply Proposition III.2.1(i) there is a constant α > 0 such that for s large enough, the density function of Z 1 is bounded below by α in [-s γ-H , s γ-H ]. Therefore,

P(|Z 1 | ≤ s γ-H ) ≥ 2αs γ-H and thus E[S γ ([0, t])] ≥ 2αt 1+γ-H .
(III.5.9)

We bound the second moment from above:

E[S γ ([0, t]) 2 ] = [0,t] 2 P(|Z u | ≤ u γ , |Z v | ≤ v γ )dudv = t 2 [0,1] 2 P |Z u | ≤ u γ t γ-H , |Z v | ≤ v γ t γ-H dudv.
By Proposition III.2.1(ii), the density function g u,v of (Z u , Z v ) is continuous and tends to 0 as |x| → ∞. Therefore, Applying (III.5.9) and (III.5.10) in (III.5.8) yields (III.5.7). Now, define the event

E[S γ ([0, t]) 2 ] ≤ t 2 [0,1] 2 dudv R 2 g u,v (x, y) 1   |x| ≤ u γ t γ-H |y| ≤ v γ t γ-H
A n,γ := S γ c 2 2 n(1+γ-H) , 2 n ≥ c 2 2 n(1+γ-H) .
By (III.5.7), it is easy to see that P (A n,γ ) ≥ c ′ > 0. Moreover, by the definition of A n,γ , one has A n,γ ⊂ S γ ([0, 2 n ]) ≥ c 2 2 n(1+γ-H) . Then it is enough to prove that A n,γ happens infinitely often which give us that S γ ([0, 2 n ]) ≥ c 2 2 n(1+γ-H) for infinitly many n. To this end, let A γ be the event that A n,γ happens infinitely often. Recall that for any sequence of events (A i ) i≥1 , one has lim n→∞ P(∪ i≥n A i ) = P(A i i. o ). In other words, one has

A γ = M ≥1 n≥M A n,γ .
(III.5.12)

We know that P (A γ ) (≥ c ′ ) is strictly positive. It remains to prove that it is in fact equal to 1. As in Section III.4.2, such a conclusion will follow by using that the time inverted process Zt = t 2H Z 1/t is distributed as the Rosenblatt process (see Proposition III.2.3). Now let Sγ (resp. Ãn,γ , Ãγ ) be the event analogous to S γ (resp. A n,γ , A γ ), but associated to Z instead of Z. So for any fixed integer n ≥ 0, we have

Sγ c 2 2 n(1+γ-H) , 2 n = Leb c 2 2 n(1+γ-H) ≤ s ≤ 2 n : |t 2H Z 1/s | ≤ s γ ,
which implies in return that Ãn,γ ∈ σ Z u : u ≤ 2 -n(1+γ-H) . As a consequence, for all M ≥ 0, one has Ãn,γ :

n ≥ M ∈ σ Z u : u ≤ 2 -M (1+γ-H) .
Recalling definition III.5.12 of A γ , we obtain that

Ãγ ∈ M ≥1 σ { n,γ : n ≥ M } .
Using (III.2.4), we deduce that

Ãγ ∈ M ≥1 σ(B u : u ≤ 2 -M (1+γ-H) ),
where (B t ) t≥0 is the Brownian motion. Therefore, Ãγ is a tail event and P( Ãγ ) = 0 or 1 by the Blumenthal 0 -1 law. Obviously, as Z and Z have the same distribution, then P( Ãγ ) = P(A γ ) ≥ c ′ > 0 and then P( Ãγ ) = P(A γ ) = 1 as desired.

III.5.3 Upper bound for Dim H (E Z (γ))

We now turn to the proof of (III.1.15). Following our discussion in Section III.4.1, and in particular the relation (III. 4.3) between Dim H (E Z (γ)) and Dim H (L Z (x)), it is enough to show (III.4.5), i.e., for every 0 ≤ γ < H,

Dim H (E Z (γ)) ≤ 1 -H, a.s.
We follow the technique in [START_REF] Nourdin | Sojourn time dimensions of fractional Brownian motion[END_REF]. Let us fix 0 ≤ γ < H, as well as η > 0 (as small as necessary). We are going to prove that Dim H (E Z (γ)) ≤ 1 -H + η. Letting η tend to zero will then give the result. Fix ρ > 1 -H + η, our aim is to prove that Dim H (E Z (γ)) ≤ ρ.

To this end, consider for every integer n ≥ 1 and i ∈ 0, ..., 2 n-1 2 n γ H the intervals

I n,i = [t n,i , t n,i+1 ) with t n,i = 2 n-1 + i2 n γ H .
And the associated event

E n,i = {∃t ∈ I n,i : |Z t | ≤ t γ } . Denote ε n,i = 2 n γ H /t n,i , so that I n,i = [t n,i , t n,i (1+ε n,i
)), and observe that the ratio between any two of the quantities 2 n( γ H -1) , ε n,i , and

t γ H -1 n,i
are bounded uniformly with respect to n and i. By self-similarity, we have that, when n becomes large,

P (E n,i ) = P (∃ t ∈ I n,i : |Z t | ≤ t γ ) = P ∃ s ∈ [1, 1 + ε n,i ] : |Z s.t n,i | ≤ (s.t n,i ) γ = P ∃ s ∈ [1, 1 + ε n,i ] : |Z s | ≤ t γ-H n,i .s γ = P ∃ s ∈ [1, 1 + ε n,i ] : |Z s | ≤ 2t γ-H n,i = P ∃ s ∈ [1, 1 + ε n,i ] : |Z s | ≤ cε H n,i = P ∃ s ∈ [1, 1 + ε n,i ] : |Z s | ≤ ε H-η n,i
.

The last estimate holds because η is a small positive real number and ε n,i tends to zero when n becomes large. By Lemma III.5.1, we deduce that P (E n,i ) ≤ cε H-η n,i and so

P (E n,i ) ≤ c2 n(γ-H) H-η H .
Now observe that E n,i is realized if and only if E Z (γ) ∩ I n,i ̸ = ∅. So, using the intervals I n,i as a covering of E Z (γ) ∩ S n , we obtain that

E ν n ρ (E Z (γ)) ≤E   ⌊2 n-1-n γ H ⌋ 0 Leb(I n,i ) 2 n ρ 1 E n,i   ≤2 ρn( γ H -1) ⌊2 n-1-n γ H ⌋ 0 P (E n,i ) ≤c2 n H-γ H (1-H+η-ρ) .
Thus, the Fubini Theorem entails E ∞ n=1 ν n ρ (E Z (γ)) < +∞ as soon as ρ > 1 -H + η. This implies that for such ρ's, the sum ∞ n=1 ν n ρ (E Z (γ)) is finite almost surely. In particular, Dim H (E Z (γ)) ≤ ρ, for every ρ > 1 -H + η. Since such a relation holds for an arbitrary (small) ρ > 0, we deduce (III.4.5) as desired.

Chapter IV Wavelet methods to study the pointwise regularity of the generalized Rosenblatt process

The content of this chapter is a copy of the paper entitled "Wavelet methods to study the pointwise regularity of the generalized Rosenblatt process", written with "Laurent Looseveldt", and to be submitted soon.

IV.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 [START_REF] Wiener | Collected works[END_REF]. 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 [START_REF] Wiener | Fourier Transforms in the Complex Domain[END_REF] using Fourier series, Lévy's representation [START_REF] Lévy | Processus stochastiques et mouvement brownien[END_REF] obtained with some techniques of interpolation theory or, more recently, Kahane's expansion [START_REF] Kahane | of Cambridge Studies in Advanced Mathematics[END_REF] in the Schauder basis.

In the last decades, the emergence of wavelet analysis allowed to obtain series expansions for many stochastic processes. Let ψ : R → R be a smooth function satisfying the admissibility condition [

79] R | ψ(ξ)| |ξ| dξ < ∞, (IV.1.1)
where ψ is the Fourier transform of ψ. As such it generates an orthonormal basis of L 2 (R). More precisely, any function f ∈ L 2 (R) can be decomposed as

f = j∈Z k∈Z c j,k ψ(2 j • -k), (IV.1.2)
where

c j,k = 2 j R f (x)ψ(2 j x -k) dx.
It is noteworthy that the expansion (IV.1.2) holds true in many function spaces. We refer to the seminal books [START_REF] Daubechies | Ten lectures on wavelets[END_REF][START_REF] Meyer | Wavelets and operators[END_REF][START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF] 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 [START_REF] Barral | From multifractal measures to multifractal wavelet series[END_REF][START_REF] Bastin | Large deviation spectra based on wavelet leaders[END_REF][START_REF] Clausel | Wavelets techniques for pointwise anti-Hölderian irregularity[END_REF][START_REF] Jaffard | Multifractal formalism for functions part I: Results valid for all functions[END_REF][START_REF] Jaffard | Wavelet techniques in multifractal analysis, fractal geometry and applications: A jubilee of Benoit Mandelbrot[END_REF][START_REF] Jaffard | Wavelet methods for pointwise regularity and local oscillations of functions[END_REF] and a practical points of view [START_REF] Arneodo | Wavelet based fractal analysis of DNA sequences[END_REF][START_REF] Arneodo | multifractales et turbulences : de l'ADN aux croissances cristallines[END_REF][START_REF] Bunde | The science of disasters: climate disruptions, heart attacks, and market crashes[END_REF][START_REF] Deliège | Mars Topography Investigated Through the Wavelet Leaders Method: a Multidimensional Study of its Fractal Structure[END_REF][START_REF] Flandrin | Time-frequency / Time-scale analysis[END_REF][START_REF] Jaffard | Function spaces vs. Scaling functions: Some issues inimage classification[END_REF][START_REF] Nicolay | Bifractality of human DNA strand-asymmetry profiles results from transcription[END_REF][START_REF] Wendt | Wavelet leader multifractal analysis for texture classification[END_REF][START_REF] Wendt | Wavelet leaders and bootstrap for multifractal analysis of images[END_REF]. Now, let us consider a probability space (Ω, A, P) and a real-valued stochastic process X defined on it. If X is smooth enough, for all ω ∈ Ω, one can apply expansion (IV.1.2) to the simple path t → X(t, ω). This way, one defines a sequence of random wavelet coefficients (c j,k (ω)) j,k∈Z . For instance, if X = B H is the fractional Brownian motion of Hurst index H ∈ (0, 1) and if ψ is a sufficiently regular wavelet, one has [START_REF] Meyer | Wavelets, generalized white noise and fractional integration: the synthesis of fractional Brownian motion[END_REF][START_REF] Jaffard | Function spaces vs. Scaling functions: Some issues inimage classification[END_REF]

B H = j∈N k∈Z 2 -Hj ξ j,k ψ H+1/2 (2 j • -k) + R, (IV.1.3)
where (R(t, •)) t∈R + is a process with almost surely C ∞ sample paths, (ξ j,k ) j∈N,k∈Z is a sequence of independent N (0, 1) random variables and ψ H+1/2 is a fractional antiderivative of ψ, see Section IV.2 for a precise definition.

In [START_REF] Esser | Slow, ordinary and rapid points for Gaussian Wavelets Series and application to Fractional Brownian Motions[END_REF], Esser and Loosveldt undertook a systematic study of Gaussian wavelet series. Thanks to (IV.1.3), it applies in particular to the fractional Brownian motion and leads to the following theorem.

Theorem IV.1.1. For all H ∈ (0, 1), there exists an event Ω H of probability 1 satisfying the following assertions for all ω ∈ Ω H and every non-empty interval I of R.

For almost every t ∈ I,

0 < lim sup s→t |B H (t, ω) -B H (s, ω)| |t -s| H log log |t -s| -1 < +∞.

Such points are called ordinary points.

There exists a dense set of points t ∈ I such that

0 < lim sup s→t |B H (t, ω) -B H (s, ω)| |t -s| H log |t -s| -1 < +∞.

Such points are called rapid points.

There exists a dense set of points t ∈ I such that

0 < lim sup s→t |B H (t, ω) -B H (s, ω)| |t -s| H < +∞.
Such points are called slow points.

Note that Theorem IV.1.1 extends some well-known results of Kahane concerning the Brownian motion [START_REF] Kahane | of Cambridge Studies in Advanced Mathematics[END_REF]. 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 → |x| H 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 log log |x| -1 = o(|x| H log |x| -1 ) if x → 0 + and thus points for which x → |x| H log |x| -1 is the pointwise modulus of continuity are refereed as rapid. On the other side, points for which x → |x| H is the pointwise modulus of continuity are referred as slow because |x| H = o(|x| H log log |x| -1 ) if x → 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 [START_REF] Dobrushin | Non-central limit theorems for nonlinear functionals of Gaussian fields[END_REF]. 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 IV.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 [START_REF] Chaurasia | Performance of synthetic Rosenblatt process under multicore architecture[END_REF] and turbulence [START_REF] Sakthivel | Retarded stochastic differential equations with infinite delay driven by Rosenblatt process[END_REF][START_REF] Lakhel | Existence, uniqueness and stability of impulsive stochastic neutral functional differential equations driven by Rosenblatt process with varyingtime delays[END_REF]. From a statistical point of view, estimating the value of the Hurst index H is important for practical applications and various estimators exist, see [START_REF] Bardet | A wavelet analysis of the Rosenblatt process: chaos expansion and estimation of the self-similarity parameter[END_REF]121]. Also, from a mathematical point of view the Rosenblatt process has received a lot of interest since its inception in [START_REF] Rosenblatt | A central limit theorem and a strong mixing condition[END_REF]. Its distribution, still not known in explicit form, was studied first in [START_REF] Albin | A note on Rosenblatt distributions[END_REF] and more recently in [START_REF] Maejima | On the distribution of the Rosenblatt process[END_REF] and [START_REF] Veillette | Properties and numerical evaluation of the Rosenblatt distribution[END_REF].

In this paper, we even consider a generalization of the Rosenblatt process, as defined and studied in [START_REF] Maejima | Selfsimilar processes with stationnary increments in the second Wiener chaos[END_REF]

R H 1 ,H 2 (t, •) = ′ R 2 K H 1 ,H 2 (t, x 1 , x 2 ) dB(x 1 )dB(x 2 ), (IV.1.4)
where ′ R 2 denotes integration over R 2 excluding the diagonal. The kernel function in (IV.1.4) is expressed, for all (t, x 1 , x 2 ) on R + × R 2 , by

K H 1 ,H 2 (t, x 1 , x 2 ) = 1 Γ H 1 -1 2 Γ H 2 -1 2 t 0 (s -x 1 ) H 1 -3 2 + (s -x 2 ) H 2 -3 2 + ds,
where Γ stands for the usual Gamma Euler function, and where for (x, α) ∈ R 2

x α + =

x α if x ≥ 0 0 otherwise. (3) Self-similarity: R H 1 ,H 2 is self-similar with exponent

H 1 + H 2 -1; that is, the pro- cesses {R H 1 ,H 2 (ct, •)} t∈R + and c H 1 +H 2 -1 R H 1 ,H 2 (t,
•) t∈R + have the same distribution for all c > 0.

In [START_REF] Ayache | Wavelet-type expansion of the generalized Rosenblatt process and its rate of convergence[END_REF], Ayache and Esmili presented a wavelet-type representation of the generalized Rosenblatt process, very similar to the one given in [START_REF] Meyer | Wavelets, generalized white noise and fractional integration: the synthesis of fractional Brownian motion[END_REF] 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 IV.1.2 which is the main result of this paper.

Theorem IV.1.2. For all H 1 , H 2 ∈ ( 1 2 , 1) such that H 1 + H 2 > 3 2
, there exists an event Ω H 1 ,H 2 of probability 1 satisfying the following assertions for all ω ∈ Ω H 1 ,H 2 and every non-empty interval I of R.

For almost every t ∈ I,

0 < lim sup s→t |R H 1 ,H 2 (t, ω) -R H 1 ,H 2 (s, ω)| |t -s| H 1 +H 2 -1 log log |t -s| -1 < +∞.
(IV.1.5)

Such points are called ordinary points.

There exists a dense set of points t ∈ I such that

0 < lim sup s→t |R H 1 ,H 2 (t, ω) -R H 1 ,H 2 (s, ω)| |t -s| H 1 +H 2 -1 log |t -s| -1 < +∞. (IV.1.6)

Such points are called rapid points.

There exists a dense set of points t ∈ I such that

lim sup s→t |R H 1 ,H 2 (t, ω) -R H 1 ,H 2 (s, ω)| |t -s| H 1 +H 2 -1 < +∞. (IV.1.7)
Such points are called slow points.

Theorem IV.1.2 shows in particular that slow, ordinary and rapid points are not specific to Gaussian processes. Remark 11. Let us compare Theorems IV.1.1 and IV.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, ω)-X(s, ω)| is always compared to a modulus of continuity of the form |t-s| α θ(|t-s|), with α corresponding to the self-similarity exponent of X and θ a potential logarithmic correction. For the ordinary points, θ is an iterated logarithm. More precisely, for the fractional Brownian motion, we have θ(|t -s|) = log log |t -s| -1 while, for the generalized Rosenblatt process, θ(|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 θ(|t -s|) = log |t -s| -1 and for the generalized Rosenblatt process we have θ(|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 IV.3.11 and IV.3.12 below. Concerning the slow points, there is no logarithmic correction, θ = 1 in both case. Unfortunately, contrary to the fractional Brownian motion, we did not manage to show the positiveness of the limit in (IV.1.7). 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 20 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 (IV.1.5) and (IV.1.6).

Our strategy to prove Theorem IV.1.2 is as follows. First, in Section IV.3 we derive upper-bounds for the oscillations |R H 1 ,H 2 (t, ω) -R H 1 ,H 2 (s, ω)| that are sharp enough to imply the finiteness of the limits (IV.1.5), (IV.1.6) and (IV.1.7). This is done by means of the wavelet-type expansion given in [START_REF] Ayache | Wavelet-type expansion of the generalized Rosenblatt process and its rate of convergence[END_REF], see Theorem IV.3.2 below. Then, in Section IV.4, we give lower bounds for the so-called wavelet-leaders, see Section IV.2, of the generalized Rosenblatt process on a given compactly supported wavelet basis. This will prove the positiveness of the limits (IV.1.5), (IV.1.6). In particular, we use different bases depending on whether we deal with the finiteness of the limits in Theorem IV.1.2 or with their strict positiveness. This is very different from [START_REF] Esser | Slow, ordinary and rapid points for Gaussian Wavelets Series and application to Fractional Brownian Motions[END_REF] where the authors always work with the same wavelet. The reason is that the expression (IV.3.3) in Theorem IV.3.2 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 are a priori no obstacles to extend our results in Section IV.4 to any Hermite process. On the contrary, extending the results of Section IV.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 IV.

IV.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 (IV.1.1) is that, if the wavelet ψ ∈ L 1 (R), its first moment always vanishes, i.e. R ψ(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 IV.1.2, we will use a wavelet-type expansion of the generalized Rosenblatt process. It is given in [START_REF] Ayache | Wavelet-type expansion of the generalized Rosenblatt process and its rate of convergence[END_REF] by the mean of the Meyer's wavelet: ψ belongs to the Schwartz class S(R), and its Fourier transform is compactly supported, see [START_REF] Lemarié | Ondelettes et bases hilbertiennes[END_REF]. In particular, for all H ∈ (1/2, 1), ψ H , the fractional antiderivative ψ H of order H -1/2 of ψ is well-defined by means of its Fourier transform as ψ H (0) = 0 and ψ

H (ξ) = (iξ) -(H-1 2 ) ψ(ξ), ∀ ξ ̸ = 0. (IV.2.1)
It also belongs to the Schwartz class S(R), see [START_REF] Ayache | Multifractional stochastic fields[END_REF][START_REF] Ayache | Wavelet-type expansion of the generalized Rosenblatt process and its rate of convergence[END_REF][START_REF] Schwartz | Théorie des distributions[END_REF] for instance. Moreover, some standard facts from distribution theory [START_REF] Schwartz | Théorie des distributions[END_REF][START_REF] Ayache | Multifractional stochastic fields[END_REF] give us the explicit formula

ψ H (t) = 1 Γ H -1 2 R (t -x) H-3 2 + ψ(x) dx.
From (IV.2.1), we see that supp( ψ H ) = supp( ψ) which is the key fact to establish the following lemma, gathering facts already proved in [START_REF] Ayache | Wavelet-type expansion of the generalized Rosenblatt process and its rate of convergence[END_REF].

Lemma IV.2.1. Let H 1 , H 2 ∈ ( 1 2 , 1). If (j 1 , j 2 , k 1 , k 2 ) ∈ Z 4 are such that |j 1 -j 2 | > 1, then the integral I k 1 ,k 2 j 1 ,j 2 := R ψ H 1 (2 j 1 x -k 1 )ψ H 2 (2 j 2 x -k 2 )
vanishes. Moreover, for all (j, k 1 , k 2 ) ∈ Z 3 , we have

I k 1 ,k 2 j+1,j = 2 -j R e -i(k 1 -2k 2 )ξ ψ H 1 (ξ) ψ H 2 (2ξ) dξ, (IV.2.2) I k 1 ,k 2 j,j = 2 -j R e -i(k 1 -k 2 )ξ ψ H 1 (ξ) ψ H 2 (ξ) dξ, (IV.2.3) I k 1 ,k 2 j,j+1 = 2 -j R e -i(2k 1 -k 2 )ξ ψ H 1 (2ξ) ψ H 2 (ξ) dξ. (IV.2.4)
In addition, for all L > 0, there exists a constant C L > 0 such that for all (j,

k 1 , k 2 ) ∈ Z 3 , |I k 1 ,k 2 j+1,j | ≤ C L 2 -j (3 + |k 1 -2k 2 |) L , |I k 1 ,k 2 j,j | ≤ C L 2 -j (3 + |k 1 -k 2 |) L , |I k 1 ,k 2 j,j+1 | ≤ C L 2 -j (3 + |2k 1 -k 2 |) L .
When dealing with the the lower bounds for the limits in Theorem IV.1.2, we use Daubechies compactly supported wavelets [START_REF] Daubechies | Orthonormal bases of compactly supported wavelets[END_REF]. Note that, if supp(Ψ) ⊆ [-N, N ], for a positive integer N , then, using the first vanishing moment, for all (j, k) ∈ N × Z and t ∈ R, one can write

c j,k = N -N f x + k 2 j -f (t) Ψ(x) dx (IV.2.5) Since Ψ is compactly supported, Ψ(2 j • -k) is localized around the dyadic interval λ j,k := k 2 j , k + 1 2 j
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 λ = λ j,k , and k = s(λ). Similarly, c λ refers to the quantity c j,k . The notation Λ j stands for the set of dyadic intervals λ of R with side length 2 -j . The unique dyadic interval from Λ j containing the point t ∈ R is denoted λ j (t). The set of dyadic intervals is Λ := ∪ j∈N Λ j . Two dyadic intervals λ and λ ′ are adjacent if there exist j ∈ N such that λ, λ ′ ∈ Λ j and dist(λ, λ ′ ) = 0. The set of dyadic intervals adjacent to λ is denoted by 3λ. In this setting, one defines the wavelet leader [START_REF] Jaffard | Wavelet techniques in multifractal analysis, fractal geometry and applications: A jubilee of Benoit Mandelbrot[END_REF] of f at t and of scale j by

d j (t 0 ) = max λ∈3λ j (t 0 ) sup λ ′ ⊆λ |c ′ λ |. (IV.2.6)
Then, if supp(Ψ) ⊆ [-N, N ], from (IV.2.5), one can write

d j (t) ≤ 2N sup s∈(t 0 -2 -j (N +2),t 0 +2 -j (N +2)) |f (s) -f (t)|∥Ψ∥ L ∞ . (IV.2.7)
When we study stochastic processes, the wavelet leaders are random variables d j (t, ω). Inequality (IV.2.7) with some easy computations implies that in order to obtain the positiveness of the limit (IV.1.5), it suffices to show that for all ω ∈ Ω H 1 ,H 2 and all open intervals I ⊆ R + , for almost every t ∈ I,

0 < lim sup j→+∞ d j (t, ω) 2 -j(H 1 +H 2 -1) log(j) . (IV.2.8)
Similarly, to prove the positiveness of the limit (IV.1.6), we just have to show that for all ω ∈ Ω H 1 ,H 2 and all open intervals I ⊆ R + , there exists a dense set of points t ∈ I such that 0 < lim sup

j→+∞ d j (t, ω) 2 -j(H 1 +H 2 -1) j .
(IV.2.9)

Remark 12. Let us mention that wavelet leaders can not be used to prove the finiteness of the limits in Theorem IV.1.2 because they do not precisely characterize the pointwise regularity, see for instance [START_REF] Kreit | Generalized pointwise Hölder spaces defined via admissible sequences[END_REF][START_REF] Loosveldt | Generalized spaces of pointwise regularity: Toward a general framework for the WLM[END_REF] for more details.

IV.3 Upper bounds for oscillations

Starting from now and until the end of the paper, we fix

H 1 , H 2 ∈ ( 1 2 , 1) such that H 1 +H 2 > 3 2 .
In this section, we show the finiteness of the limits (IV.1.5), (IV.1.6) and (IV.1.7).

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 IV.3.1. For all (j 1 , j 2 , k 1 , k 2 ) ∈ Z 4 , let ε k 1 ,k 2
j 1 ,j 2 be the second order Wiener chaos random variable defined by

2 j 1 +j 2 2 ′ R 2 ψ(2 j 1 x 1 -k 1 )ψ(2 j 2 x 2 -k 2 ) dB(x 1 )dB(x 2 ).
Remark 13. For all (j 1 , j 2 , k 1 , k 2 ) ∈ Z 4 , we have ([7, Proposition 2.3])

ε k 1 ,k 2 j 1 ,j 2 = 2 j 1 2 R ψ(2 j 1 x -k 1 )dB(x) 2 j 2 2 R ψ(2 j 2 x -k 2 )dB(x) (IV.3.1)
for j 1 ̸ = j 2 or k 1 ̸ = k 2 , and

ε k 1 ,k 1 j 1 ,j 1 = 2 j 1 2 R ψ(2 j 1 x -k 1 )dB(x) 2 -1 (IV.3.2)
for j 1 = j 2 and k 1 = k 2 . Using the fact that (2 j/2 ψ(2 j • -k)) (j,k)∈Z 2 forms an orthonormal basis of L 2 (R), and elementary properties of Wiener integral, we know that (2 j/2 R ψ(2 j xk) dB(x)) (j,k)∈Z 2 is a family of iid N (0, 1) random variables. So the random variables

ε k 1 ,k 2 j 1 ,j 2 and ε k ′ 1 ,k ′ 2 j ′ 1 ,j ′ 2
are independent as soon as

{(j 1 , k 1 ), (j 2 , k 2 )} ∩ {(j ′ 1 , k ′ 1 ), (j ′ 2 , k ′ 2 )} = ∅.
The following theorem, proved in [START_REF] Ayache | Wavelet-type expansion of the generalized Rosenblatt process and its rate of convergence[END_REF], gives the wavelet-type expansion we use in this section.

Theorem IV.3.2. Let ψ be the Meyer wavelet and I be any compact interval of R + . Almost surely, the random series

(j 1 ,j 2 ,k 1 ,k 2 )∈Z 4 2 j 1 (1-H 1 )+j 2 (1-H 2 ) ε k 1 ,k 2 j 1 ,j 2 t 0 ψ H 1 (2 j 1 x -k 1 )ψ H 2 (2 j 2 x -k 2 ) dx (IV.3.3)
converges uniformly to R H 1 ,H 2 on the interval I.

Remark 14. Any open interval in R can be written as a countable union of dyadic intervals (λ j,k ) j∈N,k∈Z . Then, to prove Theorem IV.1.2, it is sufficient to show that, for all j ∈ N, k ∈ Z, there exist an event Ω j,k of probability 1 such that, for all ω ∈ Ω j,k , almost every t ∈ λ j,k is ordinary and there exist t r ∈ λ j,k which is rapid and t s ∈ λ j,k which is slow. For the sake of simpleness in notation, we will only do the proofs in full details for λ 0,0 = [0, 1). In fact, after dilatation and translation, our proofs hold true for any arbitrary dyadic interval.

IV.3.1 Rapid points

Let us first focus on rapid points. We prove that x → |x|

H 1 +H 2 -1 log |x| -1 is almost surely a uniform modulus of continuity for R H 1 ,H 2 .
Proposition IV.3.3. There exists an event Ω rap of probability 1 such that for all ω ∈ Ω rap there exists C R (ω) > 0 such that, for all t, s ∈ (0, 1), we have

|R H 1 ,H 2 (t, ω) -R H 1 ,H 2 (s, ω)| ≤ C R (ω)|t -s| H 1 +H 2 -1 log |t -s| -1 . (IV.3.4)
Let us set, for all s, t ∈ (0, 1) and (j 1 , j 2 , k 1 , k 2 ) ∈ Z 4 ,

I k 1 ,k 2 j 1 ,j 2 [t, s] = [t,s] ψ H 1 (2 j 1 x -k 1 )ψ H 2 (2 j 2 x -k 2 ) dx.
All along this section, if s, t ∈ (0, 1) are given, n always refers to the unique positive integer such that 2

-n-1 < |t -s| ≤ 2 -n . (IV.3.5)
Our proof consists in writing

|R H 1 ,H 2 (t, •)) -R H 1 ,H 2 (s, •)| = (j 1 ,j 2 ,k 1 ,k 2 )∈Z 4 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) ε k 1 ,k 2 j 1 ,j 2 I k 1 ,k 2 j 1 ,j 2 [t, s] (IV.3.6)
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 IV.3.4. [7, Lemma 2.4.] There exist an event Ω * of probability 1 and a positive random variable C 1 with finite moment of any order, such that, for all ω ∈ Ω * and for each

(j 1 , j 2 , k 1 , k 2 ) ∈ Z 4 , |ε k 1 ,k 2 j 1 ,j 2 (ω)| ≤ C 1 (ω) log(3 + |j 1 | + |k 1 |) log(3 + |j 2 | + |k 2 |). (IV.3.7)
In view of Lemma IV.3.4, we set

L k 1 ,k 2 j 1 ,j 2 = log(3 + |j 1 | + |k 1 |) log(3 + |j 2 | + |k 2 |
). As a first step, Lemmata IV.3.5 to IV.3.9 are devoted to bound some deterministic series whose general term is

2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) L k 1 ,k 2 j 1 ,j 2 |I k 1 ,k 2 j 1 ,j 2 [t, s]|.
This first lemma will be useful to bound the subsums in the right-hand side of (IV.3.6) for j 1 < n and j 2 < n.

Lemma IV.3.5. There exists a deterministic constant C > 0 such that, for all t, s ∈ (0, 1), we have

j 1 <n j 2 <n (k 1 ,k 2 )∈Z 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) L k 1 ,k 2 j 1 ,j 2 I k 1 ,k 2 j 1 ,j 2 [t, s] ≤ C|t -s| H 1 +H 2 -1 log |t -s| -1 .
Proof. Let us start by considering, for all (j 1 , j 2 ) ∈ Z 2 , the series

R j 1 ,j 2 : t → (k 1 ,k 2 )∈Z 2 L k 1 ,k 2 j 1 ,j 2 t 0 |ψ H 1 (2 j 1 x -k 1 )ψ H 2 (2 j 2 x -k 2 )| dx and R ′ j 1 ,j 2 : t → (k 1 ,k 2 )∈Z 2 L k 1 ,k 2 j 1 ,j 2 |ψ H 1 (2 j 1 t -k 1 )ψ H 2 (2 j 2 t -k 2 )|.
The fast decay of the fractional antiderivatives of ψ allows us to write, for all

H ∈ {H 1 , H 2 } and for all x ∈ R |ψ H (x)| ≤ C(1 + |x|) -4 . (IV.3.8)
Moreover, according to [7, Lemma 4.2] for all L > 1 there exists C > 0 such that, for all j ∈ Z and x ∈ R

k∈Z log(3 + |j| + k) (3 + |2 j x -k|) L ≤ C log(3 + |j| + 2 j |x|).
(IV.3.9)

Therefore, if K is any compact set of R + , if s = sup K , for all t ∈ K, we have

|R j 1 ,j 2 (t)| ≤ C t 0 log(3 + |j 1 | + 2 j 1 |x|) log(3 + |j 2 | + 2 j 2 |x|) dx ≤ Cs log(3 + |j 1 | + 2 j 1 s) log(3 + |j 2 | + 2 j 2 s).
The same arguments can be applied to R ′ j 1 ,j 2 , which means that both series converge uniformly on any compact set of R + . From this, we can use mean value theorem: for all (j 1 , j 2 ) ∈ Z 2 there is ξ(j 1 , j 2 ) ∈ [s, t] such that

(k 1 ,k 2 )∈Z 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) L k 1 ,k 2 j 1 ,j 2 I k 1 ,k 2 j 1 ,j 2 [t, s] ≤ |t -s| (k 1 ,k 2 )∈Z 2 L k 1 ,k 2 j 1 ,j 2 |ψ H 1 (2 j 1 ξ -k 1 )ψ H 2 (2 j 2 ξ -k 2 )|. (IV.3.10)
Now, we use the fast decay of the fractional antiderivatives of ψ (IV.3.8) and inequality (IV.3.9) to bound (IV.3.10) from above: for all j 1 , j 2 < n,

(k 1 ,k 2 )∈Z 2 L k 1 ,k 2 j 1 ,j 2 |ψ H 1 (2 j 1 ξ -k 1 )ψ H 2 (2 j 2 ξ -k 2 )| ≤ C k 1 ∈Z log(3 + |j 1 | + |k 1 |) (3 + |2 j 1 ξ -k 1 |) 4 k 2 ∈Z log(3 + |j 2 | + |k 2 |) (3 + |2 j 2 ξ -k 2 |) 4 ≤ C log(3 + |j 1 | + 2 j 1 |ξ|) log(3 + |j 2 | + 2 j 2 |ξ|) ≤ C log(3 + |j 1 | + 2 j 1 ) log(3 + |j 2 | + 2 j 2 ),
as ξ ∈ (0, 1). Let us then remark that

j 1 <n 2 j 1 (1-H 1 ) log(3 + |j 1 | + 2 j 1 ) = j 1 ≤0 2 j 1 (1-H 1 ) log(3 + |j 1 | + 2 j 1 ) + n-1 j 1 =0 2 j 1 (1-H 1 ) log(3 + |j 1 | + 2 j 1 ) ≤ C + n-1 j 1 =0 2 j 1 (1-H 1 ) log(3 + |j 1 | + 2 j 1 ) ≤ C2 n(1-H 1 ) √ n, (IV.3.11)
as 1 -H 1 > 0. The same can be applied to the sum over j 2 and we finally get

j 1 <n j 2 <n (k 1 ,k 2 )∈Z 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) L k 1 ,k 2 j 1 ,j 2 I k 1 ,k 2 j 1 ,j 2 [t, s] ≤ C|t -s| j 1 <n j 2 <n 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) log(3 + |j 1 | + 2 j 1 ) log(3 + |j 2 | + 2 j 2 ) ≤ C|t -s|2 n(2-H 1 -H 2 ) n ≤ C|t -s| H 1 +H 2 -1 log |t -s| -1 .
Lemmata IV.3.7 and IV.3.8 will help finding an upper bound for the subsums in the right-hand side of (IV.3.6) with j 1 < n ≤ j 2 or j 2 < n ≤ j 1 as well as the ones where n ≤ j 1 ≤ j 2 and n ≤ j 2 ≤ j 1 . Let us define the following partition of Z, which determines the relative positions of [k 2 2 -j 2 , (k 2 + 1)2 -j 2 ) and [s, t].

Definition IV.3.6. For all j 2 ∈ N, we set

Z < j 2 (t, s) = {k 2 ∈ Z : k 2 2 -j 2 < min{t, s}}, Z > j 2 (t, s) = {k 2 ∈ Z : k 2 2 -j 2 > max{t, s}}, and Z j 2 [t, s] = Z \ (Z < j 2 (t, s) ∪ Z > j 2 (t, s)).
Remark 15. Note that we have #Z j 2 [t, s] ≤ 2 j 2 -n + 1.

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

log(3 + a + b) ≤ log(3 + a) log(3 + b).
(IV.3.12)

Lemma IV.3.7. There exists a deterministic constant C > 0 such that, for all t, s ∈ (0, 1) and j 1 ≤ j 2 , the quantities

k 1 ∈Z k 2 ∈Z < j 2 (t,s) L k 1 ,k 2 j 1 ,j 2 I k 1 ,k 2 j 1 ,j 2 [t, s] (IV.3.13) k 1 ∈Z k 2 ∈Z > j 2 (t,s) L k 1 ,k 2 j 1 ,j 2 I k 1 ,k 2 j 1 ,j 2 [t, s] (IV.3.14)
are bounded from above by

C log(3 + |j 1 | + 2 j 1 ) log(3 + |j 2 | + 2 j 2 )2 -j 2 .
Proof. Let us bound (IV.3.13), the proof for (IV.3.14) being similar. From the fast decay of the fractional antiderivatives of ψ (IV.3.8), inequalities (IV.3.9) and (IV.3.12) for j 1 ≤ j 2 , we have

(IV.3.13) ≤ C [s,t] k 1 ∈Z log(3 + |j 1 | + |k 1 |) (3 + |2 j 1 x -k 1 |) 4    k 2 ∈Z < j 2 (t,s) log(3 + |j 2 | + |k 2 |) (3 + |2 j 2 x -k 2 |) 4    dx ≤ C log(3 + |j 1 | + 2 j 1 ) log(3 + |j 2 | + 2 j 2 ) [s,t] k 2 ∈Z < j 2 (t,s) log(3 + |2 j 2 x -k 2 |) (3 + |2 j 2 x -k 2 |) 4 dx.
For all x ∈ [s, t] the mapping y → (2 + 2 j 2 x -2 j 2 min{s, t} + y) -3 is decreasing and thus

[s,t] k 2 ∈Z < j 2 (t,s) log(3 + |2 j 2 x -k 2 |) (3 + |2 j 2 x -k 2 |) 4 dx ≤ [s,t] k 2 ∈Z < j 2 (t,s) dx (3 + 2 j 2 x -k 2 ) 3 ≤ [s,t] +∞ m=0 dx (3 + 2 j 2 x -2 j 2 min{s, t} + m) 3 ≤ [s,t] +∞ 0 dxdy (2 + 2 j 2 x -2 j 2 min{s, t} + y) 3 ≤ C2 -j 2 .
(IV. 3.15) This bound leads to

(IV.3.13) ≤ C log(3 + |j 1 | + 2 j 1 ) log(3 + |j 2 | + 2 j 2 )2 -j 2 . (IV.3.16)
Lemma IV.3.8. There exists a deterministic constant C > 0 such that, for all t, s ∈ (0, 1) and j 1 ≤ j 2 , the quantities

k 1 ∈Z k 2 ∈Z j 2 [t,s] L k 1 ,k 2 j 1 ,j 2 min{s,t} -∞ ψ H 1 (2 j 1 x -k 1 )ψ H 2 (2 j 2 x -k 2 ) dx k 1 ∈Z k 2 ∈Z j 2 [t,s] L k 1 ,k 2 j 1 ,j 2 +∞ max{s,t} ψ H 1 (2 j 1 x -k 1 )ψ H 2 (2 j 2 x -k 2 ) dx
are bounded from above by

C log(3 + |j 1 | + 2 j 2 ) log(3 + |j 2 | + 2 j 2 )2 -j 2 .
Proof. Let us assume that s ≤ t, the argument for t < s being similar. As j 2 ≥ j 1 , we have, by inequality (IV.3.9),

s -∞ k 1 ∈Z log(3 + |j 1 | + |k 1 |) (3 + |2 j 1 x -k 1 |) 4   k 2 ∈Z j 2 [t,s] log(3 + |j 2 | + |k 2 |) (3 + |2 j 2 x -k 2 |) 4   dx ≤ C L s -∞ log(3 + |j 1 | + 2 j 1 |x|)   k 2 ∈Z j 2 [t,s] log(3 + |j 2 | + |k 2 |) (3 + |2 j 2 x -k 2 |) 4   dx ≤ C L s -∞ log(3 + |j 1 | + 2 j 2 |x|)   k 2 ∈Z j 2 [t,s] log(3 + |j 2 | + |k 2 |) (3 + |2 j 2 x -k 2 |) 4   dx.
For all k 2 ∈ Z j 2 [t, s], |k 2 | ≤ 2 j 2 , we have, using (IV.3.12),

log(3 + |j 1 | + 2 j 2 |x|) ≤ log(3 + |j 1 | + 2 j 2 ) log(3 + |2 j 2 x -k 2 |) and log(3 + |j 2 | + |k 2 |) ≤ log(3 + j 2 + 2 j 2 ) log(3 + |2 j 2 x -k 2 |).
Thus, it only remains us to deal with

s -∞ k 2 ∈Z j 2 [t,s] dx (3 + |2 j 2 x -k 2 |) 3 . But, for all x ≤ s and k 2 ∈ 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 (IV.3.15), we get

s -∞ k 2 ∈Z j 2 [t,s] dx (3 + k 2 -2 j 2 x) 3 ≤ C2 -j 2
(IV. 3.17) which finally leads to

k 1 ∈Z k 2 ∈Z j 2 [t,s] L k 1 ,k 2 j 1 ,j 2 s -∞ ψ H 1 (2 j 1 x -k 1 )ψ H 2 (2 j 2 x -k 2 ) dx ≤ C log(3 + |j 1 | + 2 j 2 ) log(3 + j 2 + 2 j 2 )2 -j 2 . (IV.3.18)
We get in the same way,

k 1 ∈Z k 2 ∈Z j 2 [t,s] L k 1 ,k 2 j 1 ,j 2 +∞ t ψ H 1 (2 j 1 x -k 1 )ψ H 2 (2 j 2 x -k 2 ) dx ≤ C log(3 + |j 1 | + 2 j 2 ) log(3 + j 2 + 2 j 2 )2 -j 2 .
Next Lemma will be used to bound the subsums of (IV.3.6) with j 1 < n ≤ j 2 or j 2 < n ≤ j 1 .

Lemma IV.3.9. There exists a deterministic constant C > 0 such that, for all t, s ∈ (0, 1), the quantities

R <≥n [t, s] := j 1 <n j 2 ≥n (k 1 ,k 2 )∈Z 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) L k 1 ,k 2 j 1 ,j 2 |I k 1 ,k 2 j 1 ,j 2 [t, s]| R ≥<n [t, s] := j 1 ≥n j 2 <n (k 1 ,k 2 )∈Z 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) L k 1 ,k 2 j 1 ,j 2 |I k 1 ,k 2 j 1 ,j 2 [t, s]| are bounded from above by C|t -s| H 1 +H 2 -1 log |t -s| -1 .
Proof. As R <≥n [t, s] and R ≥<n [t, s] can clearly be treated symmetrically, we restrict our attention to R <≥n [t, s]. One sees that

j 1 <n j 2 ≥n (k 1 ,k 2 )∈Z 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) L k 1 ,k 2 j 1 ,j 2 I k 1 ,k 2 j 1 ,j 2 [t, s] = j 1 <n j 2 ≥n k 1 ∈Z k 2 ∈Z < j 2 (t,s) 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) L k 1 ,k 2 j 1 ,j 2 I k 1 ,k 2 j 1 ,j 2 [t, s] (IV.3.19) + j 1 <n j 2 ≥n k 1 ∈Z k 2 ∈Z > j 2 (t,s) 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) L k 1 ,k 2 j 1 ,j 2 I k 1 ,k 2 j 1 ,j 2 [t, s] (IV.3.20) + j 1 <n j 2 ≥n k 1 ∈Z k 2 ∈Z j 2 [t,s] 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) L k 1 ,k 2 j 1 ,j 2 I k 1 ,k 2 j 1 ,j
2 j 1 (1-H 1 ) log(3 + |j 1 | + 2 j 1 ) j 2 ≥n 2 -j 2 H 2 log(3 + |j 2 | + 2 j 2 ).
The sum over over j 1 is bounded just as in (IV.3.11) while, for the sum over j 2 , we have

j 2 ≥n 2 -j 2 H 2 log(3 + |j 2 | + 2 j 2 ) ≤ j 2 ≥n 2 -j 2 H 2 log(3 + 2 j 2 +1 ) ≤ C2 -nH 2 √ n. (IV.3.22)
We bound (IV.3.20) in exactly the same way.

For (IV.3.21), let us again assume s ≤ t, then we write

I k 1 ,k 2 j 1 ,j 2 [t, s] = R ψ H 1 (2 j 1 x -k 1 )ψ H 2 (2 j 2 x -k 2 ) dx - s -∞ ψ H 1 (2 j 1 x -k 1 )ψ H 2 (2 j 2 x -k 2 ) dx - +∞ t ψ H 1 (2 j 1 x -k 1 )ψ H 2 (2 j 2 x -k 2 ) dx. (IV.3.23)
Since j 1 < n and j 2 ≥ n, recalling Lemma IV.2.1, the sum

k 1 ∈Z k 2 ∈Z j 2 [t,s] L k 1 ,k 2 j 1 ,j 2 R ψ H 1 (2 j 1 x -k 1 )ψ 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 #Z n [t, s] ≤ 2 and, for all k 2 ∈ Z n [t, s], |k 2 | ≤ 2 n .
Then, by Lemma IV.2.1 and inequality (IV.3.9), we get

k 1 ∈Z k 2 ∈Zn[t,s] L k 1 ,k 2 n-1,n I k 1 ,k 2 n-1,n ≤ C2 -n k 1 ∈Z k 2 ∈Zn[t,s] log(3 + n -1 + |k 1 |) log(3 + n + |k 2 |) (3 + |2k 1 -k 2 |) 4 ≤ C2 -n k 2 ∈Zn[t,s] log(3 + n -1 + | k 2 2 |) log(3 + n + |k 2 |) ≤ C2 -n n
Now, using Lemma IV.3.8, we also get

j 1 <n j 2 ≥n k 1 ∈Z k 2 ∈Z j 2 [t,s] 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) L k 1 ,k 2 j 1 ,j 2 s -∞ ψ H 1 (2 j 1 x -k 1 )ψ H 2 (2 j 2 x -k 2 ) dx ≤ j 1 <n j 2 ≥n 2 j 1 (1-H 1 ) 2 -j 2 H 2 log(3 + |j 1 | + 2 j 2 ) log(3 + |j 2 | + 2 j 2 ) ≤ j 1 <0 j 2 ≥n 2 j 1 (1-H 1 ) 2 -j 2 H 2 log(3 + |j 1 |) log(3 + 2 j 2 +1 ) + n-1 j 1 =0 j 2 ≥n 2 j 1 (1-H 1 ) 2 -j 2 H 2 log(3 + 2 j 2 +1 ) log(3 + 2 j 2 +1 ) ≤ C2 n(1-H 1 -H 2 ) n (IV.3.24)
The series

j 1 <n j 2 ≥n k 1 ∈Z k 2 ∈Z j 2 [t,s] 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) L k 1 ,k 2 j 1 ,j 2 +∞ t ψ H 1 (2 j 1 x -k 1 )ψ H 2 (2 j 2 x -k 2 ) dx
is bounded in exactly the same way and the conclusion follows.

It remains us to bound the subsums of (IV.3.6) with j 1 ≥ n and j 2 ≥ n. For this, let us define some random variables associated with dyadic intervals.

Definition IV.3.10. If λ is a dyadic interval of scale n, we define, for all j ≥ n, the indexation sets S 0 j (λ) := {(k (1) , K (1) , k (2) , K (2) ) ∈ Z 4 : 1) , K (1) , k (2) , K (2) ) ∈ Z 4 : 1) , K (1) , k (2) , K (2) ) ∈ Z 4 :

k (1) 2 j , K (1) 2 j , k (2) 2 j , K (2) 2 j ∈ λ}, S 1 j (λ) := {(k (
k (1) 2 j+1 , K (1) 2 j+1 , k (2) 2 j , K (2) 2 j ∈ λ}, S 2 j (λ) := {(k (
k (1) 2 j , K (1) 2 j , k (2) 2 j+1 , K (2) 2 j+1 ∈ λ}
and consider the random variables, for (k (1) , K (1) , k (2) , K (2) ) ∈ S 0 j (λ),

0 j k (2) ,K (2)
k (1) ,K (1) :=

k (1) ≤k 1 ≤K (1) k (2) ≤k 2 ≤K (2) ε k 1 ,k 2 j,j I k 1 ,k 2 j,j (IV.3.25)
, for (k (1) , K (1) , k (2) , K (2) ) ∈ S 1 j (λ), 2) ,K (2) k (1) ,K (1) :=

1 j k (
k (1) ≤k 1 ≤K (1) k (2) ≤k 2 ≤K (2) ε k 1 ,k 2 j+1,j I k 1 ,k 2 j+1,j (IV.3.26)
and, for (k (1) , K (1) , k (2) , K (2) ) ∈ S 2 j (λ),

2 j k (2) ,K (2) k (1) ,K (1) := k (1) ≤k 1 ≤K (1) k (2) ≤k 2 ≤K (2) ε k 1 ,k 2 j,j+1 I k 1 ,k 2 j,j+1 . (IV.3.27)
The idea behind the definition of these random variables is, as |t -s| ≤ 2 -n , s ∈ 3λ n (t) and thus any sum of the form

k 1 ∈Z j [t,s] k 2 ∈Z ℓ [t,s] ε k 1 ,k 2 j,ℓ I k 1 ,k 2 j,ℓ (IV.3.28)
for ℓ ∈ {j, j + 1} can be written as the sum of random variables (IV.3.25), (IV.3.26) or (IV.3.27) for some (k (1) , K (1) , k (2) , K (2) ) belonging to at most two S ℓ j (λ) (ℓ ∈ {0, 1, 2}) with λ ∈ λ n (t). Indeed, if t and s both belong to λ n (t) then we only need to rewrite (IV.3.28) in the form (IV.3.25), (IV.3.26) or (IV.3.27) for (k (1) , K (1) , k (2) , K (2) ) ∈ S ℓ j (λ n (t)); if s ∈ λ with λ ∈ 3λ n (t) \ λ n (t) then we need to consider a first sum indexed by a quadruple of S ℓ j (λ n (t)) and a second indexed by a quadruple of S ℓ j (λ).

The reason why we decide to put λ instead of 3λ in the definition of the sets S ℓ j (λ) is that if, for all n ∈ N and for all λ ∈ Λ n and j ≥ n, we define the random variable (1) ,K (1) ,k (2) ,K (2) )∈S ℓ j (λ)) ℓ j k (2) ,K (2) k (1) ,K (1) ℓ j k (2) ,K (2) k (1) ,K (1) L 2 (Ω)

Ξ j (λ) = max ℓ∈{0,1,2} sup (k
, (IV.3.29)
we want Ξ j (λ) to be independent of Ξ j (λ ′ ) as soon as λ ∩ λ ′ = ∅. Moreover, from the definitions of the random variables (IV.3.26), (IV.3.25) and (IV.3.27), the remarks below Theorem IV.3.2 and the explicit expressions (IV.2.2), (IV.2.3) and (IV.2.4), the law of Ξ j (λ) does not depend on λ ∈ Λ n but only on j -n.

The key results to estimate the random variables Ξ j are [53, Theorem 6.7 and Theorem 6.12] that we recall here.

Theorem IV.3.11. There exists a strictly positive universal deterministic constant ⋆ C such that, for every random variable X belonging to the second order Wiener chaos and for each real number y ≥ 2, one has

P(|X| ≥ y∥X∥ L 2 (Ω) ) ≤ exp(- ⋆ Cy).
Theorem IV.3.12. 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 ≥ y 0 , exp(-ay) ≤ P(|X| ≥ y) ≤ exp(-by).

Remark 16. As stated in [START_REF] Janson | Gaussian Hilbert Spaces[END_REF], the constants a, b in Theorem IV.3.12 are not universal and depend on the law of X. Note that b can be recovered from Theorem IV.3.11 and thus is universal on the unit sphere in L 2 (Ω).

Lemma IV.3.13. There exists a deterministic constant C > 0 such that, for all n ∈ N, λ ∈ Λ n , j ≥ n, ℓ ∈ {0, 1, 2} and (k (1) , K (1) , k (2) , K (2) ) ∈ S ℓ j (λ), we have

ℓ j k (2) ,K (2) k (1) ,K (1) L 2 (Ω) ≤ C2 -j-n 2
.

Proof. Following an idea from [7, Lemma 2.21], we write ℓ j k (2) ,K (2) k (1) ,K (1) L 2 (Ω)

≤ R∈{<,>,=} ℓ j k (2) ,K (2)
k (1) ,

K (1) R L 2 (Ω)
where ℓ j k (2) ,K (2) k (1) ,K (1) R is the subsum of (IV.3.26), (IV.3.25) or (IV.3.27) in which k 1 Rk 2 . By doing so, we make sure that two random variables ε k 1 ,k 2 j 1 ,j 2 and ε

k ′ 1 ,k ′ 2 j ′ 1 ,j ′ 2 appearing in this subsum are uncorrelated except when (k 1 , k 2 ) = (k ′ 1 , k ′ 2 )
. Then from Lemma IV.2.1, we have for ℓ = 0 (the argument being the same for ℓ = 1 or ℓ = 2), for all R ∈ {<, >, =}, ℓ j k (2) ,K (2) k (1) ,K (1)

R 2 L 2 (Ω) = k (1) ≤k 1 ≤K (1) k (2) ≤k 2 ≤K (2) ,k 1 Rk 2 E[(ε k 1 ,k 2 j,j ) 2 ](I k 1 ,k 2 j,j ) 2 ≤ k (1) ≤k 1 ≤K (1) k 2 ∈Z 2 -2j (3 + |k 1 -k 2 |) 8 . Since #{k 1 ∈ Z : k (1) ≤ k 1 ≤ K (1) } ≤ 2 j-n , we conclude that ℓ j k (2) ,K (2)
k (1) ,K (1) L 2 (Ω)

≤ C2 -j-n 2
.

(IV.3.30)

Lemma IV.3.14. There exist an event Ω of probability 1 and a positive random variable C 2 with finite moment of any order such that, on

Ω ∀n ∈ N, ∀ λ ⊆ [0, 1], λ ∈ Λ n , ∀j ≥ n, Ξ j (λ) ≤ C 2 (j -n + 1)n. (IV.3.31)
Proof. Let us take θ > 0 and consider, for all n ∈ N the event

A n := {∀ λ ⊆ [0, 1], λ ∈ Λ n , ∀j ≥ n, Ξ j (λ) ≤ θ(j -n + 1)n} .
If A c n stands for the complementary set of A n in Ω, we have, of course, (1) , K (1) , k (2) , K (2) ) ∈ S ℓ j (λ) we have, by Theorem IV.3.11, if θ ≥ 2,

P(A c n ) = P(∃λ ⊆ [0, 1], λ ∈ Λ n : ∃j ≥ n s. t. Ξ j (λ) ≥ θ(j -n + 1)n). But, for all λ ⊆ [0, 1], λ ∈ Λ n , j ≥ n, ℓ ∈ {0, 1, 2} and (k
P      ℓ j k (2) ,K (2)
k (1) ,K (1) ℓ j k (2) ,K (2) k (1) ,K (1) L 2 (Ω)

≥ θ(j -n + 1)n      ≤ exp(- ⋆ Cθ(j -n + 1)n).
As, for all j ≥ n, #S ℓ j (λ) ≤ 2 4(j-n) and #{λ ⊆ [0, 1] : λ ∈ Λ n } = 2 n , we get

P(A c n ) ≤ C2 n j≥n 2 4(j-n) exp(- ⋆ Cθ(j -n + 1)n) ≤ C2 n exp(- ⋆ Cθn) j≥n 2 4(j-n) exp(- ⋆ Cθ(j -n))
for a deterministic constant C > 0. Therefore, if we take θ > 4 log(2)/ ⋆ C, the conclusion follows from Borel-Cantelli Lemma.

Lemma IV.3.15. Let Ω * and Ω be the events of probability 1 given by Lemmata IV.3.4 and IV.3.14 respectively. There exists a positive random variable C 3 with finite moment of any order such that, on Ω * ∩ Ω, for all t, s ∈ (0, 1) the random variable

j 1 ≥n j 2 >n (k 1 ,k 2 )∈Z 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) ε k 1 ,k 2 j 1 ,j 2 I k 1 ,k 2 j 1 ,j 2 [t, s] (IV.3.32)
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 (IV.3.32) in two parts:

j 1 ≥n j 2 ≥j 1 (k 1 ,k 2 )∈Z 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) ε k 1 ,k 2 j 1 ,j 2 I k 1 ,k 2 j 1 ,j 2 [t, s] and j 2 ≥n j 1 >j 2 (k 1 ,k 2 )∈Z 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) ε k 1 ,k 2 j 1 ,j 2 I k 1 ,k 2 j 1 ,j 2 [t, s]. (IV.3.33)
We only focus on the first sums, as the argument is symmetric in j 1 and j 2 . As in Lemma IV.3.9 we write

j 1 ≥n j 2 ≥j 1 (k 1 ,k 2 )∈Z 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) ε k 1 ,k 2 j 1 ,j 2 I k 1 ,k 2 j 1 ,j 2 [t, s] = j 1 ≥n j 2 ≥j 1 k 1 ∈Z k 2 ∈Z < j 2 (t,s) 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) ε k 1 ,k 2 j 1 ,j 2 I k 1 ,k 2 j 1 ,j 2 [t, s] (IV.3.34) + j 1 ≥n j 2 ≥j 1 k 1 ∈Z k 2 ∈Z > j 2 (t,s) 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) ε k 1 ,k 2 j 1 ,j 2 I k 1 ,k 2 j 1 ,j 2 [t, s] (IV.3.35) + j 1 ≥n j 2 ≥j 1 k 1 ∈Z k 2 ∈Z j 2 [t,s] 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) ε k 1 ,k 2 j 1 ,j 2 I k 1 ,k 2 j 1 ,j 2 [t, s]. (IV.3.36)
To bound (IV.3.34), we use inequality (IV.3.7) and Lemma IV.3.7 to get

|(IV.3.34)| ≤ CC 1 j 1 ≥n j 2 ≥j 1 2 j 1 (1-H 1 ) 2 -j 2 H 2 log(3 + |j 1 | + 2 j 1 ) log(3 + |j 2 | + 2 j 2 ) ≤ CC 1 j 1 ≥n 2 j 1 (1-H 1 -H 2 ) j 1 ≤ CC 1 2 n(1-H 1 -H 2 ) n,
by applying twice inequality (IV. 3.22). The sum (IV.3.35) is bounded in exactly the same way.

To bound (IV.3.36), we use once again the equality (IV.3.23). First we have, by inequality (IV.3.7) and Lemma IV.3.8,

j 1 ≥n j 2 ≥j 1 k 1 ∈Z k 2 ∈Z j 2 [t,s] 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) ε k 1 ,k 2 j 1 ,j 2 s -∞ ψ H 1 (2 j 1 x -k 1 )ψ H 2 (2 j 2 x -k 2 ) dx ≤ CC 1 j 1 ≥n j 2 ≥j 1 2 j 1 (1-H 1 ) 2 -j 2 H 2 log(3 + |j 1 | + 2 j 2 ) log(3 + |j 2 | + 2 j 2 ) ≤ CC 1 2 n(1-H 1 -H 2 ) n.
(IV.3.37)

We bound

j 1 ≥n j 2 ≥j 1 k 1 ∈Z k 2 ∈Z j 2 [t,s] 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) ε k 1 ,k 2 j 1 ,j 2 +∞ t ψ H 1 (2 j 1 x -k 1 )ψ H 2 (2 j 2 x -k 2 ) dx
in the same way.

It only remains us to find an estimate for

j 1 ≥n j 2 ≥j 1 k 1 ∈Z k 2 ∈Z j 2 [t,s] 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) ε k 1 ,k 2 j 1 ,j 2 I k 1 ,k 2 j 1 ,j 2
and thus, recalling Lemma IV.2.1, we reduce the problem to first bound, for j ≥ n and ℓ ∈ {j, j + 1}, the sums

k 1 ∈Z < j (t,s) k 2 ∈Z ℓ [t,s] ε k 1 ,k 2 j,ℓ I k 1 ,k 2 j,ℓ , (IV.3.38) k 1 ∈Z > j (t,s) k 2 ∈Z ℓ [t,s] ε k 1 ,k 2 j,ℓ I k 1 ,k 2 j,ℓ , (IV.3.39) k 1 ∈Z j [t,s] k 2 ∈Z ℓ [t,s] ε k 1 ,k 2 j,ℓ I k 1 ,k 2 j,ℓ (IV.3.40)
on Ω * ∩ Ω. Let us consider (IV.3.38) with ℓ = j, the argument for ℓ = j + 1 and (IV.3.39) being similar. Using again Lemmata IV.2.1 and IV.3.4, we have on Ω * ∩ Ω, since for all

k 2 ∈ Z j [t, s], |k 2 | ≤ 2 j , for j ≥ n, |(IV.3.38)| ≤ CC k 1 ∈Z < j (t,s) k 2 ∈Z j [t,s] 2 -j (3 + |k 1 -k 2 |) 4 log(3 + j + |k 1 |) log(3 + j + |k 2 |) ≤ CC k 1 ∈Z < j (t,s) k 2 ∈Z j [t,s] 2 -j √ j (3 + k 2 -k 1 ) 4 log(3 + j + |k 1 |) ≤ CC k 1 ∈Z < j (t,s) +∞ m=0 2 -j √ j (3 + 2 j min{s, t} + m -k 1 ) 4 log(3 + j + |k 1 |) ≤ CC k 1 ∈Z < j (t,s) 2 -j j +∞ 0 dy (2 + 2 j min{s, t} + y -k 1 ) 4 log(3 + j + |k 1 |) ≤ CC 2 -j j k 1 ∈Z < j (t,s) log(3 + j + |k 1 |) (2 + 2 j min{s, t} -k 1 ) 3 ≤ CC 2 -j j log(3 + j + 2 j min{s, t}) ≤ CC 2 -j j. (IV.3.41) It follows that j≥n 2 j(2-H 1 -H 2 ) j+1 ℓ=j k 1 ∈Z < j (t,s) k 2 ∈Z ℓ [t,s] ε k 1 ,k 2 j,ℓ I k 1 ,k 2 j,ℓ ≤ CC 1 2 n(1-H 1 -H 2 ) n
and, similarly,

j≥n 2 j(2-H 1 -H 2 ) j+1 ℓ=j k 1 ∈Z > j (t,s) k 2 ∈Z ℓ [t,s] ε k 1 ,k 2 j,ℓ I k 1 ,k 2 j,ℓ ≤ CC 1 2 n(1-H 1 -H 2 ) n.
The bound for (IV.3.40) is obtained using (IV.3.31) and (IV.3.30) which lead to

j≥n 2 j(2-H 1 -H 2 ) j+1 ℓ=j k 1 ∈Z j [t,s] k 2 ∈Z ℓ [t,s] ε k 1 ,k 2 j,ℓ I k 1 ,k 2 j,ℓ ≤ CC 2 j≥n 2 j( 3 2 -H 1 -H 2 ) 2 -n 2 (j -n + 1)n ≤ CC 2 2 n( 3 2 -H 1 -H 2 ) 2 -n 2 n = CC 2 2 n(1-H 1 -H 2 ) n, as 3 2 < H 1 + H 2 .
Putting all of these together we get that (IV.3.32) is bounded from above by

C max{C 1 , C 2 }|t -s| H 1 +H 2 -1 log |t -s| -1
on Ω * ∩ Ω.

We now prove the main result of this subsection.

Proof of Proposition IV.3.3. Let us consider ω in the event Ω * ∩ Ω of probability 1, where Ω * and Ω are given by Lemmata IV.3.4 and IV.3.14 respectively. If t, s ∈ (0, 1), we write

|R H 1 ,H 2 (t, ω) -R H 1 ,H 2 (s, ω)| ≤ j 1 <n j 2 <n (k 1 ,k 2 )∈Z 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) ε k 1 ,k 2 j 1 ,j 2 (ω)I k 1 ,k 2 j 1 ,j 2 [t, s] + j 1 <n j 2 ≥n (k 1 ,k 2 )∈Z 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) ε k 1 ,k 2 j 1 ,j 2 (ω)I k 1 ,k 2 j 1 ,j 2 [t, s] + j 1 ≥n j 2 <n (k 1 ,k 2 )∈Z 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) ε k 1 ,k 2 j 1 ,j 2 (ω)I k 1 ,k 2 j 1 ,j 2 [t, s] + j 1 ≥n j 2 ≥n (k 1 ,k 2 )∈Z 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) ε k 1 ,k 2 j 1 ,j 2 (ω)I k 1 ,k 2 j 1 ,j 2 [t, s] .
The first sum is bounded from above by Lemmata IV.3.4 and IV.3.5, the second and the third one are bounded from above by Lemmata IV.3.4 and IV.3.9 and the last one is bounded from above by Lemma IV.3.15.

Remark 17. Starting from now and until the end of this section, one can reduce our attention to the process

   R ′ H 1 ,H 2 (t) = +∞ j 1 =0 +∞ j 2 =0 (k 1 ,k 2 )∈Z 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) ε k 1 ,k 2 j 1 ,j 2 I k 1 ,k 2 j 1 ,j 2 [0, •]   
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 ∈ (0, 1),

j 1 <0 j 2 <0 (k 1 ,k 2 )∈Z 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) ε k 1 ,k 2 j 1 ,j 2 I k 1 ,k 2 j 1 ,j 2 [t, s] ≤ C|t -s|, j 1 <0 j 2 >0 (k 1 ,k 2 )∈Z 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) ε k 1 ,k 2 j 1 ,j 2 I k 1 ,k 2 j 1 ,j 2 [t, s] ≤ C|t -s| -H 2 log |t -s| -1 j 1 >0 j 2 <0 (k 1 ,k 2 )∈Z 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) ε k 1 ,k 2 j 1 ,j 2 I k 1 ,k 2 j 1 ,j 2 [t, s] ≤ C|t -s| -H 1 log |t -s| -1
and we conclude because

H 1 + H 2 -1 < min{H 1 , H 2 } < 1.

IV.3.2 Ordinary points

Let us now go to the almost sure finiteness of the limit (IV.1.6) for almost every point. The main idea behind our method is that wavelets which contribute the most in

|R H 1 ,H 2 (t, •) - R H 1 ,H 2 (s, •)
| are the ones with associated dyadic intervals "close" to the interval [t, s]. Thus, we aim at proving the following Proposition.

Proposition IV.3.16. There exists an event Ω ord of probability 1 such that for all ω ∈ Ω ord , for almost every t ∈ (0, 1),

lim sup s→t |R H 1 ,H 2 (t, ω) -R H 1 ,H 2 (s, ω)| |t -s| H 1 +H 2 -1 log log |t -s| -1 < +∞.
As in [START_REF] Esser | Slow, ordinary and rapid points for Gaussian Wavelets Series and application to Fractional Brownian Motions[END_REF], for all j ∈ N, we denote by k j (t) the unique integer such that t ∈ [k j (t)2 -j , (k j (t)+ 1)2 -j ). In other words, k j (t) = s(λ j (t)). If t ∈ (0, 1) is fixed, applying Lemma IV.3.4 to the sequence of random variables (ξ

k ′ 1 ,k ′ 2 j 1 ,j 2 ) (j 1 ,j 2 ,k ′ 1 ,k ′ 2 )∈Z 4 defined by ξ k ′ 1 ,k ′ 2 j 1 ,j 2 = ε k ′ 1 +k j 1 (t) ,k ′ 2 +k j 2 (t) j 1 ,j 2
we deduce the existence of Ω * t , an event of probability 1, and C t,1 , a positive random variable with finite moment of any order, such that, for all ω ∈ Ω * t and for each (j

1 , j 2 , k 1 , k 2 ) ∈ Z 4 , one has |ε k 1 ,k 2 j 1 ,j 2 (ω)| ≤ C t,1 (ω) log(3 + |j 1 | + |k 1 -k j 1 (t)|) log(3 + |j 2 | + |k 2 -k j 2 (t)|). (IV.3.42)
In view of this fact, let us set, for t ∈ (0, 1) and (j

1 , j 2 , k 1 , k 2 ) ∈ N 2 × Z 2 L k 1 ,k 2 j 1 ,j 2 (t) = log(3 + j 1 + |k 1 -k j 1 (t)|) log(3 + j 2 + |k 2 -k j 2 (t)|).
In what follows, we show how to modify Lemmata IV.3.5 to IV.3.15 from the previous subsection, using

L k 1 ,k 2 j 1 ,j 2 (t) instead of L k 1 ,k 2 j 1 ,j 2 .
Before all, we need the following Lemma which is inspired by results from [START_REF] Esser | Slow, ordinary and rapid points for Gaussian Wavelets Series and application to Fractional Brownian Motions[END_REF] that can be extended in our case.

Lemma IV.3.17. For all L > 2 there exists a constant C L > 0 such that, for all n ∈ N and t, s ∈ (0, 1) such that

2 -n-1 < |t -s| ≤ 2 -n , for all x ∈ [s, t] 1. For all 0 ≤ j < n k∈Z log(3 + j + |k -k j (t)|) (3 + |2 j x -k|) L ≤ C L log(3 + j).
2. For all j ≥ n

k∈Z log(3 + j + |k -k j (t)|) (3 + |2 j x -k|) L ≤ C L j -n + 1 log(3 + j).
Proof. For all j ∈ N, k ∈ Z and x ∈ [s, t], observe that

|k -k j (t)| ≤ |k -2 j x| + |2 j x -2 j t| + |2 j t -k j (t)| ≤ |k -2 j x| + 2 j-n + 1. (IV.3.43)
If 0 ≤ j < n, then it follows from (IV.3.43) that |k -k j (t)| ≤ |2 j x -k| + 2 which allow us to write, thanks to inequality (IV.3.12),

log(3 + j + |k -k j (t)|) (3 + |2 j x -k|) ≤ log(3 + j) log(5 + |2 j x -k|) |2 j x -k| + 3 ≤ C log(3 + j).
where

C := sup x≥0 √ log(5+x) x+3
and we conclude using the boundedness of the function

ξ → k∈Z 1 (1 + |ξ -k|) M (IV.3.44) for all M > 1. Now, if j ≥ n, from (IV.3.43) we get |k -k j (t)| ≤ |2 j x -k| + 2 j-n+1
and thus, again by inequality (IV.3.12),

log(3 + j + |k -k j (t)|) (3 + |2 j x -k|) ≤ log(3 + 2 j-n+1 ) log(3 + j) log(3 + |2 j x -k|) |2 j x -k| + 3 ≤ C ′ j -n + 1 log(3 + j).
where

C ′ := √ 3 sup x≥0 √ log(3+x) x+3
and the conclusion comes again from the boundedness of the function in (IV.3.44) for all M > 1 Lemma IV.3.18. There exists a deterministic constant C > 0 such that, for all t, s ∈ (0, 1) we have

0≤j 1 <n 0≤j 2 <n (k 1 ,k 2 )∈Z 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) L k 1 ,k 2 j 1 ,j 2 (t) I k 1 ,k 2 j 1 ,j 2 [t, s] ≤ C|t -s| H 1 +H 2 -1 log log |t -s| -1 . Proof. If ξ ∈ [s, t],
we get from the fast decay of the fractional antiderivatives of ψ (IV.3.8) and inequality (IV.3.42), for 0 ≤ j 1 , j 2 < n,

(k 1 ,k 2 )∈Z 2 L k 1 ,k 2 j 1 ,j 2 (t)|ψ H 1 (2 j 1 ξ -k 1 )ψ H 2 (2 j 2 ξ -k 2 )| ≤ CC 1 k 1 ∈Z log(3 + j 1 + |k 1 -k j 1 (t)|) (3 + |2 j 1 ξ -k 1 |) 4 k 2 ∈Z log(3 + j 2 + |k 2 -k j 2 (t)|) (3 + |2 j 2 ξ -k 2 |) 4 .
These last two sums are bounded by the first point of Lemma IV.3.17. Using

n-1 j 1 =0 2 j 1 (1-H 1 ) log(3 + j 1 ) ≤ C2 n(1-H 1 ) log(n) (IV.3.45)
instead of (IV.3.11), we conclude, just as in Lemma IV.3.5, that the desired inequality holds.

Lemma IV.3.19. There exists a deterministic constant C > 0 such that, for all t, s ∈ (0, 1) and 0 ≤ j 1 < n ≤ j 2 , the quantities

k 1 ∈Z k 2 ∈Z < j 2 (t,s) L k 1 ,k 2 j 1 ,j 2 (t) I k 1 ,k 2 j 1 ,j 2 [t, s] (IV.3.46) k 1 ∈Z k 2 ∈Z > j 2 (t,s) L k 1 ,k 2 j 1 ,j 2 (t) I k 1 ,k 2 j 1 ,j 2 [t, s] (IV.3.47)
are bounded from above by

C j 2 -n + 1 log(3 + j 1 ) log(3 + j 2 )2 -j 2 .
Proof. Let us prove the bound for (IV.3.46), the argument for (IV.3.47) being similar. We have, by the first part of Lemma IV.3.17, for 0

≤ j 1 < n ≤ j 2 , (IV.3.46) ≤ C log(3 + j 1 ) [s,t] k 2 ∈Z < j 2 (t,s) 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 ∈ Z < j 2 (t, s) and x ∈ [s, t] we have

|k 2 -k j 2 (t)| ≤ |2 j 2 x -k 2 | + |k j 2 (t) -2 j 2 x| ≤ |2 j 2 x -k 2 | + 2 j 2 -n + 1
and, by inequality (IV.3.12),

log(3 + j 2 + |k 2 -k j 2 (t)|) ≤ C j 2 -n + 1 log(3 + j 2 ) log(3 + |2 j 2 x -k 2 |)
it just remains us to use the bound (IV.3.15) to write

(IV.3.46) ≤ C j 2 -n + 1 log(3 + j 1 ) log(3 + j 2 )2 -j 2 . (IV.3.48)
Lemma IV.3.20. There exists a deterministic constant C > 0 such that, for all t, s ∈ (0, 1) and 0 ≤ j 1 < n ≤ j 2 , the quantities

k 1 ∈Z k 2 ∈Z j 2 [t,s] L k 1 ,k 2 j 1 ,j 2 (t) min{s,t} -∞ ψ H 1 (2 j 1 x -k 1 )ψ H 2 (2 j 2 x -k 2 ) dx (IV.3.49) k 1 ∈Z k 2 ∈Z j 2 [t,s] L k 1 ,k 2 j 1 ,j 2 (t) +∞ max{s,t} ψ H 1 (2 j 1 x -k 1 )ψ H 2 (2 j 2 x -k 2 ) dx (IV.3.50)
are bounded from above by

C log(3 + j 1 ) log(3 + j 2 ) j 2 -n + 12 -j 2 .
Proof. Again we assume s ≤ t. First, using the fast decay of the fractional antiderivatives of ψ (IV.3.8), (IV.3.49) is bounded from above by

s -∞ k 1 ∈Z k 2 ∈Z j 2 [t,s] log(3 + j 1 + |k 1 -k j 1 (t)|) (3 + |2 j 1 x -k 1 |) 4 log(3 + j 2 + |k 2 -k j 2 (t)|) (3 + |2 j 2 x -k 2 |) 4 dx. (IV.3.51)
Observe that, for all

k 1 ∈ Z, k 2 ∈ Z j 2 [t, s] and x ∈ (-∞, s], we have, as j 1 < n ≤ j 2 , |2 j 1 x -k j 1 (t)| ≤ |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)| ≤ |2 j 2 x -k 2 | + 2 and therefore |k 1 -k j 1 (t)| ≤ |2 j 1 x -k 1 | + |2 j 2 x -k 2 | + 2 while |k 2 -k j 2 (t)| ≤ |k 2 -2 j 2 t| + |2 j 2 t -k j 2 (t)| ≤ 2 j 2 -n + 1.
It allows to write, thanks to inequality (IV.3.12), the boundedness of the function (IV.3.44) and inequality (IV.3.17)

|(IV.3.51)| ≤ C log(3 + j 1 ) log(3 + j 2 ) j 2 -n + 1 s -∞ k 2 ∈Z j 2 [t,s] dx (3 + |2 j 2 x -k 2 |) 3 ≤ C log(3 + j 1 ) log(3 + j 2 ) j 2 -n + 12 -j 2 .
We bound the second sums in the same way.

Lemma IV.3.21. There exists a deterministic constant C > 0 such that, for all t, s ∈ (0, 1), the quantities

0≤j 1 <n j 2 ≥n (k 1 ,k 2 )∈Z 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) L k 1 ,k 2 j 1 ,j 2 (t)|I k 1 ,k 2 j 1 ,j 2 [t, s]| j 1 ≥n 0≤j 2 <n (k 1 ,k 2 )∈Z 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) L k 1 ,k 2 j 1 ,j 2 (t)|I k 1 ,k 2 j 1 ,j 2 [t, s]| are bounded from above by C|t -s| H 1 +H 2 -1 log log |t -s| -1 .
for a determistic constant C > 0. Therefore, if we take again θ > 4 log(2)/ ⋆ C then Borel-Cantelli Lemma implies the existence of an event Ω t of probability 1 and C t,2 a positive random variable of finite moment of any order such that, on Ω t , assertion (IV.3.53) holds.

Lemma IV.3.25. If t ∈ (0, 1), let Ω * t be the event of probability 1 where inequality (IV.3.42) holds and Ω t be the event of probability 1 given by Lemma IV.3.24. There exists a positive random variable C t,3 with finite moment of any order such that, on Ω * t ∩ Ω t , for all s ∈ (0, 1) the random variable

j 1 ≥n j 2 >n (k 1 ,k 2 )∈Z 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) ε k 1 ,k 2 j 1 ,j 2 I k 1 ,k 2 j 1 ,j 2 [t, s] (IV.3.54) 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 (IV.3.33) and we only do the details for the first sum.

We deal with the series (IV.

2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) ε k 1 ,k 2 j 1 ,j 2 s -∞ ψ H 1 (2 j 1 x -k 1 )ψ H 2 (2 j 2 x -k 2 ) dx ≤ CC t,1 j 1 ≥n j 2 ≥j 1 2 j 1 (1-H 1 ) 2 -j 2 H 2 j 1 -n + 1 j 2 -n + 1 log(3 + j 1 ) log(3 + j 2 ) ≤ CC 1 2 n(1-H 1 -H 2 ) log(n). (IV.3.55)
We bound

j 1 ≥n j 2 ≥j 1 k 1 ∈Z k 2 ∈Z j 2 [t,s] 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) ε k 1 ,k 2 j 1 ,j 2 s -∞ ψ H 1 (2 j 1 x -k 1 )ψ H 2 (2 j 2 x -k 2 ) dx
on Ω * t ∩ Ω t exactly in the same way. To finish the proof, again, we have to bound (IV.3.38), (IV.3.39) and (IV.3.40) for ℓ ∈ {j, j + 1} (with j ≥ n) on Ω * t ∩ Ω t . For (IV. 3.38), in the case ℓ = j, one can note that, for all

k 2 ∈ Z j [t, s], |k 2 -k j (t)| ≤ 2 j-n + 1 and, for all k 1 ∈ Z < j (t, s), |k 1 -k j (t)| ≤ |2 j min{t, s} -k 1 | + 2 j-n + 1.
Using the same tricks as in (IV.3.41), we get, on

Ω * t ∩ Ω t |(IV.3.38)| ≤ CC t,1 (j -n + 1) log(3 + j)2 -j k 1 ∈Z < j (t,s) 1 (2 + |2 j min{s, t} -k 1 |) 3 ≤ C(j -n + 1) log(3 + j)2 -j .
The bounds for (IV.3.39) and in the case ℓ = j + 1 are obtained in the same way. Finally to bound (IV.3.40), we use (IV.3.53) and (IV.3.30) and get on Ω * t ∩ Ω t j≥n

2 j(2-H 1 -H 2 ) j+1 ℓ=j k 1 ∈Z j [t,s] k 2 ∈Z ℓ [t,s] ε k 1 ,k 2 j,ℓ I k 1 ,k 2 j,ℓ ≤ CC t,2 j≥n 2 j( 3 2 -H 1 -H 2 ) 2 -n 2 (j -n + 1) log(n) ≤ CC t,2 2 n(1-H 1 -H 2 ) log(n).
We conclude that (IV.3.54) 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 Ω * t ∩ Ω t .
We can now prove Proposition IV.3.16.

Proof of Proposition IV.3.16. Let us fix t ∈ (0, 1) and consider ω ∈ Ω * t ∩ Ω t . For all s ∈ (0, 1), we write1

|R ′ H 1 ,H 2 (t, ω) -R ′ H 1 ,H 2 (s, ω)| ≤ 0≤j 1 <n 0≤j 2 <n (k 1 ,k 2 )∈Z 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) ε k 1 ,k 2 j 1 ,j 2 (ω)I k 1 ,k 2 j 1 ,j 2 [t, s] + 0≤j 1 <n j 2 ≥n (k 1 ,k 2 )∈Z 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) ε k 1 ,k 2 j 1 ,j 2 (ω)I k 1 ,k 2 j 1 ,j 2 [t, s] + j 1 ≥n 0≤j 2 <n (k 1 ,k 2 )∈Z 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) ε k 1 ,k 2 j 1 ,j 2 (ω)I k 1 ,k 2 j 1 ,j 2 [t, s] + j 1 ≥n j 2 ≥n (k 1 ,k 2 )∈Z 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) ε k 1 ,k 2 j 1 ,j 2 (ω)I k 1 ,k 2 j 1 ,j 2 [t, s] .
We bound from above the first sum by inequality (IV. Using inequalities (IV.3.52) and (IV.3.55) and Remark 17, one can finally write that for all t ∈ (0, 1), for all ω in the event of probability 1

Ω * t ∩ Ω t lim sup s→t |R H 1 ,H 2 (t, ω) -R H 1 ,H 2 (s, ω)| |t -s| H 1 +H 2 -1 log log |t -s| -1 < +∞
and we conclude by Fubini Theorem.

IV.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 IV.3.26. There exists an event Ω slo of probability 1 such that for all ω ∈ Ω slo there exist t ∈ (0, 1) such that

lim sup s→t |R H 1 ,H 2 (t, ω) -R H 1 ,H 2 (s, ω)| |t -s| H 1 +H 2 -1 < +∞. (IV.3.56)
In [START_REF] Kahane | of Cambridge Studies in Advanced Mathematics[END_REF], Kahane described a procedure to insure the existence of slow points for the Brownian motion. This procedure was then generalized in [START_REF] Esser | Slow, ordinary and rapid points for Gaussian Wavelets Series and application to Fractional Brownian Motions[END_REF] to fit for any arbitrary fractional Brownian motion. It consists in showing that for any m > 0, almost surely, there exist µ > 0 and t ∈ (0, 1) such that, if one sets

Λ 0 j (t) = {λ ∈ Λ j : |s(λ(t)) -s(λ)| ≤ 1} (IV.3.57)
and, for all 1 ≤ l

Λ l j (t) = {λ ∈ Λ j , : 2 m(l-1) < |s(λ(t)) -s(λ)| ≤ 2 ml }, (IV.3.58)
then, for all λ ∈ Λ l j (t) we have

|ε λ | ≤ 2 l µ, (IV.3.59)
where ε λ is the random variable

2 j 2 R ψ λ (x) dB(x).
In this procedure, if µ ∈ N,for all j, l ∈ N 0 and λ ∈ Λ j , λ ⊆ [0, 1], we define

Λ j,l (λ) = {λ ′ ∈ Λ j , : |s(λ) -s(λ ′ )| ≤ 2 ml }
and the random set

S µ j,l = {λ ′ ∈ Λ j , : 2 l µ < |ε λ ′ | ≤ 2 l+1
µ}. Finally we consider the random set

I µ j = {λ ∈ Λ j , λ ⊆ [0, 1] : ∀l ∈ N 0 , Λ j,l (λ) ∩ S µ j,l = ∅},
and show that almost surely, there exists µ ∈ N such that

S µ low = j∈N 0 λ∈I µ j λ ̸ = ∅
which is equivalent to the fact that, for any J S µ low,J = j≤J λ∈I µ j λ ̸ = ∅ as (S µ low,J ) J is a decreasing sequence of compact sets. To do so, let us denote by 2S µ low,J the sets of dyadic intervals of scale J + 1 obtained by cutting in two the remaining intervals2 in S µ low,J and remark that S µ low,J+1 is obtained from 2S µ low,J by removing the dyadic intervals λ such that Λ J+1,l (λ) ∩ S µ J+1,l ̸ = ∅ for a l ∈ N 0 . But now, if ξ ∼ N (0, 1), we set, for all such a l p l (µ) = P(2 l µ < |ξ| ≤ 2 l+1 µ).

and note that, if N is the number of intervals of S µ low,J , counting the number of intervals in 2S µ low,J ∩ S µ J+1,l is a binomial random variable of parameter 2N and p l (µ) and this number is thus bounded by 2N (p l (µ) + (l + 1)

p l (µ)(1 -p l (µ)))
on an event of probability 1 -(l + 1) -2 N -1 . Therefore, to pass from S µ low,J to S µ low,J+1 we remove at most

2N +∞ l=0 (2 ml+1 + 1)(p l (µ) + (l + 1) p l (µ)(1 -p l (µ)))
intervals with probability greater than 1 -

N -1 . But if µ is large enough, as p l (µ) is of order e -(2 l µ) 2
2 l µ , one can make sure that this last term is bounded by N 2 . So, if N µ J is the random variable counting the number of subintervals of S µ low,J , we have

P(N µ J+1 ≥ 3 2 N µ J |N µ J = N ) ≥ 1 -N -1
which leads to the recursive formula Moreover, we can show that, in this case, S µ low ∩ (0, 1) ̸ = ∅. If α > 0, applying this procedure with 1 m < α gives us that any point t ∈ S µ low ∩ (0, 1) is a slow point of the fractional Brownian motion of exponent α.

P(N µ J+1 ≥ ( 3 2 ) J+1 ) ≥ (1 -( 2 3 ) J )P(N µ J ≥ ( 3
From formulas (IV.3.1) and (IV.3.2), we see that this procedure is also useful to bound the random variables appearing in the expansion (IV.3.3) of the generalized Rosenblatt process. But, from the proofs of Propositions IV.3.3 and IV.3.16 we know that this is not sufficient and we also need to give a bound for the random variables Ξ j (λ), for λ ∈ 3λ n (t), n ∈ N and j ≥ n. Such dyadic intervals are precisely the ones in the set Λ n,0 (λ n (t)) and this fact forces us to consider the following modification of the procedure. For all j ∈ N, if l ̸ = 0, the sets S µ j,l remain untouched as well as its associated probability p l (µ) while for l = 0 we set S µ j,0 = {λ ′ ∈ Λ j , λ ′ ⊆ [0, 1] : ∃j ′ ≥ j Ξ j ′ (λ ′ ) > (j ′ -j + 1)µ}, with associated probability (which only depends on µ) p 0 (µ) = P(∃j ′ ≥ j Ξ j ′ (λ) > (j ′ -j + 1)µ).

As Ξ j ′ (λ 1 ) is independent of Ξ j ′ (λ 2 ) as soon as λ 1 ∩ λ 2 = ∅, for all J ∈ N, if N is again the number of dyadic intervals of S µ low,J , the number of such intervals in 2S µ low,J ∩ S µ J+1,0 is still a binomial random variable of parameter 2N and p 0 (µ). Therefore if µ is large enough, using Theorems IV.3.11 and IV.3.12, one can still affirm

2N +∞ l=0 (2 ml+1 + 1)(p l (µ) + (l + 1) p l (µ)(1 -p l (µ))) ≤ N 2
and the end of the procedure is saved: equality (IV. In order to use notations (IV.3.57) and (IV.3.58), here after λ 1 (resp. λ 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 ψ λ 1 (resp. ψ λ 2 ) will be the associated antiderivative of wavelet ψ H 1 (2

j 1 • -k 1 ) (resp. ψ H 2 (2 j 2 • -k 2 )) and I λ 1 ,λ 2 [t, s] will stand for I k 1 ,k 2 j 1 ,j 2 [t, s].
Finally, ε λ 1 ,λ 2 will stand for ε k 1 ,k 2 j 1 ,j 2 . If t ∈ (0, 1), let (y λ (t)) λΛ be the sequence defined by

y λ (t) = 2 l if λ ∈ Λ l j (t).
Note that, if we apply the preceding procedure, we find Ω slo an event of probability 1 such that, for all ω ∈ Ω slo , there exists µ for which S µ low ∩ (0, 1) ̸ = ∅. Then, if t belong to this set, we have, thanks to inequality (IV.3.59) and equalities (IV.3.1) and (IV.3.2)

|ε λ 1 ,λ 2 (ω)| ≤ Cµ 2 y λ 1 (t)y λ 2 (t), (IV.3.62)
for a deterministic constant C > 0. Again, we need to adapt the Lemmata from previous sections with this alternative upper bound.

Lemma IV.3.27. There exists a deterministic constant C > 0 such that, for all t, s ∈ (0, 1) we have

0≤j 1 <n 0≤j 2 <n λ 1 ∈Λ j 1 ,λ 2 ∈Λ j 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) y λ 1 (t)y λ 2 (t) |I λ 1 ,λ 2 [t, s]| ≤ C|t -s| H 1 +H 2 -1 .
Proof. If ξ ∈ [s, t] and λ ∈ λ l j (t), for 0 ≤ j < n and l ≥ 1,

|2 j ξ -s(λ)| ≥ |s(λ(t)) -s(λ)| -2 > 2 m(l-1) -2
and so, using the fast decay of the fractional antiderivatives of ψ (IV.3.8) and the definition of (y λ ) λ∈Λ , we get for 0 ≤ j 1 , j 2 < n

λ 1 ∈Λ j 1 ,λ 2 ∈Λ j 2 y λ 1 (t)y λ 2 (t)|ψ λ 1 (ξ)ψ λ 2 (ξ)| = (l 1 ,l 2 )∈N 2 0 λ 1 Λ l 1 j 1 (t) λ 2 Λ l 2 j 2 (t) y λ 1 (t)y λ 2 (t)|ψ λ 1 (ξ)ψ λ 2 (ξ)| ≤ C (l 1 ,l 2 )∈N 0 λ 1 Λ l 1 j 1 (t) λ 2 Λ l 2 j 2 (t) 2 l 1 +l 2 (3 + |2 j 1 ξ -k 1 |) 4 (3 + |2 j 2 ξ -k 2 |) 4 ≤ C (l 1 ,l 2 )∈N 0 λ 1 Λ l 1 j 1 (t) λ 2 Λ l 2 j 2 (t) 2 l 1 +l 2 2 -m(l 1 +l 2 ) (3 + |2 j 1 ξ -k 1 |) 3 (3 + |2 j 2 ξ -k 2 |) 3 ≤ C k 1 ∈Z 1 (3 + |2 j 1 ξ -k 1 |) 3 k 2 ∈Z 1 (3 + |2 j 2 ξ -k 2 |) 3 ≤ C.
(IV. In what follows, we use these notations instead of the one given in Definition IV.3.6:

Λ < j 2 (t, s) = {λ 2 ∈ Λ j 2 : s(λ 2 ) ∈ Z < j 2 (t, s)}, Λ > j 2 (t, s) = {λ 2 ∈ Λ j 2 : s(λ 2 ) ∈ Z > j 2 (t, s)}, Λ j 2 [t, s] = {λ 2 ∈ Λ j 2 : s(λ 2 ) ∈ Z j 2 [t, s]}.
Lemma IV.3.28. There exists a deterministic constant C > 0 such that, for all t, s ∈ (0, 1) and 0 ≤ j 1 < n ≤ j 2 , the quantities

λ 1 ∈Λ j 1 λ 2 ∈Λ < j 2 (t,s) y λ 1 (t)y λ 2 (t) |I λ 1 ,λ 2 [t, s]| (IV.3.64) λ 1 ∈Λ j 1 λ 2 ∈Λ > j 2 (t,s) y λ 1 (t)y λ 2 (t) |I λ 1 ,λ 2 [t, s]| (IV.3.65) are bounded by C2 1 m (j 2 -n) 2 -j 2 .
Proof. Again, we prove the bound for (IV.3.64), the reasoning for (IV.3.65) being similar. Let us remark that, if j 2 ≥ n x ∈ [s, t] and λ j 2 (x) ∈ Λ l j 2 (t) then, the construction and the definition of (y λ (t)) λ∈Λ gives that 1) .

l ≤ 1 m (j 2 -n), as |s -t| ≤ 2 -n , if λ ∈ Λ l 2 j 2 (x) then |y λ | ≤ 2 l 2 2 l+1 µ while, by definition, if l 2 ≥ 1 3 + |2 j 2 x -s(λ)| ≥ 2 + 2 m(l 2 -
Therefore, if we set

D l j 2 (t) = λ∈Λ l j 2 (t) λ,
we have

(IV.3.64) ≤ λ 1 ∈Λ j 1 λ 2 ∈Λ < j 2 (t,s) y λ 1 (t)y λ 2 (t) [s,t] |ψ λ 1 (x)ψ λ 2 (x)| dx ≤ 0≤l≤ 1 m (j 2 -n) λ 1 ∈Λ j 1 λ 2 ∈Λ < j 2 (t,s) y λ 1 (t)y λ 2 (t) D l j 2 (t) |ψ λ 1 (x)ψ λ 2 (x)| dx.
(IV.3.66)

But, for all x ∈ D l j 2 , using the same method as in (IV.3.63), but splitting the sums according to the set Λ l 1 j 1 (x) and Λ l 2 j 2 (x) on which y λ 1 (t)y λ 2 (t) ≤ 2 l+l 1 +l 2 +1 we get

λ 1 ∈Λ j 1 λ 2 ∈Λ < j 2 (t,s) |ε λ 1 ,λ 2 ||ψ λ 1 (x)ψ λ 2 (x)| ≤ C2 l+1 λ 1 ∈Λ j 1 1 (3 + |2 j 1 x -k 1 |) 3 λ 2 ∈Λ < j 2 (t,s) 1 (3 + |2 j 2 x -k 2 |) 3 ≤ C2 l+1 λ 2 ∈Λ < j 2 (t,s) 1 (3 + |2 j 2 x -k 2 |) 3 .
(IV. 3.67) Finally, using the techniques in (IV.3.15), we get

(IV.3.64) ≤ C2 1 m (j 2 -n) 2 -j 2 .
Lemma IV.3.29. There exists a deterministic constant C > 0 such that, for all t, s ∈ (0, 1) and 0 ≤ j 1 < n ≤ j 2 , the quantities

λ 1 ∈Λ j 1 λ 2 ∈Λ j 2 [t,s] y λ 1 (t)y λ 2 (t) min{s,t} -∞ ψ H 1 (2 j 1 x -k 1 )ψ H 2 (2 j 2 x -k 2 ) dx (IV.3.68) λ 1 ∈Λ j 1 λ 2 ∈Λ j 2 [t,s] y λ 1 (t)y λ 2 (t) +∞ max{s,t} ψ H 1 (2 j 1 x -k 1 )ψ H 2 (2 j 2 x -k 2 ) dx (IV.3.69) are bounded by C2 1 m (j 2 -n) 2 -j 2 .
Proof. Again, we assume s ≤ t. If x ∈ (-∞, s] is such that λ j 1 (x) ∈ Λ l j 1 (s), we have, for all λ 1 ∈ Λ l 1 j 1 (x) and λ 2 ∈ Λ j 2 [t, s] ∩ Λ l 2 j 2 (s) (with j 1 < n ≤ j 2 ), 1) .

y λ 1 (t)y λ 2 (t) (3 + |2 j 1 x -k 1 |) 4 (3 + |2 j 2 x -k 2 |) 4 ≤ C 2 1 m (j 2 -n)+l+l 1 +l 2 +1 µ 2 (3 + |2 j 1 x -k 1 |) 4 (3 + |2 j 2 x -k 2 |) 5 ≤ C 2 1 m (j 2 -n)+l+1 (3 + |2 j 1 x -k 1 |) 3 (3 + |2 j 2 x -k 2 |) 4 ≤ C 2 1 m (j 2 -n) (3 + |2 j 1 x -k 1 |) 3 (3 + |2 j 2 x -k 2 |) 3 (IV.3.70) because 3 + |2 j 2 x -k 2 | = 3 + k 2 -2 j 2 x ≥ 2 + 2 j 1 (s -x) ≥ 2 m(l-
Thus we get, using the fast decay of the fractional antiderivatives of the wavelet before splitting the integral over (-∞, s] into the integral over the sets (-∞, s] ∩ D l j 1 (s), in the same way as in (IV.3.66), using (IV.3.70) and finally the boundedness of the function (IV.3.44) for M = 3 and inequality (IV.3.17)

λ 1 ∈Λ j 1 λ 2 ∈Λ j 2 [t,s] s -∞ |ψ λ 1 (x)ψ λ 2 (x)| dx ≤ C2 1 m (j 2 -n) s -∞ λ 1 ∈Λ j 1 λ 2 ∈Λ j 2 [t,s] dx (3 + |2 j 1 x -k 1 |) 3 (3 + |2 j 2 x -k 2 |) 3 ≤ C2 1 m (j 2 -n) s -∞ λ 2 ∈Λ j 2 [t,s] dx (3 + |2 j 2 x -k 2 |) 3 ≤ C2 1 m (j 2 -n) 2 -j 2 .
In the same way we get

λ 1 ∈Λ j 1 λ 2 ∈Λ j 2 [t,s] y λ 1 (t)y λ 2 (t) +∞ max{t,s} ψ H 1 (2 j 1 x -k 1 )ψ H 2 (2 j 2 x -k 2 ) dx ≤ C2 -j 2 . (IV.3.71)
Lemma IV.3.30. There exists a deterministic constant C > 0 such that, for all t, s ∈ (0, 1), the quantities

0≤j 1 <n j 2 ≥n λ 1 ∈Λ j 1 ,λ 2 ∈Λ j 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) y λ 1 (t)y λ 2 (t)|I λ 1 ,λ 2 [t, s]| j 1 ≥n 0≤j 2 <n λ 1 ∈Λ j 1 ,λ 2 ∈Λ j 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) y λ 1 (t)y λ 2 (t)|I λ 1 ,λ 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 IV.3.9 excepted that we use Lemmata IV.3.28 and IV.3.29 instead of Lemmata IV.3.7 and IV.3.8 respectively. It leads on one side us to consider the sums n-1

j 1 =0 2 j 1 (1-H 1 ) +∞ j 2 =n 2 1 m (j 2 -n) 2 -j 2 H 2 + 2 n(1-H 1 -H 2 )
which are bounded by

C2 n(1-H 1 -H 2 ) ≤ C|t -s| H 1 +H 2 -1 because 1 m < H 2 .
On the other side, if we write I λ 1 ,λ 2 for I k 1 ,k 2 j 1 ,j 2 in Lemma IV.2.1, we have, from it,

λ 1 ∈Λ n-1 λ 2 ∈Λn[t,s] y λ 1 (t)y λ 2 (t)I λ 1 ,λ 2 ≤ C2 -n +∞ l 1 =0 λ 1 ∈Λ l 1 n-1 (t) λ 2 ∈Λn[t,s] 2 l 1 (3 + |2k 1 -k 2 |) 4 ≤ C2 -n λ 1 ∈Λ n-1 λ 2 ∈Λn[t,s] 1 (3 + |2k 1 -k 2 |) 3 ≤ C2 -n . (IV.3.72)
Lemma IV.3.31. There exists a deterministic constant C > 0 such that, for all t, s ∈ (0, 1) and n ≤ j 1 ≤ j 2 the quantities (IV.3.64) and (IV.3.65) are bounded by

C2 1 m (j 1 -n) 2 1 m (j 2 -n) 2 -j 2 .
Proof. The proof is essentially the same as for Lemma IV.3.28 excepted that, now, as

n ≤ j 1 ≤ j 2 , we remark that if x ∈ D l j 2 (t) for a 0 ≤ l ≤ 1 m (j 2 -n) then x ∈ D l ′ j 1 (t) for a 0 ≤ l ′ ≤ 1 m (j 1 -n). Lemma IV.3.32.
There exists a deterministic constant C > 0 such that, for all t, s ∈ (0, 1) and n ≤ j 1 ≤ j 2 the quantities (IV.3.68) and (IV.3.69) are bounded by

C2 1 m (j 1 -n) 2 1 m (j 2 -n) 2 -j 2 .
Proof. The proof is essentially the same as for Lemma IV.3.29 and the only modification is the same as in the proof of Lemma IV.3.31. This time, the bound for the random variables Ξ j (λ) is already considered in the construction and we can directly go to the proof of the main Proposition of this subsection.

Proof of Proposition IV.3.26. 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 Ω slo of probability 1 such that, for all ω ∈ Ω slo , there is µ ∈ N for which S µ low ∩ (0, 1) ̸ = ∅. Then, if ω ∈ Ω slo and t ∈ S µ low (ω) ∩ (0, 1) and s ∈ (0, 1), we write

|R ′ H 1 ,H 2 (t, ω) -R ′ H 1 ,H 2 (s, ω)| ≤ 0≤j 1 <n 0≤j 2 <n λ 1 ∈Λ j 1 ,λ 2 ∈Λ j 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) ε λ 1 ,λ 2 (ω)I λ 1 ,λ 2 [t, s] + 0≤j 1 <n j 2 ≥n λ 1 ∈Λ j 1 ,λ 2 ∈Λ j 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) ε λ 1 ,λ 2 (ω)I λ 1 ,λ 2 [t, s] + j 1 ≥n 0≤j 2 <n λ 1 ∈Λ j 1 ,λ 2 ∈Λ j 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) ε k 1 ,k 2 j 1 ,j 2 (ω)I λ 1 ,λ 2 [t, s] + j 1 ≥n j 2 ≥n λ 1 ∈Λ j 1 ,λ 2 ∈Λ j 2 2 j 1 (1-H 1 ) 2 j 2 (1-H 2 ) ε λ 1 ,λ 2 (ω)I λ 1 ,λ 2 [t, s] .
λ 1 ∈Λ < j (t,s) λ 2 ∈Λ j [t,s] ε λ 1 ,λ 2 (ω)I λ 1 ,λ 2 ≤ C2 -j 2 2 m (j-n) µ 2 λ 1 ∈Λ < j (t,s) λ 2 ∈Λ j [t,s] 1 (3 + |2k 1 -k 2 |) 3 ≤ C2 -j 2 2 m (j-n) µ 2 .
The same bound holds when we consider the sums over λ 1 ∈ Λ > j (t, s) or λ 2 ∈ Λ j+1 [t, s], i.e. for(IV.3.38) and (IV. 3.39). Finally the construction and especially (IV.3.61) insures us that

λ 1 ∈Λ j [t,s] λ 2 ∈Λ j [t,s] ε λ 1 ,λ 2 (ω)I λ 1 ,λ 2 ≤ C(j -n + 1)2 -j-n 2 µ.
Therefore, the last term in (IV.3.73) is bounded from above by Cµ 2 j 1 ≥n

2 j 1 (1-H 1 ) 2 1 m (j 1 -n) j 2 ≥j 1 2 -j 2 H 2 2 1 m (j 2 -n) + j≥n 2 j( 3 2 -H 1 -H 2 ) (j -n + 1)2 -n 2 ≤ Cµ 2 j 1 ≥n 2 j 1 (1-H 1 -H 2 ) 2 2 m (j 1 -n) + 2 n( 3 2 -H 1 -H 2 ) 2 -n 2 ≤ Cµ 2 2 n(1-H 1 -H 2 ) ≤ Cµ 2 |t -s| H 1 +H 2 -1
and thus inequality (IV.3.56) holds.

IV.4 Lower bounds for wavelet leaders

In this section, we show that the limits (IV.1.5) and (IV.1.6) are strictly positive. In [START_REF] Ayache | Different possible behaviors of wavelet leaders of the Brownian motion[END_REF], 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 context 3 [START_REF] Ayache | Lower bound for local oscillations of Hermite processes[END_REF], 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 IV.4.5 below. All along this section, in order to ease notations we set

C H 1 ,H 2 := 1 Γ H 1 -1 2 Γ H 2 -1 2 and for s, x 1 , x 2 ∈ R f H 1 ,H 2 (s, x 1 , x 2 ) = (s -x 1 ) H 1 -3/2 + (s -x 2 ) H 2 -3/2 +
Let Ψ be a wavelet with compact support included in [-N, N ]. Using formula (IV.2.5) at t = k/2 j , the wavelet coefficient c j,k of the generalized Rosenblatt process is given by

c j,k = N -N R H 1 ,H 2 x + k 2 j -R H 1 ,H 2 k 2 j Ψ(x)dx = c H 1 ,H 2 N -N Ψ(x) ′ R 2 x+k 2 j k 2 j f H 1 ,H 2 (s, x 1 , x 2 ) ds dB(x 1 ) dB(x 2 ) dx = c H 1 ,H 2 ′ R 2 N -N Ψ(x) x+k 2 j k 2 j f H 1 ,H 2 (s, x 1 , x 2 ) ds dx dB(x 1 ) dB(x 2 ) = c H 1 ,H 2 ′ A N -N Ψ(x) x+k 2 j k 2 j f H 1 ,H 2 (s, x 1 , x 2 ) ds dx dB(x 1 ) dB(x 2 )
3 In [6], Ayache does not consider wavelets at all but directly work on Wiener-Itô integrals

where A := -∞, k+N 2 j 2 , because, as soon as x ∈ [-N, N ] and s ∈ [k2 -J , (k + N )2 -j ],

f (s, x 1 , x 2 ) vanishes for all x 1 , x 2 outside of A.

Definition IV.4.1. Given an integer M ≥ 0, c j,k can be written as following

c j,k = c j,k M + Cc j,k M where c j,k M = c H 1 ,H 2 ′ λ M j,k N -N Ψ(x) x+k 2 j k 2 j f H 1 ,H 2 (s, x 1 , x 2 ) ds dx dB(x 1 ) dB(x 2 ) (IV.4.1) with λ M j,k := k -N M 2 j , k + N 2 j 2 and Cc j,k M = c H 1 ,H 2 ′ A\λ M j,k N -N Ψ(x) x+k 2 j k 2 j f H 1 ,H 2 (s, x 1 , x 2 ) ds dx dB(x 1 ) dB(x 2 ).
Remark 18. Let us highlight the fact that using time change of variable for Wiener-Itô integrals [START_REF]Stochastic differential equations: an introduction with applications[END_REF]Theorem 8.5.7], for all j, k, we have c j,k M is equal in law to the random variable

c H 1 ,H 2 2 -j(H 1 +H 2 -1) ′ I M N -N ψ(x) x 0 f H 1 ,H 2 (s, x 1 , x 2 ) dsdx dB(x 1 )dB(x 2 )
with I M = (-M N, N ] 2 , while Cc j,k M is equal in law to the random variable

c H 1 ,H 2 2 -j(H 1 +H 2 -1) ′ I ′ M N -N ψ(x) x 0 f H 1 ,H 2 (s, x 1 , x 2 ) dsdx dB(x 1 )dB(x 2 )
with

I ′ M = (-∞, N ] 2 \ (-M N, N ] 2 .
Definition IV.4.2. For all (j, k) ∈ N × Z and M ∈ N we define the random variables

ε j,k M := c j,k M 2 -j(H 1 +H 2 -1) and Cε j,k M := Cc j,k M 2 -j(H 1 +H 2 -1) .
Remark 19. Note that ε j,k M and ε j ′ ,k ′ M are independent when

λ M j,k ∩ λ M j ′ ,k ′ = ∅. (IV.4.2)
Indeed, if (f j ) j is a sequence of real-valued step functions on R 2 \{(x, x) : x ∈ R} which converge to the integrand with respect to dB(x 1 )dB(x 2 ) in (IV.4.1) then ′ R 2 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 ∈ λ M j,k . Thus ε j,k M is measurable with respect to the σ-algebra generated by these increments

σ M j,k := σ {B(t 2 ) -B(t 1 ) : t 1 , t 2 ∈ λ M j,k } .
Using the independence of the increments of the Brownian motion, one concludes that σ M j,k and σ M j ′ ,k ′ are independent as soon as condition (IV.4.2) is met and so the same holds for ε j,k M and ε j,k ′ M . Moreover, ε j 1 ,k 1 M , . . . , ε jn,kn M are independent when the following condition is satisfied

λ M j i ,k i ∩ λ M j l ,k l = ∅ for all 1 ≤ i < l ≤ n. (IV.4.3)
This leads to defining the following condition.

Definition IV.4.3. Let n ≥ 2. We say λ j 1 ,k 1 , . . . , λ jn,kn satisfy condition (C M ) if (IV.4.3) is satisfied.

From Remark 18, we know that ( ε j,k M ) λ∈Λ is a family of identically distributed second order Wiener chaos random variables. Moreover, ε j 1 ,k 1 M , . . . , ε jn,kn M are independent as soon as λ j 1 ,k 1 , . . . , λ jn,kn satisfies (C M ). The following proposition provides a lower bound (independent of M ) for the tail behavior of the random variable ε j,k M .

Proposition IV.4.4. Let M ∈ N and y ∈ R + . If M and y are large enough, then the exists a deterministic constant c 2 > 0 (independent of M ) such that

P | ε j,k M | > y ≥ exp (-c 2 y) (IV.4.4)
for all (j, k) ∈ N × Z Proof. Fix y ∈ R + (large enough). Our aim is to prove the existence of lower bound for P | ε λ M | > y which is independent of M . To this end, we start by proving the following lemma Lemma IV.4.5. There exist three strictly positive deterministic constants

C Ψ,H 1 ,H 2 , C ′ Ψ,H 1 ,H 2 and C * Ψ,H 1 ,H 2 such that for all (j, k) ∈ N × Z and M ≥ 2 one has C Ψ,H 1 ,H 2 2 -j(H 1 +H 2 -1) ≤ c j,k M 2 ≤ C ′ Ψ,H 1 ,H 2 2 -j(H 1 +H 2 -1) Cc j,k M 2 ≤ C * Ψ,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 ≥ H 2 . We define the functions

Φ 1 : (x 1 , x 2 ) → N -N Ψ(x) x 0 f H 1 ,H 2 (s, x 1 , x 2 )ds dx, Φ 2 : (x 1 , x 2 ) → N -N Ψ(x) x 0 f H 1 ,H 2 (s, x 2 , x 1 )ds dx,
and the symmetric function

4 Φ = 1 2 (Φ 1 + Φ 2 ) .
By Remark 18 we have, using the "Wiener isometry"5 [113, Section 5],

c j,k M 2 = √ 2c H 1 ,H 2 2 -j(H 1 +H 2 -1) ∥Φ∥ L 2 (I M )
and thus it suffices to take

C Ψ,H 1 ,H 2 := √ 2c H 1 ,H 2 ∥Φ∥ L 2 ([-N,N ] 2 ) C ′ Ψ,H 1 ,H 2 := √ 2c H 1 ,H 2 ∥Φ∥ L 2 ((-∞,N ] 2 )
Now, still using Remark 18 and "Wiener isometry" we have

Cc j,k M 2 = √ 2c H 1 ,H 2 2 -j(H 1 +H 2 -1) ∥Φ∥ L 2 (I ′ M ) ≤ √ 2c H 1 ,H 2 2 -j(H 1 +H 2 -1) ∥Φ 1 ∥ L 2 (I ′ M )
. Also as

I ′ M = (-∞, N ] 2 \ (-M N, N ] 2 ⊂ R × (-∞, -M N ] (-∞, -M N ] × R, we write ∥Φ 1 ∥ 2 L 2 (I ′ M ) = I ′ M N -N Ψ(x) x 0 (s -x 1 ) H 1 -3 2 + (s -x 2 ) H 2 -3 2 + ds dx 2 dx 1 dx 2 ≤ I ′ M N -N |Ψ(x)| [0,x]
(s -x 1 )

H 1 -3 2 + (s -x 2 ) H 2 -3 2 + ds dx 2 dx 1 dx 2 ≤ R -M N -∞ N -N |Ψ(x)| [0,x] (s -x 1 ) H 1 -3 2 + (s -x 2 ) H 2 -3 2 + ds dx 2 dx 1 dx 2 + R -M N -∞ N -N |Ψ(x)| [0,x] (s -x 1 ) H 1 -3 2 + (s -x 2 ) H 2 -3 2 + ds dx 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 → y H 1 -3/2 is decreasing, one gets

R -M N -∞ N -N |Ψ(x)| [0,x] (s -x 1 ) H 1 -3 2 + (s -x 2 ) H 2 -3 2 + ds dx 2 dx 1 dx 2 ≤ -M N -∞ (-N -x 1 ) 2H 1 -3 dx 1 × R N -N |Ψ(x)| [0,x] (s -x 2 ) H 2 -3/2 + ds dx 2 dx 2 .
Proof. Let us fix t ∈ (0, 1) and j ∈ N. For any λ ∈ S j,k j (t),m , there exists a unique decreasing finite sequence (I n ) 0≤n≤m of decreasing dyadic intervals in the sense of inclusion such that I 0 = λ j,k j (t) , I m = λ and I n ∈ S j,k j (t),n . Then, define the sequence (T n ) 1≤n≤m of unique dyadic intervals such that I n-1 = I n ∪ T n . Note that for all 1 ≤ n ≤ m, T n ∈ 3I n . Moreover, as (I n ) 0≤n≤m is decreasing, (T n ) 1≤n≤m are pairwisely disjoint. Furthermore, for every n ∈ {1, ..., m}, there exist T ′ n = λ jn,kn ∈ S Tn,⌊log 2 N M ⌋+2 such that

k n -N M 2 jn , k n + N 2 jn ⊂ T n .
As a consequence, the associated random variables ε

T ′ n M 1≤n≤m
are independent as the dyadic intervals (T ′ n ) 1≤n≤m satisfies condition (C M ) in Definition IV.4.3. Next, for a constant C > 0 to be chosen later, we set

E j,m (t) = ω ∈ Ω : max 1≤n≤m ε T ′ n M ≥ C log(2m) .
Note that, as the random variables ε T ′ n M 1≤n≤m are independent,

P (E j,m (t)) = 1 - m n=1 P ε T ′ n M < C log(2m)
Recalling (IV.4.4), and the fact that log(1 -x) ≤ -x if x ∈ (0, 1), one gets, for m is large enough,

P (E j,m (t)) ≥ 1 -(1 -exp(-Cc 2 log(2m)) m = 1 -1 - 1 2m Cc 2 m ≥ 1 -exp m (2m) Cc 2 = 1 -exp m 1-Cc 2 2 Cc 2 .
Finally, choosing C such that 0 < Cc 2 < 1, one obtain that p∈N P (E 2 p ,2 p (t)) = +∞.

Knowing that the events E 2 p ,2 p (t) are independent for all p ∈ N, one concludes using Borel-Cantelli Lemma that

P lim sup m→+∞ E 2 m ,2 m (t) = 1
It follows that for a fixed t ∈ R, almost surely, there are infinitely many j ∈ N such that max

λ ′ ∈ S λ,⌊log 2 N M ⌋+2 λ ∈ 3λ j (t) ε λ ′ M (ω) ≥ C log j.
Concerning the "non-independent part" of the wavelet coefficients, one can state the following Lemma.

Lemma IV.4.8. There is a deterministic constant C ′ > 0 such that, for all M ∈ N and for all t ∈ (0, 1), there exists Ω t,2 ⊂ Ω with probability 1 such that for all ω ∈ Ω t,2 there exists J ∈ N such that, for all j ≥ J,

max λ ′ ∈ S λ,⌊log 2 (N M )⌋+2 λ ∈ 3λ j (t) Cε λ ′ M (ω) ≤ C ′ M max{H 1 ,H 2 }-1 log j.
Proof. Let us fix t ∈ (0, 1). For any C ′ > 0, for all j sufficiently large and λ ∈ 3λ j (t), we have, by Theorem IV.3.11,

P ∃λ ′ ∈ S λ,⌊log 2 N M ⌋+2 : Cε λ ′ M ≥ C ′ M max{H 1 ,H 2 }-1 log j ≤ λ ′ ∈S λ,⌊log 2 N M ⌋+2 P Cε λ ′ M ) ≥ C ′ M max{H 1 ,H 2 }-1 log j ≤ λ ′ ∈S λ,⌊log 2 N M ⌋+2 P Cε λ ′ M ≥ C ′ (C * Ψ,H 1 ,H 2 ) -1 ∥ Cε λ ′ M ∥ L 2 (Ω) log j ≤ 4N M exp(- ⋆ CC ′ (C * Ψ,H 1 ,H 2 ) -1 log j) Thus, for C ′ > C * Ψ,H
C -C ′ M max{H 1 ,H 2 }-1 > 0.
Let us fix t ∈ (0, 1) and consider ω ∈ Ω t,1 ∩ Ω t,2 , where the events, of probability 1, Ω t,1 and Ω t,2 are given by the same Lemmata. For all J ∈ N, by Lemma IV.4.7, there exist j ≥ J and λ ′ (j) ⊆ 3λ j (t) of scale j

′ = j + ⌊log N M ⌋ + 2 such that c λ ′ (j) M (ω) ≥ C2 -j ′ (H 1 +H 2 -1) log j.
If J is large enough, we also have, for all such j ≥ J, by Lemma IV.4.8,

Cc λ ′ (j) M (ω) ≤ C ′ M max{H 1 ,H 2 }-1 2 -j ′ (H 1 +H 2 -1) log j.
From this we deduce that

d j (t, ω) ≥ c λ ′ (j) (ω) ≥ c λ ′ (j) M (ω) -Cc λ ′ (j) M (ω) ≥2 -j ′ (H 1 +H 2 -1) log j C -C ′ M max{H 1 ,H 2 }-1 ≥2 -j(H 1 +H 2 -1) (4N M ) 1-H 1 -H 2 log j C -C ′ M max{H 1 ,H 2 }-1
Therefore, (IV.4.6) holds true for all t ∈ (0, 1) and ω ∈ Ω t,1 ∩ Ω t,2 . The conclusion follows then from Fubini Theorem.

IV.4.2 Rapid Points

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

Proposition IV.4.9. There exists Ω * 2 ⊂ Ω with probability 1 such that, for all ω ∈ Ω * 2 , there exist t ∈ (0, 1) such that

lim sup j→+∞ d j (t, ω) 2 -j(H 1 +H 2 -1) j > 0. (IV.4.7)
As in the previous subsection, we start by working with the random variables ε λ M .

Lemma IV.4.10. There exists a deterministic constant C > 0 such that for all M there is Ω 2 ⊂ Ω with probability 1 such that for all ω ∈ Ω 2 there exist t ∈ (0, 1) such that

lim sup j→+∞ ε λ j (t) M (ω) j ≥ C. (IV.4.8)
Proof. Let us fix a ∈ (0, 1) and C > 0 to be chosen later on. For every (j, l) ∈ N × 0, . . . , ⌊2 j(1-a) ⌋ -1 , we set S M j,l = l⌊2 aj /(2N M )⌋, . . . , (l + 1)⌊2 aj /(2N M )⌋ -1 and consider the event

E M j,l = ω ∈ Ω : max k∈S M j,l ε j,2kN M M (ω) ≥ Cj
Let j 0 be the smallest integer such that ⌊2 aj /(2N M )⌋ ≥ 1. If we assume that

Ω * 2 = J≥j 0 j≥J l∈{0,...,⌊2 j(1-a) ⌋-1} E M j,l (IV.4.9)
is an event of probability 1 and we consider ω ∈ Ω * 2 . For every j ≥ j 0 , denote by

G M j (ω) := k ∈ {0, . . . , 2 j -1} : ε j,k M (ω) ≥ Cj . (IV.4.10)
Moreover, for every n ≥ j 0 , one considers

O M n (ω) := j≥n U M j (ω), where U M j (ω) := k∈G M j (ω) k 2 j , k + 1 2 j . (IV.4.11)
If one proves that O M n (ω) is dense in (0, 1), then by Baire's theorem the set ∩ n≥j 0 O M n (ω) is non-empty and let t be an element of this set. Then for every n ≥ j 0 , there is j ≥ n such that ε λ j (t) M (ω) ≥ Cj, and so desired statement (IV.4.8) is true.

We still have to prove two points:

1. O M n (ω) is dense in (0, 1).

Ω *

2 is an event of probability 1.

Indeed, starting with statement 1, consider t ∈ (0, 1), j ≥ j 0 and k such that λ j (t) = λ j,k . Then, we have two cases:

Case 1 : There is l ∈ {0, . . . , ⌊2 j(1-a) ⌋ -1} such that k ∈ l⌊2 aj ⌋, . . . , (l + 1)⌊2 aj ⌋ -1 Using (IV.4.9) and (IV.4.10), there is k ′ ∈ {l⌊2 aj /(2N M )⌋, . . . , (l +1)⌊2 aj /(2N M )⌋-1} such that 2k ′ N M ∈ G j (ω). Then, by (IV.4.11),

2N M k ′ 2 j , 2N M k ′ + 1 2 j ⊂ O M n (ω).
which is at is at most 2 -j (⌊2 aj ⌋ + 2N M ⌊2 aj /(2N M )⌋) from t. Finally, we get that t is at a distance at most 22 j(a-1) of U M j (ω).

Case 2 : k ∈ {⌊2 j(1-a) ⌋⌊2 ja ⌋, . . . , 2 j -1}. Again by (IV.4.9) and (IV.4.10), there is

k ′ ∈ S M j,l such that 2k ′ N M ∈ G M j (ω)
, and similarly, we get that t is at a distance at most c2 j(a-1) of U M j (ω), 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 Ω * 2 has a probability 1, it is enough to prove that

P    C    l∈{0,...,⌊2 j(1-a) ⌋-1} E M j,l       (IV.4.12)
is the general term of a convergent series, then the result follows by Borel-Cantelli Lemma. Note that the variables ε j,2N M k M , k ∈ S M j,l and l ∈ {0, . . . , ⌊2 j(1-a) ⌋ -1}, are independent because for every k

̸ = k ′ , |2N M k -2N M k ′ | ≥ 2N M and so λ M j,2M N k ∩ λ M j,2M N k ′ = ∅. Consequently, one has P    C    l∈{0,...,⌊2 j(1-a) ⌋-1} E M j,l       =1 -P    l∈{0,...,⌊2 j(1-a) ⌋-1} C E M j,l    =1 - l∈{0,...,⌊2 j(1-a) ⌋-1}   1 - k∈S M j,l P ε j,2N M k M < Cj   =1 -1 -(1 -P (|ε| ≥ Cj)) ⌊2 aj /(2N M )⌋ ⌊2 j(1-a) ⌋ ≤1 -exp 2 j(1-a) log(1 -p j ) (IV.4.13)
where ε is a random variable belonging to the Wiener chaos of order 2 distributed according to the ( ε λ ) λ∈Λ and p j = (1 -P (|ε| ≥ Cj)) ⌊2 aj /(2N M )⌋ . Remark that p j is a positive term that tends to 0 as j → +∞. Indeed, using the fact that log(1 -x) ≤ -x if x ∈ (0, 1) together with (IV.4.4), there exists J ∈ N such that for all j ≥ J,

0 ≤ p j ≤ (1 -exp (-C c 2 j)) ⌊2 aj /(2N M )⌋ ≤ exp - 2 aj 2N M exp (-C c 2 j) ≤ exp -C ′ exp (log 2 aj ) exp (-C c 2 j) ≤ exp (-C ′ exp j(a log 2 -C c 2 )) (IV.4.14)
where C ′ depends only on N , M and a and c 2 is the constant given in (IV.4.4). It is enough to choose C such that a log 2 -C c 2 > 0 to deduce that and so p j → 0 as j → +∞. Similarly, one can get for all j ≥ J 0

≤ 2 j(1-a) p j ≤ exp (-C ′ exp j(log 2 -C c 2 ))
which indeed shows that 2 j(1-a) p j tends to 0 as j → +∞. Now, using the fact that log(1 -x) = -x + o(x) and exp (x) = 1 + x + o(x) as x → 0, together with (IV.4.13) we obtain that for all δ > 0 If we consider ω belonging to this event of probability 1, first, from Proposition IV.3.3 there exists C R > 0 such that, for all t, s ∈ (0, 1)

P    C    l∈{0,...,⌊2 j(1-a) ⌋-1} E M j,l       ≤ 2 j(
|R H 1 ,H 2 (t, ω) -R H 1 ,H 2 (s, ω)| ≤ C R |t -s| H 1 +H 2 -1 log |t -s| -1 (IV.5.1)
while, for almost every t o ∈ (0, 1), from Propositions IV.3.16 and IV.4.6

0 < lim sup s→to |R H 1 ,H 2 (t o , ω) -R H 1 ,H 2 (s, ω)| |t o -s| H 1 +H 2 -1 log log |t o -s| -1 < +∞.
Nevertheless, from Proposition IV.4.9 we also know that there exists t r ∈ (0, 1) such that

0 < lim sup s→tr |R H 1 ,H 2 (t r , ω) -R H 1 ,H 2 (s, ω)| |t r -s| H 1 +H 2 -1 log |t r -s| -1
which, combined with (IV.5.1), gives that, for all such a t r ,

0 < lim sup s→tr |R H 1 ,H 2 (t r , ω) -R H 1 ,H 2 (s, ω)| |t r -s| H 1 +H 2 -1 log |t r -s| -1 < +∞.
Moreover, from Proposition IV.3.26, we also know that one can find t σ ∈ (0, 1) such that lim sup

s→tσ |R H 1 ,H 2 (t σ , ω) -R H 1 ,H 2 (s, ω)| |t σ -s| H 1 +H 2 -1 < +∞.
The conclusion follows by Remark 14.

Remark 20. Unfortunately, our method does not allow us to affirm the positiveness of the limit (IV.1.7), at the opposite of limits (IV.1.5) and (IV.1.6). Indeed, as for almost every

ω ∈ Ω lim sup s→t |R H 1 ,H 2 (t, ω) -R H 1 ,H 2 (s, ω)| |t -s| H 1 +H 2 -1
is finite for some t, we would need to show its positiveness for all t and thus the positiveness of the limit lim sup

j→+∞ d j (t, ω) 2 -j(H 1 +H 2 -1) (IV.5.2)
for all t.

Concerning the random variables ( ε λ M ) λ , one can obtain a positive result6 . Indeed, from [53, Theorem 6.9 and Remark 6.10] we know that there exists an universal deterministic constant γ ∈ [0, 1) such that, for each random variable X in the Wiener chaos of order 2

P |X| ≤ 1 2 ∥X∥ 2 ≤ γ.
ℓ) n ) 1≤ℓ≤ℓ 0 in S Tn,⌊log 2 (ℓ 0 N M )⌋+2 such that, for all 1 ≤ ℓ ≤ ℓ 0 k (ℓ) n -N M 2 j (ℓ) n , k (ℓ) n + N 2 j (ℓ) n ⊆ T n and, if ℓ ̸ = ℓ ′ , T ℓ n ∩ T ℓ ′ n = ∅. (
Therefore, the dyadic intervals (T ℓ n ) 1≤n≤m,1≤ℓ≤ℓ 0 satisfy condition (C M ) in Definition IV.4.3. From this, for all S ∈ S j,k,m we define the Bernouilli random variable

B j,k,m (S) = 1≤n≤m,1≤ℓ≤ℓ 0 1 {| ε T ℓ n M |<2 -1 C Ψ,H 1 ,H 2 }
for which, by Proposition IV.4.5, we have, using the independence of the random variables

( ε T ℓ n M ) 1≤n≤m,1≤ℓ≤ℓ 0 , E[B j,k,m (S)] ≤ γ mℓ 0 . Therefore, if we define the random variable G j,k,m = S∈S j,k,m B j,k,m (S)
then E[G j,k,m ] ≤ (2γ ℓ 0 ) m and it follows from inequality (IV.5.3) and Fatou Lemma that

E lim inf m→+∞ G j,k,m = 0. As a consequence, Ω 1 = j∈N,0≤k<2 j {ω : lim inf m→+∞ G j,k,m (ω) = 0}
is an event of probability 1. Now if ω ∈ Ω 1 and t ∈ (0, 1), we take j ∈ N and k = k j (t) and since, for all m, G j,k j (t),m has values in {0, . . . , 2 m } we conclude that there are infinitely many m for which, for every S ∈ S j,k j (t),m , B j,k,m (S) = 0. Considering such a m and S = λ j+m (t) then we first remark that, for all 1 ≤ n ≤ m, I n = λ j+n (t) and thus T n ∈ 3λ j+n (t). Now, as B j,k,m (λ j+m (t)) = 0, one can find 1 ≤ n ≤ m and 1 ≤ ℓ ≤ ℓ 0 such that

| ε T ℓ n M (ω)| ≥ 2 -1 C Ψ,H 1 ,H 2 .
Thus we have showed that, for all ω ∈ Ω 1 and t ∈ (0, 1) there exist infinitely many j

′ ∈ N such that max λ ′ ∈ S λ,⌊log 2 (ℓ 0 N M )⌋+2 λ ∈ 3λ j ′ (t) ε λ ′ M (ω) ≥ 2 -1 C Ψ,H 1 ,H 2 .
To pass to the wavelet leaders, in the spirit of Propositions IV.4.6 and IV.4.9, we would need to get from Borel-Cantelli Lemma an upper bound of

max λ ′ ∈ S λ,⌊log 2 (ℓ 0 N M )⌋+2 λ ∈ 3λ j ′ (t) Cε λ ′ M (ω)
for all j sufficiently large on an event of probability 1 which does not depend on t. Then, as

P       ∃λ ∈ Λ j , λ ⊆ [0, 1] : max λ ′′ ∈ S λ ′ ,⌊log 2 (ℓ 0 N M )⌋+2 λ ′ ∈ 3λ Cε λ ′ M (ω) ≥ C ′ M max{H 1 ,H 2 }-1 j       ≤ 2 j 4ℓ 0 N M exp(- ⋆ CC ′ (C * Ψ,H 1 ,H 2 ) -1 j), if C ′ > log(2)C * Ψ,H 1 ,H 2 /
⋆ C this probability is the general term of some convergent series and in this case one can affirm the existence of an event Ω ′ 1 of probability 1 such that, for all ω ∈ Ω ′ 1 there exist J ∈ N such that, for all j ≥ J, for all λ

∈ Λ j , λ ⊆ [0, 1], max λ ′′ ∈ S λ ′ ,⌊log 2 (ℓ 0 N M )⌋+2 λ ′ ∈ 3λ Cε λ ′ M (ω) ≤ C ′ M max{H 1 ,H 2 }-1 j.
It seems to be the sharper upper bound that we can hope to find with our constraints and the fact that we don't have any independence property to take advantage of when dealing with the random variables Cε λ M . This is insufficient to consider properly limit (IV.5.2). Nevertheless, if, instead of working with an uniform constant M we make it depends on the scale j by setting

M j = (4C ′ C -1 Ψ,H 1 ,H 2 j) 1 1-max{H 1 ,H 2 } , where C ′ > log(2)C * Ψ,H 1 ,H 2 / ⋆ C is the same constant as in Lemma IV.4.11, λ M j j,k := k -N M j 2 j , k + N 2 j 2 , c j,k M j = c H 1 ,H 2 ′ λ M j j,k N -N Ψ(x) x+k 2 j k 2 j f H 1 ,H 2 (s, x 1 , x 2 ) ds dx dB(x 1 ) dB(x 2 ) and Cc j,k M j = c j,k -c j,k M j
then Proposition IV.4.5 stills holds if we replace M by M j with j sufficiently large and, by directly adapting what precedes one can find on event Ω * 1 of probability 1 such that, for all ω ∈ Ω * 1 and t ∈ (0, 1) there exist infinitely many j ∈ N such that

7 max λ ′ ∈ S λ,⌊log 2 (ℓ 0 N M j )⌋+2 λ ∈ 3λ j (t) ε λ ′ M j (ω) ≥ 2 -1 C Ψ,H 1 ,H 2 .
7 The random variables ε λ ′ Mj and Cε λ ′ Mj are defined in an obvious way.

while there exist J ∈ N such that, for all j ≥ J, for all λ ∈ Λ j , λ ⊆

[0, 1], max λ ′′ ∈ S λ ′ ,⌊log 2 (ℓ 0 N M j )⌋+2 λ ′ ∈ 3λ Cε λ ′ M j (ω) ≤ C ′ (M j ) max{H 1 ,H 2 }-1 j ≤ 4 -1 C Ψ,H 1 ,H 2 .
As a consequence, as in Proposition IV.4.6, for all J ∈ N there exist j ≥ J with

d j (t, ω) ≥ 2 -j(H 1 +H 2 -1) (4C ′ C -1 Ψ,H 1 ,H 2 j) 1-H 1 -H 2 1-max{H 1 ,H 2 } (4ℓ 0 N ) 1-H 1 -H 2 4 -1 C Ψ,H 1 ,H 2
which allows to state that, for all t ∈ (0, 1) and ω ∈ Ω 1 , lim sup j→+∞ d j (t, ω)

2 -j(H 1 +H 2 -1) j 1-H 1 -H 2 1-max{H 1 ,H 2 } > 0
, and thus, for all ω ∈ Ω 1 and for all t ∈ (0, 1),

lim sup s→t |R H 1 ,H 2 (t, ω) -R H 1 ,H 2 (s, ω)| |t -s| H 1 +H 2 -1 (log |t -s| -1 ) 1-H 1 -H 2 1-max{H 1 ,H 2 } > 0.
In particular, we find an almost sure uniform lower modulus of continuity for the generalized Rosenblatt process, similar to the one established in [START_REF] Kerchev | Local times and sample path properties of the Rosenblatt process[END_REF] for the Rosenblatt process. However, we are not able to judge the optimality of this modulus, which seems to be a difficult problem, as already stated in [START_REF] Ayache | Lower bound for local oscillations of Hermite processes[END_REF]Remark 1.2].

An interesting corollary of Remark 20 and Proposition IV.3.3 is the fact that, almost surely, the pointwise Hölder exponent of the generalized Rosenblatt process is everywhere H 1 + H 2 -1 and, in particular, it is nowhere differentiable.

Similarly, one can also take (M

j = (4C ′ C -1 Ψ,H 1 ,H 2 log(j) 1 1-max{H 1 ,H 2
} ) j , where C ′ is this time the same constant that in Lemma IV.4.8 and show, precisely like in this Lemma, that there exists a deterministic constant C ′ > 0 such that, for all t ∈ (0, 1) there exists Ω t,2 ⊂ Ω with probability 1 such that for all ω ∈ Ω t,2 there exist J ∈ N such that, for all j ≥ J,

max λ ′ ∈ S λ,⌊log 2 (ℓ 0 N M j )⌋+2 λ ∈ 3λ j (t) Cε λ ′ M j (ω) ≤ 4 -1 C Ψ,H 1 ,H 2 .
and conclude in the same way that there exists an event of probability 1 such that, for all ω in this event and for almost every t ∈ (0, 1)

lim sup s→t |R H 1 ,H 2 (t, ω) -R H 1 ,H 2 (s, ω)| |t -s| H 1 +H 2 -1 (log log |t -s| -1 ) 1-H 1 -H 2 1-max{H 1 ,H 2 } > 0.

Chapter V Potential methods and projection theorems for macroscopic Hausdorff dimension

The content of this chapter is a copy of the paper entitled "Potential methods and projection theorems for macroscopic Hausdorff dimension", written with "Stéphane Seuret", and to be submitted soon.

V.1 Introduction

Fractal geometry provides a general framework for studying sets possessing either irregular or self-reproducing (deterministic or random, self-similar or self-affine) properties. Most definitions of fractal dimensions of sets included in R d are based on the local properties (also known as microscopic) of the set. Taking into consideration that many statistical physics models are built on discrete spaces, Barlow and Taylor [START_REF] Barlow | Defining fractal subsets of Z d[END_REF][START_REF] Barlow | Fractional dimension of sets in discrete spaces[END_REF] introduced a new notion of dimension to study unbounded "fractal-like" sets on discrete space. This so-called macroscopic Hausdorff dimension (see Definition V.2.2 below) has proved to be useful in quantifying the behavior at infinity of several objects, beyond the transient range of random walks in Z d which was the original motivation of Barlow and Taylor in [START_REF] Barlow | Defining fractal subsets of Z d[END_REF]. Macroscopic Hausdorff dimension is actually defined for every set (not only discrete) in R d [START_REF] Barlow | Defining fractal subsets of Z d[END_REF]. It is a discrete analog of Hausdorff dimension, and the word macroscopic comes from the fact that this dimension ignores the local structure of the sets. At the same time, the macroscopic Hausdorff dimension assesses the asymptotic behavior at infinity of the sets, so it is very relevant when one is interested in the description of infinite objects, how they fill the space "at large scale". The macroscopic Hausdorff dimension was a key tool used by Xiao et Zheng [START_REF] Xiao | Discrete fractal dimensions of the ranges of random walks in Z d associate with random conductances[END_REF] in studying the range of a random walk in random environment. It is related to [START_REF] Khoshnevisan | On the macroscopic fractal geometry of some random sets[END_REF] where Khoshnevisan and Xiao are concerned with the macroscopic geometry of other random sets. In [START_REF] Khoshnevisan | Intermittency and multifractality: a case study via parabolic stochastic PDEs[END_REF], Khoshnevisan, Kim and Xiao found out a multifractal behavior for the macroscopic dimension of tall peaks of solutions to stochastic PDEs. Georgiou et el [START_REF] Georgiou | The dimension of the range of a transient random walk[END_REF] solved Barlow and Taylor question [START_REF] Barlow | Defining fractal subsets of Z d[END_REF]Problem,p. 145] by qualifying the range of an arbitrary transient random walk. The macroscopic Hausdorff dimension was also useful for studying the large scale structure of sojourn sets associated to the Brownian motion [START_REF] Seuret | On sojourn of Brownian motion inside moving boundaries[END_REF], the fractional Brownian motion [START_REF] Nourdin | Sojourn time dimensions of fractional Brownian motion[END_REF][START_REF] Daw | A uniform result for the dimension of fractional Brownian motion level sets[END_REF], and the Rosenblatt process [START_REF] Daw | Fractal dimensions of the Rosenblatt process[END_REF].

In this paper we are interested in building various methods for estimating the macroscopic Hausdorff dimension. Recalling the fact that macroscopic Hausdorff dimension is a discrete analog of the Hausdorff dimension, we start by stating the estimating methods used for the Hausdorff dimension. In most cases, when estimating the Hausdorff dimension of a set F , the difficult part consists in finding a suitable lower bound for dim H (F ). Various methods exist to find lower bounds for the standard Hausdorff dimension, and it is a natural question to ask whether these methods have their counterparts for the macroscopic Hausdorff dimension. The two usual techniques are the mass distribution principle and the potential theoretic method.

The mass distribution principle, see for instance [38, page 67], states that if a set F ⊂ R d and a Borel finite measure µ are such that µ(F ) = 1 and µ(B(x, r)) ≤ Cr s for every x ∈ R d and r > 0, then the s-dimensional Hausdorff measure H s (F ) is larger than µ(F )/C, and so F has at least Hausdorff dimension s.

The potential theoretic method is based on an integral analysis: if for some probability measure µ, µ(F ) = 1 and the integral

(R d ) 2 dµ(x)dµ(y) ∥x -y∥ s 2
is finite, then again F has at least Hausdorff dimension s. In addition to bounding the Hausdorff dimension from below, the potential theoretic method plays a key role in proving the projection theorem.

The first aim of this paper is to establish similar results for the macroscopic Hausdorff dimension. This happens to be very easy for the mass distribution principle, and follows essentially from previous works. It is much more challenging for the potential theoretic method, and a careful analysis is needed.

As an application of the new potential theoretic method, we obtain a Marstrand-like projection theorem, describing the dimension of almost all projections on lines of sets F ∈ R 2 . Dealing with the dimensions of projections of Borel sets is a line of research that has a long history. It started with the investigation by Marstrand [START_REF] Marstrand | Some fundamental geometrical properties of plane sets of fractional dimensions[END_REF] of the projection theorem associated to the Hausdorff dimension. He dealt with orthogonal projections on linear subspaces and proved that for every Borel set

E ⊂ R 2 , dim H (proj V E) = min{dim H E, 1}
for almost every 1-dimensional subspaces V , where proj V denotes the orthogonal projection onto V and dim H E denotes the Hausdorff dimension of E. Afterwards Marstrand's results was proved by Kaufman but using potential theoretic methods [START_REF] Kaufman | On Hausdorff dimension of projections[END_REF]. Subsequently in 1975 Mattila extended these results to Borel sets E ⊂ R n and almost all V in the Grassmannian G(n, m) [START_REF] Mattila | Hausdorff dimension, orthogonal projections and intersections with planes[END_REF]. We prove analog results for the macroscopic Hausdorff dimension, using the potential theory method we developed above. A key ingredient when working with the standard Hausdorff dimension is the existence of s-sets, i.e. sets E ⊂ R d with Hausdorff dimension dim H (E) = s and such that its s-Hausdorff measure H s (E) is finite. We introduce a similar notion for the macroscopic Hausdorff dimension. Definition V.2.3. Let s ≥ 0. A set E ⊂ R d is called a macroscopic s-set when Dim H (E) = s and ν s (E) < +∞.

We prove the existence of macroscopic s-sets. Theorem V.2.4. Let E ⊂ R d be such that ν s (E) = +∞. Then there exists a macroscopic s-set E such that E ⊂ E. This extraction theorem is a key ingredient at various places in our proofs.

V.2.2 Methods to find lower bounds for Dim H (E)

For every set B and every measure µ, µ |B stands for the restriction of µ on B, i.e. µ |B (A) = µ(A ∩ B).

As recalled above, the mass distribution principle is a powerful, albeit simple, tool allowing to find a lower bound of the Hausdorff dimension by considering measures supported on the set, see [38, page 67]. We prove a similar result for the macroscopic Hausdorff dimension Dim H . Proposition V.2.5 (Macroscopic mass distribution principle). Let E be a Borel subset of R d and s > 0. Suppose that there exists a Radon measure µ on R d such that µ(E) = +∞ and a constant c > 0 such that for all n ∈ N , x ∈ S n and 1 ≤ r ≤ The proof of the macroscopic mass distribution principle is not complicated. Although it was not exactly stated before as we write it, it essentially follows directly from previous results, and so it is not so innovative. This is not the case for the potential method below. Let us first introduce the macroscopic s-energy of a measure. Definition V.2.6. Let s ≥ 0, and let µ be a finite mass distribution on R d . The macroscopic (µ, s)-potential at a point x is defined as

ϕ s µ (x) := R d dµ(y) ∥x -y∥ s 2 ∨ 1 . (V.2.4)
The macroscopic s-energy of µ is

I s (µ) := R d ϕ s µ (x)dµ(x) = (R d ) 2
dµ(x)dµ(y) ∥x -y∥ s 2 ∨ 1 .

(V.2.5)

In the case of standard Hausdorff dimension, in the integrals (V.2.4) and (V.2.5), the quantity ∥x -y∥ s 2 ∨ 1 is simply ∥x -y∥ s 2 . This modification is justified by the fact that Dim H is not concerned with local behavior, so we are not interested in small interactions ∥x -y∥ 2 < 1.

Theorem V.2.7. Let E be a subset of R d . 2. If ν s (E) = +∞, then for all 0 < ε < s there exists a Radon measure µ ε on R d such that µ ε (E) = +∞ and n≥0 2 n(s-ε) I s-ε (µ ε |Sn ) < +∞.

The potential theoretic methods we demonstrated in Theorem V.2.7 are very comparable to the ones established for the standard Hausdorff dimension [START_REF] Falconer | Fractal geometry[END_REF]Theorem 4.13]. Unlike the standard Hausdorff dimension case, for the macroscopic Hausdorff dimension, we consider the measure µ is define on R d , and we focus on the restriction of µ on every annulus S n . For this reason, we deal with sums over n.

V.2.3 Application to projections

Projection theorems for Hausdorff dimensions have recently regained a lot of attention after some breakthroughs by M. Hochman and P. Shmerkin [START_REF] Hochman | Local entropy averages and projections of fractal measures[END_REF] and others, who used these theorems to tackle many longstanding questions in geometric measure theory and dynamical systems. It is quite satisfactory that they have natural counterparts in terms of macroscopic Hausdorff dimensions, as stated in the following theorem. As in the standard Hausdorff dimension case, the proof is based on a subtle use of the potential method and Theorem V.2.7.

It can be expected that Theorem V.2.8 can be extended in higher dimensional spaces, and that both Theorem V.2.7 and Theorem V.2.8 are useful in other situations that the one we describe here.

The structure of the paper is as follows. The main three results, Theorems V.2.4, V.2.7 and V.2.8 are established in Sections V.4, V.5, and V.6 respectively. Some necessary technical properties of the macroscopic Hausdorff dimension are proved in Section V.3.

V.3 First properties of Macroscopic Hausdorff Dimension V.3.1 An alternative definition for the macroscopic Hausdorff dimension

We will use an alternative, easier to handle with, definition for the macroscopic Hausdorff dimension, based on a simple modification of the ν s n quantities. We restrict ourselves to covers centered on integer points, with integer radii. We show that, up to a constants, this does not modify the values of the quantities involved in the computations, and the value of the macroscopic Hausdorff dimension is left unchanged.

We introduce for E ⊆ R d and n ≥ 0, the set of proper covers of E restricted to S n by Due to the fact that the x i are (multi)-integers, as well as the r i , the above infimum (V. [START_REF] Arneodo | multifractales et turbulences : de l'ADN aux croissances cristallines[END_REF] Lemma V.3.2 shows in particular that the convergence/divergence properties of ν s (E) and ν s (E) are identical.

C n (E) = {B(x i , r i )} m i=1 : m ∈ N, x i ∈ Z d ∩ S n , r i ∈ N * , E ∩ S n ⊂ m i=1 B(x i , r i ) .
The main advantage of dealing with ν s (E) is the existence of optimal proper s-covers, i.e. covers {B i = B(x i , r i )} m i=1 ∈ C n (E) such that ν s n (E) = m i=1 r i 2 n s . These optimal covers exists because x i and r i are positive integers.

In our further analysis, the size of the balls of optimal covers will matter, justifying the following definition. The quantity β s n (E) will be important, in particular for Theorem V.2.7 about potential methods and for the projection Theorem V.2.8.

Using the previous lemma, there exists a finite family (B i = B(x i , r i )) i=1,...,m of disjoint balls, all elements of B n , such that B∈Bn B ⊂ m i=1 5B i . The finiteness of the family comes from the boundedness of S n and the fact that the balls all have a diameter greater than 1. Up to a small translation of each x i by a vector of length at most √ d/2, one can assume that x i ∈ Z d and that With the translations that we added, some balls B ∈ B n may intersect, but this does not affect our argument.

Using the definition of ν s n (E), one finally gets where the last equality comes from the disjointness of the B i 's.

The following proposition guarantees that given a measure µ on a set E, there exists a smaller set F ⊂ E such that the measure µ has a controlled local scaling behavior on F . Observe that the same proof works if C n (E) and ν s n (E) are replaced respectively by C n (E) and ν s n (E).

V.4 Subsets of finite macroscopic measure

In this section, we prove a stronger version than Theorem V. Observe that we can either work with ν s or ν s , since ( ν s (E) < +∞) ⇔ (ν s (E) < +∞). We choose to work with ν s , and in this case β s n ( E) is defined without ambiguity. We start with three technical lemmas, that will later help us extract a macroscopic s-set and prove the projection theorem. This is a standard exercise, we prove it for completness.

Proof. Let ε > 0. For n ≥ 2 and ε > 0, one has Proof. We assume without loss of generality that 0 ≤ a n , b n < 1 for every n, and that (a n ) n∈N is a non-increasing sequence.

An A n-1 dx x 1+ε ≥ An A n-1 dx A 1+ε n = a n A 1+ε n . Then, 1 ε 1 A ε 1 ≥ 1 ε 1 A ε 1 - 1 A ε n = An A 1 dx x 1+ε ≥
For j ≥ 0, let us call D j = {n ≥ 0 : 2 -j-1 ≤ a n < 2 -j }, and B j = n∈D j b n . We call d j = max(D j ), which is finite since a n → 0. Observe that the integer sets D j are arranged in increasing order: d j + 1 = min(D j+1 ). Also, one has 2 -j B j , so that +∞ j=0 2 -j B j = +∞. We put n 1 = 0, j 1 = 1, and c n = 0 for every n ∈ D 0 ∪ D 1 .

Remark that n≥d 1 +1 a n b n ≥ 1/2 j≥2 2 -j B j = +∞.

Let us call n 2 the first integer n such that n 2 n=d 1 +1 a n b n > 1/2. Observing that for n ≥ d 1 + 1, a n b n ≤ 2 -1 , one necessarily has 1/2 < n 2 n=d 1 +1 a n b n < 1. We call j 2 the unique integer such that n 2 ∈ D j 2 , and we put c n = b n for every n ∈ {d 1 + 1, ..., n 2 }, and c n = 0 for every n ∈ {n 2 + 1, ..., d j 2 }. By construction, 1/2 < j 2 j=j 1 +1 n∈D j a n c n < 1.

We iterate the construction. Assume that we have built two finite sequences of integers (n k ) k=1,...,p and (j k ) k=1,...,p such that:

1. for k = 1, ..., p -1, j k+1 > j k , and for k = 1, ..., p, n k ∈ D j k This concludes the proof.

The same lines of computations can certainly be adapted to impose n≥0 a n c n = +∞ and n≥0 h(a n )c n < +∞ for any map h : R + → R + such that h(x) = o(x) when x → 0 + . As a first step toward Theorem V.4.1, we reduce the problem to sets that can be covered by small sets only.

V.5.2 Second part of Theorem V.2.7

This is the most delicate part. Assume now that ν s (E) = +∞, and fix 0 < ε < s.

Our goal is to build a Radon measure µ ε on R d such that µ ε (E) = +∞ and n≥0 2 n(s-ε) I s-ε (µ ε |Sn ) < +∞. We are going to build each measure µ ε n = µ ε |Sn . For this, we use the results we previously proved. By Theorem V.4.1, there exists a macroscopic s-set E 1 ⊂ E such that lim n→+∞ sup t∈[0,d] β t n (E 1 ) = 0, Dim H (E 1 ) = s and ν s (E 1 ) = +∞.

Proof. The fact that J x,k is an interval is obvious. Let x = (u, v) ∈ S k . We study the case where x 1 ≥ 0, the case x 1 < 0 being symmetric. Using polar coordinates, one has x = (r cos θ 0 , r sin θ 0 ) for some 2 k-1 ≤ r ≤ 2 k and θ 0 ∈ [-π 2 , π 2 ]. Then the projection of x on L θ is given by: proj θ x = (r cos(θ -θ 0 ) cos θ, r cos(θ -θ 0 ) sin θ).

Recall (V. From the proof, it also follows that |J x,n | ∼ C2 n-k when n/k is quite small. Let us study (V.6.11). One has By Lemma V.6.5, the interval J n,x ∩ J n,y has length smaller than C 0 2 n-k . So the integral above is taken over an interval of length at most C 0 2 n-k . Moreover, as s < 1, the integral reaches its largest value when θ close to (V.6.12)

where C > 0 is some positive constant. Then going back to I 1 and using V.6.12, one gets

I 1 ≤ C n≥0 2 n(s-ε) k≥n 2 (n-k)(1-s+ε) (E∩S k ) 2 dµ ε k (x) dµ ε k (y) ∥x -y∥ ε 2 ∨ 1 = C n≥0 k≥n
2 n+k(s+ε-1)

(E∩S k ) 2 dµ ε k (x) dµ ε k (y) ∥x -y∥ s-ε 2 ∨ 1 = C n≥0 2 n(s+ε-1) n k=0 2 k (E∩Sn) 2 dµ ε n (x) dµ ε n (y) ∥x -y∥ s-ε 2 ∨ 1 ≤ 2C n≥0 2 n(s-ε) I s-ε (µ ε n ) < +∞,
which is finite by (V.6.4).

Moving to I 2 , the same manipulations as above for I 1 yield As before, by Lemma V.6.5, |J k,x | ≤ 2 n-k and |J j,y | ≤ 2 n-j for all x ∈ S k ∩ C θ n and y ∈ S j ∩ C θ n ). Then, as k ≥ j + 1, the same argument as in (V.6.12) yields π 0 1 x∈C θ n (θ)1 y∈C θ n (θ) |τ x-y • e θ | s-ε dθ ≤ C2 (n-k) (1-s+ε) .

(V. 6.13) for some C > 0.

Next, we make use of equation (V.6.7) : indeed, it is not possible that µ ε j and µ ε j+1 are simultaneously non-zero. Hence, for x ∈ S k and y ∈ S j such that j < k and µ ε j and µ ε k not both equal to zero, then necessarly |k -j| ≥ 2 and 2 k-2 ≤ ∥x -y∥ 2 ≤ 2 k+1 . This implies in particular that E∩S j E∩S k dµ ε k (x) dµ ε j (y) ∥x -y∥ s-ε 2 ≤ C2 -k(s-ε) µ ε k (E ∩ S k ) µ ε j (E ∩ S j ), (V. 6.14) the inequality being in fact close to be sharp.

Finally, combining (V.6.14) and (V.6.13)), one gets that for some C ′ > 0,

I 2 ≤ C ′ n≥0 2 n(s-ε) k>j≥n 2 (n-k)(1-s+ε) 2 -k(s-ε) µ ε k (E ∩ S k ) µ ε j (E ∩ S j ) = C ′ n≥0 2 n k>j≥n 2 -k µ ε j (E ∩ S j )µ ε k (E ∩ S k ) = C ′ j≥0 j n=0 2 n µ ε n (E ∩ S n ) k≥n+1 2 -k µ ε k (E ∩ S k ) ≤ C ′ n≥0 2 n µ ε n (E ∩ S n ) k≥n+1 2 -k µ ε k (E ∩ S k ) ≤ C ′ n≥0 2 -n µ ε n (E ∩ S n ) n k=0 2 k µ ε k (E ∩ S k ) .
This last double sum is finite, because the set E was chosen so that (V.6.5) holds true. This concludes the proof.

Figure I. 1

 1 Figure I.1 shows sample paths of fractional Brownian motion for various H. The 1 2 -indexed fractional Brownian motion is simply the standard Brownian motion. As we can see in the figure, the smoothness of the path increases with H.

Figure I. 1 :

 1 Figure I.1: Simulation of a Fractional Brownian motion of Hurst index H = 0.25, 0.5, 0.75. [108]

2 1H

 2 -1 and for all t ≥ 0, P sup x,y∈K |L(x, t) -L(y, t)| |x -y| β < ∞ = 1. (I.1.6) I.1.1.

  or finite collection of sets with diameter equal to δ covering F , i.e. F ⊂ ∪ i U i . Recall that for a set U ⊂ R d , the diameter of U is defined as |U | = sup{∥x -y∥ 2 : x, y ∈ U }. Moreover, N δ (F ) denotes the smallest number of exact δ-covers of F . The lower and upper box dimension of F are defined respectively by: dim B (F ) = lim inf δ→0 log N δ (F ) -log δ (I.1.15) dim B (F ) = lim sup δ→0 log N δ (F ) -log δ . (I.1.16) Obviously, dim B (F ) ≤ dim B (F ), and if these are equal, their common value refers to the box dimension and is denoted by dim B (F ) = lim δ→0 log N δ (F ) -log δ . (I.1.17)

Figure I. 2 ,

 2 the Hausdorff measure jumps from ∞ to zero. If s = dim H (F ), H s (F ) ∈ [0, +∞]. Moreover, if 0 < H s (F ) < +∞, F is said to be an s-set.

Figure I. 2 :

 2 Figure I.2: The Hausdorff measure H s (F ) of a set F as a function of s [38].

  s ∈ [0, d] such that lim sup r→0 log(H s r,θ (F )) -log r = 0. As a consequence, the intermediate dimensions are defined as in [22]. Definition I.1.20. Let F ⊂ R d be bounded. For 0 ≤ θ ≤ 1, the lower θ-intermediate dimension is dim θ (F ) = the unique s ∈ [0, d] such that lim inf r→0 log H s r,θ (F ) -log r = 0 . (I.1.27)

  r→0 log(C s,m r,θ (F )) -log r = s and a unique s ∈ [0, d] such that lim sup r→0 log(C s,m r,θ (F )) -log r = s. Now for m ∈ {1, ..., d}, the lower intermediate dimension profiles of F ⊂ R d are dim θ,m (F ) = the unique s ∈ [0, m] such that lim inf r→0 log C s,m r,θ (F ) -log r = s , (I.1.31) and the upper intermediate dimension profiles are dim θ,m (F ) = the unique s ∈ [0, m] such that lim sup r→0 log C s,m r,θ (F ) -log r = s . (I.1.32)

  .1.5) where E ⊂ R + and E ⊂ R are Borel sets. These sets , due to self-similarity property of Z, may look like a fractal, see, e.g., Figure III.1, for the sojourn set of the Rosenblatt process. In order to describe such sets quantitatively one can use a type of fractal dimension.

Figure III. 1 :

 1 Figure III.1: Simulation of a Rosenblatt process of Hurst index H = 0.6. In red -the sojourn set E Z (γ) for γ = 0.6.

Moreover, a recent result [ 57 ,

 57 Theorem 1.4] describes the scaling behavior of the local time of Z: Proposition III.2.6. The local time L(x, [0, t]) is jointly continuous with respect to (x, t) and has finite moments. For a finite closed interval I ⊂ (0, ∞), let L * (I) = sup x∈R L(x, I).

r 1 ,

 1 ≤ |x| < r θ , r θ(m-s)+s |x| m r θ ≤ |x|. (III.3.5) Using this kernel we define the capacity of a compact set E ⊂ R N as C s,m r,θ (E) := inf µ∈M(E) ϕ s,m r,θ (x -y)dµ(x)dµ(y) -(III.3.6) where M(E) is the set of probability measures supported in E. Now for 0 < m ≤ N , the lower intermediate dimension profiles of E ⊂ R N are dim θ,m (E) = the unique s ∈ [0, m] such that lim inf r→0 log C s,m r,θ (E) -log r = s , (III.3.7) and the upper intermediate dimension profiles are dim θ,m (E) = the unique s ∈ [0, m] such that lim sup r→0 log C s,m r,θ (E) -log r = s . (III.3.8)

  H with probability one. Starting with the upper bound, we follow the technique used for [9, Theorem 5] -an upper bound result for the classical Hausdorff dimension of level sets associated to fractional Brownian sheet. But in fact, the covers used are of equal length and so this technique gives an upper bound for the Box dimension.

  1.2 and (III.1.15) of Theorem III.1.3 . Recalling Definitions III.1.3 and III.1.4, for a fixed γ > 0 and any x ∈ R, the level set L Z (x) is ultimately included in E Z (γ):

≤

  Ct 2+2γ-2H . (III.5.11)

  3.42) and Lemma IV.3.18, the second and the third sums by inequality (IV.3.42) and Lemma IV.3.21 and the last sum by Lemma IV.3.25.

2 )

 2 J ), ∀J ∈ N 0 , see [36, Lemma 3.6 and Theorem 3.7.]. Finally, we deduce P µ J∈N 0 (N µ J ≥ 1) = 1. (IV.3.60)

3 . 63 )

 363 It leads, just as in Lemmata IV.3.5 and IV.3.18, to the desired estimate.

(IV. 3 . 73 )

 373 As inequality (IV.3.62) holds, we use Lemma IV.3.27 to bound the first sum, and Lemma IV.3.30 to bound the second and the third one. For the last sum, from inequality (IV.3.62) and Lemmata IV.3.31 and IV.3.32, it just remains us to find bound for the random variables (IV.3.38), (IV.3.39) and (IV.3.40)with ℓ ∈ {j, j + 1} on Ω slo . For (IV.3.38) with ℓ = j , we have, as in (IV.3.72) and then (IV.3.41)

As 0 ≤ γ < 1 ,

 1 of course, one can find ℓ 0 ∈ N such that γ ℓ 0 < 2 -1 . (IV.5.3) Let us go back to the construction starting the proof of Lemma IV.4.7. If the dyadic interval λ j,k and m ∈ N are fixed and S ∈ S j,k,m we define the sequences of dyadic intervals (I n ) 0≤n≤m and (T n ) 1≤n≤m in the same way: I 0 = λ j,k , I m = S and, for all 1 ≤ n ≤ m, I n-1 = I n ∪ T n . Now, for any 1 ≤ n ≤ m, there are ℓ 0 dyadic intervals (T ℓ n = λ j (ℓ) n ,k

Definition V. 2 . 2 .

 22 For every s ≥ 0 and E ⊂ R d , defineν s (E) = n≥1 ν s n (E).The macroscopic Hausdorff dimension of E ⊂ R d is defined byDim H (E) = inf {s ≥ 0 : ν s (E) < +∞} . (V.2.3)One easily checks that Dim H (E) ∈ [0, d] for all E ⊂ R d , that Dim H (E) = 0 when E is bounded, and that an alternative definition for Dim H (E) isDim H (E) = sup {s ≥ 0 : ν s (E) = +∞} ,where sup ∅ = 0 by convention. It is also standard that Dim H (f (E)) ≤ Dim H (E) for every Lipschitz mapping f : R d → R d .

  2 n , µ |Sn (B(x, r)) ≤ c r 2 n s .Then, for all n ∈ N, ν s n (E) ≥µ |Sn (E) cand Dim H (E) ≥ s.

1 .

 1 If there exists a Radon measure µ on R d such that µ(E) = +∞ and if n≥0 2 ns I s (µ |Sn ) < +∞, then ν s (E) = +∞ and Dim H (E) ≥ s.

Theorem V. 2 . 8 .

 28 Let E ⊂ R 2 be a Borel set. Define L θ as the straight line passing through 0 with angle θ, and proj θ E as the orthogonal projection of E onto L θ . (a) If Dim H (E) < 1, then Dim H (proj θ E) = Dim H (E) for Lebesgue almost every θ ∈ [0, π]. (b) If Dim H (E) ≥ 1, then Dim H (proj θ E) = 1 for Lebesgue almost every θ ∈ [0, π].

Definition V. 3 . 1 .:

 31 For every s ≥ 0, n ≥ 0 andE ⊂ R d , define {B i = B(x i , r i )} m i=1 ∈ C n (E) (V.3.1)andν s (E) = n≥1 ν s n (E). (V.3.2)

Definition V. 3 . 3 .

 33 For E ⊂ Z d , n ∈ N and 0 < s < d, define β s n (E) := max max 1≤i≤p r i 2 n : (B(x i , r i )) p i=1is an s-optimal proper cover of E ∩ S n .

  (B i ) ≤ (5(2 + √ d/2)) s c µ n (S n ),

Proposition V. 3 . 6 . 1 F.

 361 Let E ⊂ R d be a Borel set. Then, for every 0 < s ≤ d there exists a constant c s > 0 (depending only on s) and a set∅ ̸ = F ⊂ E such that for every n ≥ 1, ) ≤ ν s n (F ) ≤ ν s n (E) (b) ν s n (F ∩ B (x, r)) ≤ c s r 2 n s for all x ∈ Z d ∩ S n and r ≥ 1. Proof. Let E ⊂ R d and set for every n ≥ n := x ∈ E ∩ S n : max r≥1 ν s n (E ∩ B(x, r)) (r/2 n ) s > 5(5(2 + √ d/2)) s .Using Proposition V.3.4 (b) applied to the set F n and the measureµ n (A) = ν s n (E ∩ A), one gets µ n (F n ) ≤ (5(2 + √ d/2)) s 5 -1 (2 + √ d/2)) -s µ n (S n ) = 1 5 µ n (E).Then µ n (E \ F n ) ≥ 4 5 µ n (E), i.e. as soon as E ∩ S n is not empty, (E \ F n ) ∩ S n ̸ = ∅. Finally, the set F = n≥0 E \ F n satisfies the two conditions mentioned above, with the constantc s = 5(5(2 + √ d/2)) s .V.3.3 Proof of the mass distribution principle : Proposition V.2.5For n ∈ N, let {B(x i , r i )} m i=1 ∈ C n (E), thenµ |Sn (E ∩ S n ) ≤ µ |Sn m i=1 B(x i , r i ) ≤ m i=1 µ |Sn (B(x i , r i )) ≤ cTaking infimum over all proper covers {B(x i , r i )} m i=1 ∈ C n (E), one gets µ |Sn (E ∩ S n ) c ≤ ν s n (E).Then ν s (E) ≥ n≥0 µ |Sn (E) c = µ(E) c = +∞ and so Dim H (E) ≥ s.

2 . 4 ,

 24 more precisely:Theorem V.4.1. Let E ⊂ R d such that ν s (E) = +∞.Then there exists a macroscopic s-set E such that E ⊂ E and lim n→+∞ sup t∈[0,d] β t n ( E) = 0.

Lemma V. 4 . 2 .

 42 Let (a n ) n≥1 be a bounded sequence of positive real numbers, such that lim n→+∞ A n := +∞ k=1 a k = +∞. For every ε > 0,

1 . 3 .

 13 and the series converges. Similarly, ln(A n ) -ln(A 1 ) = Since A n → +∞ as n → +∞, the series n k=2 a k A k-1 diverges. Also, since (a n ) is bounded,A n ∼ A n-1 and the series Let (a n ) n≥1 be a positive sequence converging to zero, (b n ) n≥1 be a bounded sequence of positive real numbers, such that n≥1 a n b n = +∞. Then, there exists a sequence (c n ) n≥1 such that:1. either c n = b n , or c n = 0, 2. n≥1 a n c n = +∞,3.n≥1 a 2 n c n < +∞.

2 . 2 -

 22 for k = 1, ..., p, c n = b n if n ∈ {d j k-1 + 1, ..., n k }, andc n = 0 if n ∈ {n k + 1, ..., d j k }, 3. for k = 1, ..., p, one has 1/(k + 1) < j k j=j k-1 +1 n∈D j a n c n < 2/k. (V.4.1)Let us call n p+1 the first integer such thatn p+1 n=dp+1 a n b n > 1/(p + 2). Observing that for n ≥ d p + 1, a n b n ≤ 2 -jp ≤ 1/(p + 1) (since j p ≥ p), one necessarily has 1/(p + 2) < n p+1n=dp+1 a n b n < 1/(p + 2) + 1/(p + 1) ≤ 2/(p + 1). We call j p+1 the unique integer such that n p+1 ∈ D j p+1 , and we put c n = b n for every n ∈ {d p + 1, ..., n p+1 }, and c n = 0 for every n ∈ {n p+1 + 1, ..., d j p+1 }. Clearly, these n p+1 and j p+1 satisfy the recurrence properties. Now, gathering the information, we deduce by (V.4.1) that n≥0a n c n = +∞ k=1 j k j=j k-1 +1 n∈D j a n c n ≥ +∞ k=1 1/(k + 1) = +∞and, using that a n ≤ 2 -j when n ≥ D j , and that j k-1 ≥ k -1, k+1 /(k + 1) < +∞.

Proposition V. 4 . 4 . 1 For 1 ≥ 1 ≥

 44111 Let E ⊂ R d such that ν s (E) = +∞. Then, there exists a set E ⊂ E such that ν s ( Ē) = +∞ and lim n→+∞ sup t∈[0,d] β t n ( E) = 0.Proof. It is an application of Lemma V.4.2.Call A n = n k=1 ν s k (E) and α n = A -1 n . By assumption, α n → 0 when n → +∞. For every n ≥ 1, S n can be covered by at most 2α -1 n balls of diameter 2 n α 1/d n . Call A n such a family of sets. One obviously hasν s n (E) ≤ A∈An ν s n (E ∩ A)V.5 Potential Methods V.5.1 First part of Theorem V.2.7Consider E ⊂ R d , and assume that there exists a Radon measure µ on R d such that µ(E) = +∞ and n≥0 2 ns I s (µ |Sn ) < +∞. We prove that ν s (E) = +∞, which implies that ν s (E) = +∞ and Dim H (E) ≥ s.For n ∈ N, we write µ n = µ |Sn , and defineϕ s µn := R d dµ n (y) ∥x -y∥ s 2 ∨ 1 and E n = x ∈ E ∩ S n : max r≥1 µ n (B(x,r)) every x ∈ E c n , there exists an integer r x such thatµ n (B(x, r x )) B(x,rx) dµ n (y) ∥x -y∥ s 2 ∨ µ n (B(x, r x )) (x)dµ n (x) ≥ 1 2 ns µ n (E c n ), which implies that n≥0 µ n (E c n ) ≤ n≥0 2 ns I s (µ n ) < +∞.But asE ∩ S n = E n ∪ E c n and n≥0 µ n (E ∩ S n ) = +∞, then n≥0 µ n (E n ) = +∞.Moreover, by Proposition V.3.4 a), one has ν s n (E n ) ≥ µn(En) 2 s . Finally, ν s (E) = n≥0 ν s n (E n ) = +∞ which gives that Dim H (E) ≥ s.

6 . 1 )

 61 , one sees that for 0≤ n ≤ k, x ∈ C θ n ⇐⇒ 2 n-1 ≤ r cos(θ -θ 0 ) ≤ 2 n ⇐⇒ 2 n-1 r ≤ cos(θ -θ 0 ) ≤ min 1, 2 n r ⇐⇒ θ ∈ θ 0 + arccos 2 n-1 r , θ 0 + arccos min 1, 2 n r mod π.The Taylor development arccos(y) = π 2 -y + o(y) together with the fact that 2 k-1 ≤ r ≤ 2 k yields that |J n,x | = 2 n-k (1 + o(1)).

2 where 2 ∨ 1 ,

 221 n (u) dµ ε,θ n (v) |u -v| s-ε ∨ 1 dθ E∩S j ∩C θ n E∩S k ∩C θ n dµ ε k (x) dµ ε j (y) |x • e θ -y • e θ | s-ε ∨ 1 dθ := I 1 + 2I dµ ε k (y) |(x -y) • e θ | s-ε ∨ 1 dθ I 2 = π 0 n≥0 2 n(s-ε) k>j≥n E∩S j ∩C θ n E∩S k ∩C θ n dµ ε k (x) dµ ε j (y) |(x -y) • e θ | s-ε ∨ 1 dθ. Starting with I 1 , one has dµ ε k (y) |(x -y) • e θ | s-ε ∨ 1 dθ -y) • e θ | s-ε ∨ 1 dµ ε k (x) dµ ε k (y) dθ -y) • e θ | s-ε ∨ 1 dθdµ ε k (x) dµ k (y) n (θ)1 y∈C θ n (θ) |τ x-y • e θ | s-ε dθ dµ ε k (x) dµ ε k (y) ∥x -y∥ s-εwhere τ x-y is the unit vector in the direction of x -y. By Lemma V.6.5, when x ∈ S k one has1 x∈C θ n (θ) = 1 Jn,x (θ). Then π 0 1 x∈C θ n (θ)1 y∈C θ n(θ) |τ x-y • e θ | s-ε dθ = Jn,x∩Jn,y dθ |cos( τ x-y , e θ )| s-ε .

  n (θ)1 y∈C θ n (θ) |τ x-y • e θ | s-ε dθ ≤ π 2 +C 0 2 n-k π 2 -C 0 2 n-k dθ |cos(θ)| s-ε ≤ C 0 2 n-k -C 0 2 n-k dθ |θ| s-ε = C2 (n-k)(1-s+ε) .

0 1

 0 dµ ε j (y) |(x -y) • e θ | s-ε ∨ 1 dθ. j E∩S k 1 C θ n (x)1 C θ n (y) |(x -y) • e θ | s-ε dµ ε k (x) dµ ε j (y) dθ = n≥0 2 n(s-ε) k>j≥n E∩S j E∩S k π Jn,x (θ)1 Jn,y (θ) |τ x-y • e θ | s-ε dθ dµ ε k (x) dµ ε j (y) ∥x -y∥ s-ε 2 .

  Moreover, by (I.1.21) if H s (F ) is finite, then letting δ → 0, one can observe that H t (F ) = 0. As a result, for a given set F ⊂ R d , there exists a critical value of s at which H s (F ) jumps from ∞ to 0. This critical value is called the Hausdorff dimension of F . Rigorously speaking dim H

		).	(I.1.21)
	We define the s-dimensional Hausdorff measure by
	H s (F ) = lim δ→0	H s δ (F ),	(I.1.22)
	where the value of the limit belongs to [0, ∞].

Table I

 I

	Dimension	Name	Cover	Size of covers Values Limit
	dim H

.1 compares the covering procedure between the mentioned fractal dimensions, while Table

I

.2 compares how these dimensions measures different types of sets.

Table I .

 I 1: Overview of the types of fractal dimensions. For the pixel density the cover consists of the integer points in the ball at distance less than 1 from F .

	Dimension	Name	Discrete sets Bounded sets

  For E ⊂ R and θ ∈ [0, 1], let dim θ (E) and Dim H (E) denote the θ-intermediate and macroscopic Hausdorff dimensions of E. Then, for any x ∈ R and 0

	and
	dim θ (Z(E)) =	1 H	dim θ,H (E) ,	(I.2.8)
	where dim θ,H (•) and dim θ,H (•) are the lower and upper θ-intermediate dimension pro-
	files respectively. For the precise technical definitions of these two objects see (I.1.31)
	and (I.1.32) in Section I.1.2.1.4.
	Then, we describe the size of the level sets L Z (x) in terms of intermediate dimensions
	and macroscopic Hausdorff dimension. The following holds:
	Theorem I.2.3 (L. Daw, G. Kerchev (2021)).
	2.2 (L. Daw, G. Kerchev (2021)). Let θ ∈ (0, 1] and E ⊂ R + be compact.
	Then almost surely:
	dim θ (Z(E)) =	1 H	dim θ,H (E) ,	(I.2.7)

Table III .

 III 2: Table of fractal dimensions and densities of random sets associated with the fractional Brownian motion with γ ∈ [0, H).

  103, Proposition 2.5(xii)]). Therefore, it is enough to show that for all t ∈ R n + :

		n
	E exp i	ξ j Z t j dξ < ∞.
	R d	j=1
	At this point we recall [57, Lemma 2.1, 2.2].
	Lemma III.2.2. Let L 2 G (R) be a weighted space with norm ∥f ∥ 2

  . It depends on two parameters H 1 , H 2 ∈ ( 1 2 , 1) which are such that H 1 + H 2 > 3 2 . The generalized Rosenblatt process {R H 1 ,H 2 (t, •)} t∈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

  Note that the (standard) Rosenblatt process is the process {R H,H (t, •)} t∈R + for H ∈ (3/4, 1). The generalized Rosenblatt process {R H 1 ,H 2 (t, •)} t∈R + is non-Gaussian, belongs to the second Wiener chaos, and has the following basic properties:(1) Continuity: the trajectories of the Rosenblatt process R H 1 ,H 2 are continuous.(2) Stationary increments: R H 1 ,H 2 has stationary increments; that is, the distribution of the process {R H 1 ,H 2 (t + s, •) -R H 1 ,H 2 (s, •)} t∈R + does not depend on s ≥ 0.

  2.1 for instance. Notations used through this paper are rather standard except, maybe, that if s, t are two real numbers, [s,t] stands for

t s if s ≤ t and -t s = s t otherwise.

  3.34) and (IV.3.35) in the same way that in Lemma IV.3.15 but using inequality (IV.3.42) and Lemmata IV.3.22 and IV.3.23 and finally inequality (IV.3.52).

For (IV.

3.36)

, first, by Lemma IV.3.23 and inequality (IV.3.42), we have, on Ω * t ∩ Ω t j 1 ≥n j 2 ≥j 1 k 1 ∈Z k 2 ∈Z j 2 [t,s]

  3.60) still holds. Now, if t ∈ S µ low ∩ (0, 1) we know that ∀ n ∈ N, ∀ λ ∈ 3λ n (t) , ∀j ≥ n , Ξ j (λ) ≤ (j -n + 1)µ. (IV.3.61) Let us remark that, as for all λ ∈ Λ n , |ε 2 λ | ≤ 2Ξ n (λ)+1, we still have, in this case, for all λ ∈ 3λ n (t), |ε λ | ≤ Cµ, 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 .

  Proof of Proposition IV.4.6. The constant C and C ′ of Lemmata IV.4.7 and IV.4.8 being deterministic and independent of M , on can choose M large enough such that

1 ,H 2 / ⋆ C, the conclusion follows by Borel-Cantelli Lemma.

  Proof of Theorem IV.1.2. Let us denote by Ω R the event obtained by taking the intersection of all the events of probability 1 induced by Propositions IV.3.3, IV.3.16, IV.3.26, IV.4.6 and IV.4.9.

1-a)

(δ(p j + δp j ) + p j + δp j ) for j large enough. Using the upper bound in (IV.

4.14)

, one can finally conclude that (IV.4.12) is indeed the general term of a convergent series.

  .1) in ν s n (E) is reached for some cover{B i = B(x i , r i )} m i=1 ∈ C n (E). Observe that ν s n is still sub-additive, i.e. ν s n (A ∪ B) ≤ ν s n (A) + ν s n (B) for every sets A and B. Lemma V.3.2. For every n ≥ 0, every set E ⊂ R d , one has In particular, one still has Dim H(E) = inf {s ≥ 0 : ν s (E) < +∞} = sup {s ≥ 0 : ν s (E) = +∞} . (V.3.4) Proof. The fact that C n (E) ⊂ C n (E) implies directly that ν s n (E) ≤ ν s n (E). Now, let {B(x i , ri )} m i=1 ∈ C n (E). Each ball B(x i , ri ) is included in a ball B(x i , ri + √ d),wherex i ∈ Z d ∩ E n . So B x i , ri + √ d

						ν s n (E) ≤ ν s n (E) ≤ (2 +	√	d) s ν s n (E).	(V.3.3)
	ri +	√	d + 1 ≤ (2 +	√	m i=1 d)r i (since ri ≥ 1), one has	∈ C n (E), and using that ri +	√	d ≤
					m i=1	ri + 2 n √	d	s	≤ (2 +	√	d) s	m i=1	ri 2 n

s

.

This holds for any cover {B(x

i , ri )} m i=1 ∈ C n (E), so ν s n (E) ≤ (2 + √ d) s ν s n (E).

A mass distribution µ on R d is a measure such that 0 < µ(R d ) < ∞.

We recall that R ′ H1,H2 is defined in Remark 17.

The 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) ].

The function Φ is in the fact the symmetrization of Φ 1 .

For f a symmetric function in L 2 (R 2 ) , and I 2 (f ) the second order Wiener-Itô integral of f . One hasE(I m (f )) 2 = 2!||f || L 2 (R 2 ) .

This result is again a generalization of[START_REF] Ayache | Different possible behaviors of wavelet leaders of the Brownian motion[END_REF] Lemma 3.3.] where most of the modifications comes from the fact that we are working in the Wiener chaos of order 2

Acknowledgements

Throughout the writing of this dissertation I have received a great deal of support and assistance.

List of Publications

The following list of publications contains the work presented in this thesis:

1. L. Daw Proof. The proof is exactly the same as the one of Lemma IV.3.9 excepted that we use Lemmata IV.3.19 and IV.3.20 instead of Lemmata IV.3.7 and IV.3.8 respectively and that we conclude using again (IV.3.45) instead of (IV. 3.11) and

(IV. 3.52) instead of (IV. 3.22).

Lemma IV. 3.22. There exists a deterministic constant C > 0 such that, for all t, s ∈ (0, 1) and n ≤ j 1 ≤ j 2 , the quantities (IV. 3.46) and (IV.3.47) are bounded from above by

Proof. The proof is exactly the same as for Lemma IV.3.19 except that, here, we use the second part of Lemma IV.3.17 instead of the first one.

Lemma IV.3.23. There exists a deterministic constant C > 0 such that, for all t, s ∈ (0, 1) and n ≤ j 1 ≤ j 2 the quantities (IV. 3.49) and (IV.3.50) are bounded from above by

Proof. The proof is exactly the same as for Lemma IV.3.20 except that, here, we use the second part of Lemma IV.3.17 instead of the first one.

Just as we did for the rapid points, it remains us to bound the random variables Ξ j (λ) (IV. 3.29). 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 ∈ N the random variables Ξ j (λ) for j ≥ n and λ ∈ 3λ n (t). We thus have the following result.

Lemma IV.3.24. For all t ∈ (0, 1), there exist an event Ω t of probability 1 and a positive random variable C t,2 with finite moment of any order such that, on Ω t , ∀n ∈ N, ∀λ ∈ 3λ n (t), ∀j ≥ n, Ξ j (λ) ≤ C t,2 (j -n + 1) log(n).

(IV.3.53)

Proof. If t ∈ (0, 1) is fixed and θ > 0, let us define the event A n (t) = {∀λ ∈ 3λ n (t) ∀j ≥ n, Ξ j (λ) ≤ θ(j -n + 1) log(n)}.

Similarly to Lemma IV.3.14, we get

Concerning the first integral, we have, as

while, using again the "Wiener isometry",

where B H 2 denotes the fractional Brownian motion with parameter H 2 . As a result, there exists a positive constant C * Ψ,H 1 ,H 2 such that, as we suppose

By Lemma IV.4.5, one can remark that as M → +∞, (

By Theorem IV.3.12, there exists a constant c 1 > 0 such that, for all λ ∈ Λ and y sufficiently large

Then, for all M ∈ N, we have, for all such λ and y

Using Lemma IV.4.5 and Theorem IV.3.11 one has

Thus, if M is large enough, one has, as 1 -

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

with c 2 := 2c 1 . In the sequel, we will implicitly always consider such large enough M .

In the following two subsections, Lemmata IV.4.7 and IV.4.10 follow the lines of Lemmata 3.6 and 3.8 in [START_REF] Ayache | Different possible behaviors of wavelet leaders of the Brownian motion[END_REF] respectively, with some subtle modifications as the authors in [START_REF] Ayache | Different possible behaviors of wavelet leaders of the Brownian motion[END_REF] deal with 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.

IV.4.1 Ordinary Points

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

Proposition IV.4.6. There exists Ω * 1 ⊂ Ω with probability 1 such that for all ω ∈ Ω * 1 and Lebesgue almost every t ∈ (0, 1) one has

To this end, as a first step, let us state the following lemma concerning the random variable ε λ M . If λ = λ j,k is a dyadic interval and m ∈ N, S λ,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 λ j,k , formally speaking

There is a deterministic constant C > 0 such that the following holds: for all M ∈ N and for all t ∈ (0, 1), there exists Ω t,1 ⊂ Ω with probability 1 such that for all ω ∈ Ω t,1 there are infinitely many j ∈ N such that

Concerning the random variable Cε λ M , one can give an almost sure upper bound.

Lemma IV.4.11. There exists a deterministic constant C ′ > 0 such that for all M there is Ω ′ 2 ⊂ Ω with probability 1 such that for all ω ∈ Ω ′ 2 there exist J ∈ N such that, for all j ≥ J, for all λ ∈ Λ j , λ ⊆ [0, 1],

Proof. If C ′ > 0, for all j sufficiently large, we have, by Theorem IV.3.11

and thus, if

C, the conclusion follows by Borel-Cantelli Lemma.

Proof of Proposition IV.4.9. Again, one can choose M large enough such that

where C and C ′ are the constant given by Lemmata IV.4.10 and IV.4.11 respectively. Let us consider ω ∈ Ω * 2 := Ω 2 ∩ Ω ′ 2 where the evnets, of probability 1, Ω 2 and Ω ′ 2 are giving by the same Lemmata. We use the same notations as in them. First there exist t ∈ (0, 1) such that for all J ∈ N there exist j ≥ n such that

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

In this case, as in IV.4.6 we have that for all J great enough, there is j ≥ J such that

and so one can conclude that (IV.4.7) holds true for all ω ∈ Ω * 2 .

IV.5 Proof of the main Theorem

Theorem IV.1.2 is then a straightforward consequence of Propositions IV.3.3, IV.3.16, IV.3.26, IV.4.6 and IV.4.9.

V.2 Definitions and statements of the results

Here and in the reset of the paper, let (R d , ∥.∥ 2 ) be the d-dimensional Euclidean space equipped with the L 2 -norm.

V.2.1 The macroscopic Hausdorff dimension

For x ∈ R d and r > 0, B(x, r) denotes the Euclidean ball with center x and radius r.

Let us recall the definition of the Barlow-Taylor macroscopic Hausdorff dimension Dim H (E) of a set E ⊆ R d , developed in [START_REF] Barlow | Fractional dimension of sets in discrete spaces[END_REF][START_REF] Barlow | Defining fractal subsets of Z d[END_REF].

Define, for all integer n ∈ N, the n-th shell of R d by S 0 = B(0, 1) and

Like the standard Hausdorff dimension, the macroscopic Hausdorff dimension Dim H (E) aims at describing how a set E can be efficiently covered by balls. Since Dim H is concerned only with large scale behaviors, Barlow and Taylor proposed to study the covers of the intersections E ∩ S n by balls, for every n ∈ N, and the balls used to cover the sets E ∩ S n will all be of diameter at least 1. Again this is justified by the fact that this dimension is supposed to describe discrete sets (so small balls are not relevant).

To this end, let us introduce, for E ⊆ R d , the set of covers of E restricted to S n defined by

Finally, for s ≥ 0 and n ∈ N, set

for every sets A and B, but is not a measure (because of the constraints on r i ).

The existence of optimal covers is not guaranteed. We will deal with this issue in Section V.3.

We are now ready to define the Barlow-Taylor macroscopic Hausdorff dimension.

V.3.2 Some preliminary results

We first prove two propositions that will be needed later. Proposition V.3.4. Let µ n be a Borel measure on S n , E ⊂ R d be a Borel set and 0 < c < +∞ be a constant.

Then,

which is true for all covers {B(x i , r i )} m i=1 ∈ C n (E). Finally, taking the infimum over all elements of C n (E), one gets

b) Consider the family of balls

Now, we invoke the following 5r-covering Lemma [START_REF] Evans | Measure theory and fine properties of functions[END_REF]Lemma 4.8].

Lemma V.3.5. Let B be a family of balls in R N and suppose that sup B∈B d(B) < ∞.

Then there exists a countable sub-family of disjoint balls B 0 of B such that

Thus there must exist

n , so by Definition V.3.3, for every t > 0

Actually, this implies more: necessarily

Finally, we prove Theorem V.4.1.

Proof. Let E be such that ν s (E) = +∞. By Proposition V.4.4, one also assumes that lim n→+∞ sup s∈[0,d] β s n ( E) = 0. This fact will not be used in this proof only, but will be key in the next section.

Observe that since for every n ν s n (E) ≤ 1, then

The idea consists in replacing E by a set E such that ν s n ( E) ∼ b n ν s n (E), such that n≥1 ν s n ( E) < +∞ but b n is "as large as possible". Lemma V.4.2 helps to build such a sequence.

First, for every ε > 0, denote by

k By Lemma V.4.2, one knows that B ε n → 0 as n → ∞, for every ε > 0. We build iteratively a non-increasing sequence (ε n ) n≥0 ⊂ R + , and a sequence of integers (n k ) k≥1 .

Consider n 1 as the smallest positive integer such that B 1 4

Next we proceed by induction to build (ε n ) n≥0 and (n k ) k≥1 .

Assume that n 1 < n 2 < ... < n p are defined.

Define n p+1 as the smallest integer such that

Then by construction of ε n , one has:

(ii) By (V.4.2), and the fact that B

Next, we construct a set E ⊂ E such that for all n ∈ N, one has

To achieve this, observe that by (V.2.1), S n contains a finite number of lattice points, and denote by M n,d their cardinality. These points are denote by x i for i ∈ {1, . . . , M n,d }.

Consider the following function:

where g n (0) = 0 by convention. It is clear that g n is non-decreasing, and ranges from 0 to ν s n (E). Moreover, for all m ∈ {1, . . . , M n,d -1}, if {B(y j , r j )} p j=1 is an s-optimal cover of m i=1 E ∩ B(x i , 1), then (B(y j , r j )) p j=1 , B(x m+1 , 1) is a proper cover of

(not necessarily optimal). Using these two covers, one gets

Hence, g n has only small increments.

Then by construction, E ⊂ E, and for all n ∈ N one has

And so, by (V.4.5),

To complete the proof, it is enough to show that for all ε > 0, ν s-ε ( E) = +∞. To this end, fix ε > 0, and let (B(x i , r i )) m i=1 be an optimal (s -ε)-cover of E ∩ S n , and assume that for this specific cover, β s-ε n ( E) is reached, i.e. there exists i ∈ {1, ..., m} such that

Two cases are separated.

On the one hand, if

where the fact that ν s n (E) ≤ 1 has been used in the last step.

On the other hand, if

Finally, using the fact that ε n → 0 together with the lower bounds (V.4.8) and (V.4.9), one gets that for every large n, ν s-ε n ( E) ≥

This holds for every ε > 0, so Dim H E = s.

Consider an optimal (s -ε 2 )-cover {B(x i , r i )} m i=1 of E 1 ∩ S n . One sees that

where we used that β

Setting

, one then sees that the sequences (a n ) n≥1 and (b n ) n≥1 satisfies the assumptions of Lemma V.4.3. Consider the sequence (c n ) n≥1 given by this Lemma, and define the set E 2 ⊂ E 1 as follows: for every n ≥ 1,

It is immediate from the construction and Lemma V.4.

Finally, by Proposition V.3.6, there exists

Then by our construction and (V.5.2), one has

We are left to prove that

. By (V.5.3), one has

Using the fact that B(x, 2 n+1 ) =

r=1 B(x,r)\B(x,r-1)

One the one hand, by (V.5.2), µ ε n (B(x, 1))

. On the other hand,

r=2 B(x,r)\B(x,r-1)

for some constant C. So

Moving to the integral, one gets

Finally, recalling (V.5.1), (V.5.2), (V.5.3) and the definition of µ ε n , one has

as desired.

V.6 Projection of a Set

In this section we are considering the orthogonal projection of sets in R 2 and we aim at proving the projection Theorem V.2.8 for the macroscopic Hausdorff dimension.

Let us introduce some notations.

For every θ ∈ [0, 2π], call e θ = (cos θ, sin θ) the vector with angle θ, and L θ the straight line in R 2 with angle θ passing through the origin.

Then, recall that proj θ : R 2 → L θ is the orthogonal projection onto L θ .

V.6.1 Case where Dim H (E) ≥ 1

Let us start by proving item b) of Theorem V.2.8, assuming that item a) is proved.

By Theorem V.4.1, for every p ≥ 2, there exists E p ⊂ E such that Dim H (E p ) = 1-1/p. For each set E p , by item a), there exists a set Θ p ⊂ [0, π] of full Lebesgue measure such that for every θ ∈ Θ p , Dim H (proj θ (E p )) = 1 -1/p. In particular, this implies that Dim H (proj θ (E)) ≥ 1 -1/p. Consider now the set Θ = p≥2 Θ p . The above arguments show that Θ is still of full Lebesgue measure in [0, π], and that for every θ ∈ Θ, Dim H (proj θ (E)) ≥ 1. Since obviously Dim H (proj θ (E)) is always less than 1 (since it is included in L θ ), the result follows.

V.6.2 First extractions when Dim

Writing L θ = {λe θ : λ ∈ R}, we can define the n-th shells inside L θ as S θ n = {v = (x, y) ∈ L θ : ∥v∥ 2 ∈ [2 n-1 , 2 n ]}. Identifying L θ with R, the results we obtained before in dimension 1 apply to L θ and S θ n . We are going to project 2-dimensional measures onto the lines L θ . For this, let us define for every n ≥ 0 the cylinders

We are going to prove that for every 0 < ε < s, the set

has full Lebesgue measure. The conclusion then follows using the same argument as the one used to prove item b). More precisely, from the properties above, Θ := p≥1 Θ s- Fix 0 < ε < s.

Applying Theorem V.2.7(2), there exists a Borel measure µ ε supported by E such that

where µ ε n is a simplified notation for µ ε |Sn . Observe that in fact, via the finer Theorem V.4.1 and Proposition V.4.4, we can impose that lim n→+∞ µ ε n (E ∩ S n ) = 0.

We need to impose an additional condition on µ ε , namely that n≥0

This is achieved thanks to the following lemma.

Lemma V.6.1. Let (a n ) n≥1 and (b n ) n≥1 be two positive sequences converging to zero, such that n≥1 a n = +∞ and n≥1 a n b n = +∞. There exists a sequence (c n ) n≥1 such that:

1. either c n = a n , or c n = 0,

2.

n≥1 c n = +∞,

3.

n≥1 c n b n < +∞.

Proof. Again, without loss of generality, we assume that 0 < a n , b n < 1. Let us call D j = {n ≥ 0 : 2 -j-1 ≤ b n < 2 -j }, for j ≥ 0.

Put c n = 0 for every n ∈ D 0 ∪ D 1 , and n 0 = 0, j 0 = 1.

We know that j≥2 n∈D j a n b n = +∞. We go through each D j in increasing order. Consider the first couple (n 1 , j 1 ) such that n 1 ∈ D j 1 and

, and c n = 0 for all n ∈ {n ∈ D j 1 : n > n 1 }. By our choice,

We then iterate the process: assume that we have built two finite sequences of integers (n k ) k=1,...,p and (j k ) k=1,...,p such that 1. for k = 1, ..., p -1, j k+1 > j k , and for k = 1, ..., p,

We know that j≥jp+1 n∈D j a n b n = +∞. Consider the first couple (n p+1 , j p+1 ) such that n p+1 ∈ D j p+1 and p+1) . Put c n = a n for all n ∈ j p+1 -1 j=jp D j ∪ {n ∈ D j p+1 : n ≤ n p+1 }, and c n = 0 for all n ∈ {n ∈ D j p+1 : n > n p+1 }. Then, since for all the selected integers n, a n b n ≤ 2 -j p+1 ≤ 2 -(p+1) , (V.6.6) holds true.

Collecting the information, on one hand one has by (V.6.6)

On the other hand, since j k ≥ k + 1, one sees that for each n ∈ D j for j ∈ {j k-1 , ...j k }, b n ≤ 2 -k , so again by (V.6.6),

Setting a n = µ ε n (E), then (a n ) n≥0 tends to zero when n tends to infinity. Define then

Since n k=0 2 k ∼ 2 n , (b n ) n≥0 is a generalized Caesaro mean associated with the sequence (a n ) n≥0 , and converges to zero when n tends to infinity.

So either

n≥1 a n b n < +∞, and (V.6.5) is true, or n≥1 a n b n = +∞ and we are exactly in the situation of Lemma V.6.1: there exists a sequence (c n ) n≥1 such that:

Setting E = n≥0:an=cn E ∩ S n , by construction one has µ ε ( E) = n≥1 c n = +∞, and since

hence (V.6.5) is obtained for E. This property will be used at the very end of the proof of Proposition V.6.4 only. It is obvious that if Theorem V.2.8 is proved for this smaller set E, it is also true for the original set.

Finally, observe that, replacing E by n≥0 E ∩ S 2n or n≥0 E ∩ S 2n+1 , one can assume in addition to (V.6.3), (V.6.4) and (V.6.5) that if S n ̸ = ∅, then S n-1 = S n+1 = ∅.

(V.6.7)

To resume this section, we have proved that the original set E contains a subset, still denoted by E for simplification, and a measure µ ε supported by E such that (V.6.3), (V.6.4), (V.6.5) and (V.6.7) simultaneously hold. where x.e θ denotes the scalar product. Since e θ is unitary, we identify x.e θ with proj θ x, the orthogonal projection of x onto L θ .

Definition V.6.2. The projected measure µ ε,θ is defined as µ ε,θ = n≥1 µ ε,θ n , where

(V.6.8)

Note that each µ ε,θ n is a measure supported on proj θ E ∩ S θ n . We are going to prove that for almost all θ ∈ [0, π], n≥0 µ ε,θ n (proj θ E) = +∞ and n≥0 2 n(s-ε) I s-ε (µ ε,θ n ) < ∞. (V.6.9)

for almost all θ ∈ [0, π]. Then item a) of Theorem V.2.7 will allow us to conclude that the set Θ s-ε defined by (V.6.2) has full Lebesgue measure, as announced. This is the purpose of the next two propositions.

Proposition V.6.3. For every θ ∈ [0, π], µ ε,θ (proj θ E) = +∞. (V.6.10)

Proof. This simply follows from the observation that

since the union of the (C θ n ) n≥1 cover R 2 (there are small overlaps (their borders) between the C θ n ). Hence the result.

So the first part of (V.6.9) is proved.

Let us move to the second part. Observe that even if µ ε,θ (proj θ E) = +∞, it is likely that proj θ E has dimension less than Dim H (E). A trivial example is when the s-dimensional set E is included in a straight line of angle ϕ passing through 0, and θ = ϕ + π/2. Proposition V.6.4. One has E θ n≥0 2 n(s-ε) I s-ε (µ ε,θ n ) < +∞. (V.6.11)

Proof. Remark that if (V.6.11) is proved, then n≥0 2 n(s-ε) I s-ε (µ ε,θ n ) < +∞ for Lebesgue almost every θ ∈ [0, π], so (V.6.9) and item a) of Theorem V.2.8 are proved.

We start with the following lemma. Lemma V.6.5. There exists a constant C 0 > 0 such that the following holds. Let x ∈ S k for some k ≥ 0. For all 0 ≤ n ≤ k, the set J x,n = {θ ∈ [0, π] : x ∈ C θ k } is an interval modulo π, and |J x,n | ≤ C 0 2 n-k .