Convergence of $p$-Stable Random Fractional Wavelet Series and Some of its Properties

TL;DR

This paper studies the convergence and geometric properties of a $p$-stable fractional wavelet series, revealing new insights into its behavior and self-similarity in the context of wavelet analysis and fractional operators.

Contribution

It introduces conditions for convergence of $p$-stable wavelet series involving fractional integrals and explores their geometric and self-similar properties.

Findings

Series converges under specific conditions involving $p$-stability and fractional integrals.

Identifies geometric properties related to self-similarity of the series.

Extends analysis to modified fractional integral operators.

Abstract

For appropriate orthonormal wavelet basis $\{\psi_{j\,k}^e \}_{j\in\mathbb{Z}\,k\in\mathbb{Z}^d\,e\in\{0,1\}^d}$, constants $p$ and $\gamma$, if $\mathcal{I}_{\gamma}$ denotes the Riesz fractional integral operator of order $\gamma$ and $(\eta_{j\,k\,e})_{j\in\mathbb{Z} k\in\mathbb{Z}^d \,e\in\{0,1\}^d}$ a sequence of independent identically distributed symmetric $p$-stable random variables, we investigate the convergence of the series $\sum\limits_{j\,k\,e} \eta_{j\,k\,e} \mathcal{I}_{\gamma} \psi_{j\,k\,}^e$. Similar results are also studied for modified fractional integral operators. Finally, some geometric properties related to self similarity are studied.

Figures

Equations

$$X_{\gamma }=\sum _I{\eta }_I{\mathcal{I}}_{\gamma }{\psi }_I,$$

$$\underset{(x_1,...x_d)\in {\mathbb{R}}^d}{\sup }\prod _{i=1}^d{\left|x_i\right|}^{{\alpha }_i}\left|\frac{\partial }{\partial x_1^{{\beta }_1}}...\frac{\partial }{\partial x_d^{{\beta }_d}}\varphi (x_1,...x_d)\right|<\infty \forall {\alpha }_j{\beta }_j\in \mathbb{N},$$

$$\hat{f}(\lambda )=\sum _{k\in {\mathbb{Z}}^d}\hat{f}(k)\varphi (\lambda -k)$$

$$\tilde{f}\in L^2\left({\left[-\frac{1}{2},\frac{1}{2}\right]}^d\right)\subset L^1\left({\left[-\frac{1}{2},\frac{1}{2}\right]}^d\right)$$

$$\sum _{k\in {\mathbb{Z}}^d}{a}_k{e}^{-2\pi ix.k},$$

$$\mathop{\phi}\limits^{\vee}(x)=\left\{\begin{array}[]{l}1,\left|{x_{i}}\right|<x_{0}\\ 0,\left|{x_{i}}\right|\geq 1-x_{0}\\ \end{array}\right.\,.$$

$$\underset{R\to \infty }{\lim }\underset{\lambda \in {\mathbb{R}}^d}{\sup }\left|\widehat{S_R}(\lambda )-\hat{f}(\lambda )\right|=0,$$

$$\widehat{S_R}(\lambda )=\sum _{k\in D_R}\hat{f}(k)\varphi (\lambda -k).$$

$$({\mathcal{I}}_{\gamma }f)(x)=\frac{1}{C_{\gamma }}\underset{{\mathbb{R}}^d}{\int }\frac{f(y)}{{\left|x-y\right|}^{d-\gamma }}dy$$

$${\Vert {\mathcal{I}}_{\gamma }f\Vert }_{L^q({\mathbb{R}}^d)}\le C_{pq}{\Vert f\Vert }_{L^p({\mathbb{R}}^d)}.$$

$$\widehat{{\mathcal{I}}_{\gamma }f}(\lambda )=(2\pi {)}^{-\gamma }{\left|\lambda \right|}^{-\gamma }\hat{f}(\lambda )$$

$$⟨{\mathcal{I}}_{\gamma }f,g⟩=⟨f,{\mathcal{I}}_{\gamma }g⟩.$$

$${\mathcal{K}}_{\gamma }f(x)=\underset{{\mathbb{R}}^d}{\int }(k_{\gamma }(x-y)-k_{\gamma }(y))f(y)dy=\underset{{\mathbb{R}}^d}{\int }{K}_{\gamma }(x,y)f(y)dy$$

$${\Vert K_{\gamma }(x,.)\Vert }_{L^p({\mathbb{R}}^d)}=C_{p\gamma d}|x{|}^{\gamma -\left(1-\frac{1}{p}\right)d}.$$

$${\Vert K_{\gamma }(x,.)\Vert }_{L^p({\mathbb{R}}^d)}^p=\underset{\{|y|<2|x|\}}{\int }|K_{\gamma }(x,y){|}^{p}dy+\underset{\{|y|\ge 2|x|\}}{\int }|K_{\gamma }(x,y){|}^{p}dy.$$

$$|K_{\gamma }(x,y)|\le C|x-y{|}^{\gamma -d-1}|x|,$$

$${\mathcal{K}}_{\gamma }f(x)=\frac{1}{(2\pi {)}^{\gamma }}\underset{{\mathbb{R}}^d}{\int }\left(\frac{e^{-2\pi i\lambda x}-1}{{\left|\lambda \right|}^{\gamma }}\right)\hat{f}(\lambda )d\lambda .$$

$$\widehat{({\mathcal{I}}_{\gamma }{K}_{\beta }(x,.))}(\lambda )=\widehat{K_{\beta +\gamma }(x,.)}(\lambda )=\frac{1}{(2\pi {)}^{\gamma +\beta }}\left(\frac{e^{-2\pi i\lambda x}-1}{{\left|\lambda \right|}^{\beta }}\right)\frac{1}{|\lambda {|}^{\gamma }}.$$

$${\mathcal{K}}_{\gamma }({\mathcal{I}}_{\beta }f)(x)={\mathcal{K}}_{\beta +\gamma }f(x)=\underset{{\mathbb{R}}^d}{\int }{K}_{\beta +\gamma }(x,y)f(y)dy.$$

$$\widehat{{\mathcal{J}}_{s}f}(\lambda )=(1+|\lambda {|}^2{)}^{s/2}\hat{f}(\lambda ).$$

$${\Vert {\mathcal{J}}_{s}f\Vert }_{L^p({\mathbb{R}}^d)}\le C_p{\Vert f\Vert }_{L^p({\mathbb{R}}^d)}.$$

$$H_s^p({\mathbb{R}}^d)=\left\{f\in {\mathcal{S}}^{\prime }({\mathbb{R}}^d):{\mathcal{J}}_{s}f\in L^p({\mathbb{R}}^d)\right\}.$$

$${\Vert K_{\gamma }(x,.)\Vert }_{H_s^p({\mathbb{R}}^d)}\sim {\Vert K_{\gamma }(x,.)\Vert }_{L^p({\mathbb{R}}^d)}+{\Vert (K_{\gamma -s}(x,.)\Vert }_{L^p({\mathbb{R}}^d)}.$$

$$\mathcal{F}{L^p}_w=\left\{f\in {\mathcal{S}}^{\prime }({\mathbb{R}}^d):\hat{f}(1+|.{|}^2{)}^{-d}\in L^p({\mathbb{R}}^d)\right\}$$

$$\underset{{\mathbb{R}}^d}{\int }\frac{|g(\lambda )|}{(1+|\lambda |{)}^m}d\lambda =\underset{{\mathbb{R}}^d}{\int }\frac{|g(\lambda )|}{(1+|\lambda |{)}^m}\frac{(1+|\lambda {|}^2{)}^{\frac{d}{p}}}{(1+|\lambda {|}^2{)}^{\frac{d}{p}}}d\lambda$$

$$\le {\left(\underset{{\mathbb{R}}^d}{\int }|g(\lambda ){|}^p(1+|\lambda {|}^2{)}^{-d}d\lambda \right)}^{\frac{1}{p}}{\left(\underset{{\mathbb{R}}^d}{\int }\frac{(1+|\lambda {|}^2{)}^{\frac{dq}{p}}}{(1+|\lambda |{)}^{mq}}d\lambda \right)}^{\frac{1}{q}}<\infty ,$$

$$|⟨f_n,\phi ⟩-⟨f,\phi ⟩|=|⟨{\hat{f}}_n-\hat{f},{\phi }^{\vee }⟩|=\left|\underset{{\mathbb{R}}^d}{\int }({\hat{f}}_n(\lambda )-\hat{f}(\lambda )){\phi }^{\vee }(\lambda )d\lambda \right|$$

$$=\left|\underset{{\mathbb{R}}^d}{\int }({\hat{f}}_n(\lambda )-\hat{f}(\lambda )){\phi }^{\vee }(\lambda )\frac{(1+|\lambda {|}^2{)}^{\frac{d}{p}}}{(1+|\lambda {|}^2{)}^{\frac{d}{p}}}d\lambda \right|$$

$$\le {\left(\underset{{\mathbb{R}}^d}{\int }|{\hat{f}}_n(\lambda )-\hat{f}(\lambda ){|}^p\frac{1}{(1+|\lambda {|}^2{)}^d}d\lambda \right)}^{\frac{1}{p}}{\left(\underset{{\mathbb{R}}^d}{\int }|{\phi }^{\vee }(\lambda ){|}^q(1+|\lambda {|}^2{)}^{\frac{dq}{p}}d\lambda \right)}^{\frac{1}{q}},$$

$${\Vert f\Vert }_{\mathcal{F}{L^p}_w}^p\le C_{pd}\sum _{k\in {\mathbb{Z}}^d}|\hat{f}(k){|}^p(1+|k{|}^2{)}^{-d}.$$

Full text

I Introduction

Uncoupled representations of random processes are of practical interest. A classical example for Gaussian processes is the Karhunen-Lóeve (KL) representation. Motivated in part by applications in signal and image processing [2, 18, 19, 20], a usual requirement for a random process defined on $\mathbb{R}^{d}$ is to be self similar (see section II-B) in some specified sense, since there exists several related notions in the literature. This property, in the case $d=2$, is of certain relevance for characterizing textures. For the finite variance case, several KL like representations for the family of $\dfrac{1}{f}$ of self-similar and related processes were proposed, e.g. [2, 4, 13, 19] among others. In this case, these representations have in general the form:

[TABLE]

where $\mathcal{I}_{\gamma}$ is some fractional integration operator, $\{\psi_{I}\}_{I}$ is an orthonormal basis of $L^{2}(\mathbb{R}^{d})$ or other Hilbert space of functions and the $\{\eta_{I}\}_{I}$ is a sequence of finite variance identically distributed random variables, in most cases Gaussian. The parameter $\gamma$ is usually linearly related to the self-similarity Hurst parameter $H$ of the process, [3]. Apart from applications, series like (1) and its geometric properties were extensively studied in the case of Fourier Gaussian random series, see for example [9]. Considering this sum as a generalized random process in the sense of Gelfand and Vilenkin [5], Chapter 3, p. 237, if the $\eta_{I}$’s are Gaussian and $\mathcal{I}_{\gamma}$ is the Riesz fractional integration operator (Definition 3) then this sum converges a.s. in the sense of distributions, i.e. in $\mathcal{D}^{\prime}(\mathbb{R}^{d})$ to a self-similar process as defined here in Section II-B in terms of equality in probability law between $X_{\gamma}$ and a re-scaled version of it: $a^{\delta}X_{\gamma}(a\,.\,)$ for some $\delta\in\mathbb{R}$. In this particular case, $X_{\gamma}$ is a fractional Gaussian noise (See Theorem III.2). These type of representations have received some interest because of its simplicity for modeling certain random signals (see e.g. [19]), since one only needs to know the probability distribution of the coefficients $\eta_{I}$ and the parameter $\gamma$ or similar. On the other hand, the finite variance requirement may be a constraint in some applications. A first attempt to overcome this limitation, retaining at the same time some of the properties of interest of $X_{\gamma}$, is to substitute the $\eta_{I}$’s with non Gaussian $p$-stable random variables, $p\in(0,2)$, [16]. However, it may become a non trivial task to check which properties are preserved for this case. For example, besides self similarity, in [15] is proved that it is not possible to represent a $p$-stable stationary random process by a series like (1).

Here, we prove that for appropriate parameters $\gamma\leq\dfrac{d}{2}$ and $p$, if we consider $\{\psi_{I}\}_{I}$ a suitable wavelet basis, the series (1) stills converges a.s. in $\mathcal{D}^{\prime}(\mathbb{R}^{d})$, and if we change $\mathcal{I}_{\gamma}$ by a modified operator, then it converges to an ordinary process for the case $\dfrac{d}{2}<\gamma\leq\dfrac{d}{2}+1$. If $p=2$ the limit of the series (1) is self similar of parameter $\dfrac{d}{2}+\gamma$, and in the case $p\neq 2$, although its limit is not necessarily self similar, we can prove that the distribution function of the re-scaled process $a^{\frac{d}{2}+\gamma}X_{\gamma}(a\,.\,)$ is, in some sense, properly stochastically dominated. In the Gaussian case of $p=2$, the series of equation (1) converges to a fractional Gaussian noise, for which an integrated version of it gives the well known fractional Brownian motion, and its $d$-dimensional analogues, with their known “fractal” properties. We shall see that, for appropriate parameters $p$ and $\gamma$, that integrated versions of the process $X_{\gamma}$ have a graph with Hausdorff dimension greater than $d$, justifying the possible use of the process defined by (1) as a model of a fractal process still for $p\neq 2$.

II Auxiliary results and definitions.

II-A Function spaces, Fourier transforms and Wavelets.

In the following, if $p\in[1,\infty]$ and $\mu$ is the Borel measure over $\mathbb{R}^{d}$, the corresponding Lebesgue spaces of the equivalence classes of functions will be denoted by $L^{p}(\mathbb{R}^{d},d\mu)$, and if $\mu$ is the usual Lebesgue measure, we will write shortly $L^{p}(\mathbb{R}^{d})$. When $p=2$ it becomes a Hilbert space and the $L^{2}(\mathbb{R}^{d})$ inner product will be denoted by $\langle{\,.\,,.\,}\rangle$. If $x\in\mathbb{C}^{d}\;(d\in\mathbb{N})$ we will denote its usual norm by $\left|x\right|$ and the support of a function $f$ is defined by $supp(f)=\overline{\left\{x\,:f(x)\neq 0\right\}}$. The Schwartz class of functions $\mathcal{S}(\mathbb{R}^{d})$ is defined as the linear space of smooth functions rapidly decreasing at infinity, together with its derivatives. This means that $\phi\in\mathcal{S}(\mathbb{R}^{d})$ whenever $\phi\in C^{\infty}\left({\mathbb{R}^{d}}\right)$ and

[TABLE]

endowed with its usual topology. We will denote $\mathcal{D}(\mathbb{R}^{d})$ the space of functions which are in $C^{\infty}\left({\mathbb{R}^{d}}\right)$ and have compact support. Both spaces are topological vector spaces, for more details see [7], Chapter 2, p. 109. Their duals are denoted as: $\mathcal{S}^{\prime}(\mathbb{R}^{d})$ (Tempered distributions) and $\mathcal{D}^{\prime}(\mathbb{R}^{d})$ (distributions) respectively. Clearly: $\mathcal{D}(\mathbb{R}^{d})\subset\mathcal{S}(\mathbb{R}^{d})$ and then $\mathcal{S}^{\prime}(\mathbb{R}^{d})\subset\mathcal{D}^{\prime}(\mathbb{R}^{d})$. The Fourier Transform $\widehat{f}$ of $f\in\mathcal{S}(\mathbb{R}^{d})$ is defined as $\widehat{f}\left(\lambda\right)=\int\limits_{\mathbb{R}^{d}}{f\left(x\right)}e^{-2\pi i\lambda.x}dx\,.$ It is a known fact that $\widehat{f}$ also belongs to the space $\mathcal{S}(\mathbb{R}^{d})$. The Fourier transform can be defined, as usual as a linear map over $L^{1}(\mathbb{R}^{d})$, as an isometry on $L^{2}(\mathbb{R}^{d})$ or over the class of tempered distributions. The inverse Fourier transform $\mathop{f}\limits^{\vee}$ is defined in an analogous way. For further references on Fourier transforms and series, see for example [7].

Below, we will need a variant of the classical Shannon, Nyquist and Kotelnikov sampling theorem.

Theorem** II.1****.**

If $f\in L^{2}(\mathbb{R}^{d})$ is such that $supp(f)\subset[-x_{o},x_{o}]^{d}$ with $x_{o}<\dfrac{1}{2}$. Then there exists $\phi\in\mathcal{S}(\mathbb{R}^{d})$ such that

[TABLE]

Proof.

Let $\widetilde{f}(x)=\sum\limits_{k\in\mathbb{Z}^{d}}{f(x+k)}$ be the periodization of $f$. Then, $\widetilde{f}$ verifies

[TABLE]

and therefore $\widetilde{f}$ has Fourier series given by

[TABLE]

and then $\mathop{\lim}\limits_{R\to\infty}\sum\limits_{k\in D_{R}}{a_{k}e^{-2\pi ix.k}}=\widetilde{f}$ a.e. and in $L^{1}\left(\left[-\dfrac{1}{2},\dfrac{1}{2}\right]^{d}\right)$ (and in $L^{2}$) norm for a suitable domain ${D_{R}}\in\mathbb{R}^{d}$. Next, we can take $\phi\in\mathcal{S}(\mathbb{R}^{d})$ such that

[TABLE]

Defining $S_{R}(x)=\mathop{\phi}\limits^{\vee}(x)\left({\sum\limits_{k\in D_{R}}a_{k}e^{-2\pi ix.k}}\right)$, then $f=\widetilde{f}\;\mathop{\phi}\limits^{\vee}$ and $\mathop{\lim}\limits_{R\to\infty}\left\|{S_{R}-f}\right\|_{L^{1}(\mathbb{R}^{d})}=0$. This implies

[TABLE]

but (see e.g. [7], Exercise 3.6.4, p.236) $a_{k}=\widehat{f}(k)$, so that

[TABLE]

Then (2) follows immediately from this. ∎

In the following we will use fractional integral operators, for which some of their properties are reviewed. We begin with a definition ([8], Chapter 6, p. 2 or [17], Chapter 5, p. 117):

Definition II.2**.**

Let $0<\alpha<d$. For $f\in\mathcal{S}(\mathbb{R}^{d})$ we define its Riesz Potential:

[TABLE]

where $C_{\gamma}=\dfrac{{\pi^{d/2}\,2^{\alpha}\,\Gamma\left({\dfrac{\gamma}{2}}\right)}}{{\Gamma\left({\dfrac{d}{2}-\dfrac{\gamma}{2}}\right)}}$.

Riesz potentials have the following scaling property: for every $a\neq 0$: $\mathcal{I}_{\gamma}(f(a\,.\,))=|a|^{-\gamma}(\mathcal{I}_{\gamma}f)(a\,.\,)$, i.e. $\mathcal{I}_{\gamma}(f(a\,y\,))(x)=|a|^{-\gamma}(\mathcal{I}_{\gamma}f(y))(a\,x\,)$. A crucial result for this integral operator is the following, [8], Chapter 6, p.3 :

Theorem** II.3****.**

*(Hardy, Littlewood and Sobolev) Let $0<\gamma<d$, $1\leq p<q<\infty$ and $\dfrac{1}{q}=\dfrac{1}{p}-\dfrac{\gamma}{d}$ then:

(a) For all $f\in L^{p}(\mathbb{R}^{d})$, the integral that defines $\mathcal{I}_{\gamma}f$ converges a.e.

(b)If $p>1$ then*

[TABLE]

Note that, in the appropriate sense, the Fourier Transform of $\mathcal{I}_{\gamma}f$ is given by:

[TABLE]

and it is easy to check that for $f\in\mathcal{S}(\mathbb{R}^{d})$ and $\alpha+\beta<d$ then $\mathcal{I}_{\alpha}(\mathcal{I}_{\beta}f)=\mathcal{I}_{\alpha+\beta}(f)$. Furthermore, if $\rm\Delta\it f=\sum\limits_{j=1}^{d}{\dfrac{{\partial^{2}f}}{{\partial x_{j}^{2}}}}$ is the Laplacian of $f$ , then $\Delta(\mathcal{I}_{\gamma}f)=\mathcal{I}_{\gamma-2}f$. Finally, $\mathcal{I}_{\gamma}$ can be thought as defined by the convolution with the locally integrable function $k_{\gamma}(x)=\dfrac{1}{C_{\gamma}}\dfrac{1}{\left|{x}\right|^{d-\gamma}}$, and is formally self adjoint, in the sense that for every $f,g\,\in\,\mathcal{S}(\mathbb{R}^{d})$:

[TABLE]

Considering again $k_{\gamma}$, we can define a fractional integral operator for $f\in L^{p}(\mathbb{R}^{d})$, in the following way:

[TABLE]

The modified kernel $K_{\gamma}(x,y)=k_{\gamma}(x-y)-k_{\gamma}(y)$ is easier to control, and we sketch the proof of the following lemma:

Lemma** II.4****.**

*If $1<p<\infty$ and $0<d\left({1-\dfrac{1}{p}}\right)<\gamma<d\left({1-\dfrac{1}{p}}\right)+1$, then $K_{\gamma}(x,\,.\,)\in L^{p}(\mathbb{R}^{d})$ and moreover:

(i) There exists a positive constant $C_{p\,\gamma\,d}$ such that for each $x\in\mathbb{R}^{d}$:*

[TABLE]

(ii) For every $x,x^{\prime}\in\mathbb{R}^{d}$: $\left\|{K_{\gamma}(x,\,.\,)-K_{\gamma}(x^{\prime},\,.\,)}\right\|_{L^{p}(\mathbb{R}^{d})}=\left\|{K_{\gamma}(x-x^{\prime},\,.\,)}\right\|_{L^{p}(\mathbb{R}^{d})}$.

Proof.

(Sketch) Since

[TABLE]

The condition $d\left({1-\dfrac{1}{p}}\right)<\gamma$ gives the appropriate exponent for the boundedness of the first integral. In addition, since $\gamma<d\left({1-\dfrac{1}{p}}\right)+1$ and considering that for some positive constant $C$

[TABLE]

if $|y|>2|x|$, then the second integral is also finite. Hence, the map $x\mapsto\left\|{K_{\gamma}(x,\,.\,)}\right\|_{L^{p}(\mathbb{R}^{d})}$ is well defined and by a change of variable, we obtain that it is an homogeneous function depending only on $|x|$, from which assertion (i) follows. Assertion (ii) is also obtained by a change of variable. ∎

For fixed $x\in\mathbb{R}^{d}$, we note that in the Fourier domain $\mathcal{K}_{\gamma}$ can be characterized, in an appropriate sense, [2], Chapter 3, p. 45, by:

[TABLE]

Some formal manipulations show that from equations (5) and (7), for suitable parameters $\beta$ and $\gamma$, we have:

[TABLE]

and

[TABLE]

For $s\in\mathbb{R}$ another related operator $\mathcal{J}_{s}f$ is defined, formally, by its Fourier transform as:

[TABLE]

Theorem** II.5****.**

[8]**, Chapter 6, p. 8. If $s<0$ and $p\geq 1$, $\mathcal{J}_{s}:L^{p}(\mathbb{R}^{d})\longrightarrow L^{p}(\mathbb{R}^{d})$ defines a continuous linear operator, i.e. there exists $C_{p}>0$ such that

[TABLE]

For $1<p<\infty$, and $s\in\mathbb{R}$, we introduce the Sobolev spaces $H^{p}_{s}(\mathbb{R}^{d})$:

[TABLE]

These are Banach spaces of tempered distributions with the norm defined by $\left\|{f}\right\|_{H^{p}_{s}(\mathbb{R}^{d})}=\left\|{J_{s}f}\right\|_{L^{p}(\mathbb{R}^{d})}$. Moreover, [14], p.168, if $s\geq 0$, this norm is equivalent to $\left\|{f}\right\|_{L^{p}(\mathbb{R}^{d})}+\left\|{(|\,.\,|^{s}\widehat{f})^{\vee}}\right\|_{L^{p}(\mathbb{R}^{d})}$. Recalling again equation (7) the equivalence of norms for $K_{\gamma}(x,\,.\,)$ takes the following form which will be useful in the sequel:

[TABLE]

In the particular case $s=-d$, only when $p=2$, the $H^{p}_{s}(\mathbb{R}^{d})$ spaces coincide with the following $\mathcal{F}{L^{p}}_{w}$ spaces, which are introduced for auxiliary purposes.

Proposition** II.6****.**

For $1\leq p\leq 2$, the space

[TABLE]

is a Banach space with the norm defined by $\left\|{f}\right\|_{\mathcal{F}{L^{p}}_{w}}=\left\|{\widehat{f}(1+|\,.\,|^{2})^{-d}}\right\|_{L^{p}(\mathbb{R}^{d})}$. Moreover convergence in $\mathcal{F}{L^{p}}_{w}$ implies convergence in $\mathcal{S}^{\prime}(\mathbb{R}^{d})$.

Proof.

Observe that if we define $w(\lambda)=(1+|\lambda|^{2})^{-d}$, then $f\in\mathcal{F}{L^{p}}_{w}$ if and only if $\widehat{f}\in\;L^{p}(\mathbb{R}^{d},w\,d\lambda)$. Let $(f_{n})_{n\in\mathbb{N}}$ be a Cauchy sequence en $\mathcal{F}{L^{p}}_{w}$ which is equivalent to $(\widehat{f}_{n})_{n\in\mathbb{N}}$ being a Cauchy sequence in $L^{p}(\mathbb{R}^{d},w\,d\lambda)$, and then there exists a unique $g\in L^{p}(\mathbb{R}^{d},wd\lambda)$ such that $\left\|{\widehat{f}_{n}-g}\right\|_{L^{p}(\mathbb{R}^{d},wd\lambda)}\longrightarrow 0$, when $n\longrightarrow\infty$. We shall verify that $g\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ and therefore taking $f:=g^{\vee}\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ we are done. For this take $\dfrac{1}{p}+\dfrac{1}{q}=1$ and $m>d\left(1+2\dfrac{q}{p}\right)$ then by Hölder’s inequality:

[TABLE]

[TABLE]

thus, see e.g. [7], Exercise 2.3.1, p.122, $g\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ and therefore $f\in\mathcal{F}{L^{p}}_{w}$. Finally, $\mathop{f_{n}\longrightarrow f}\limits_{n\longrightarrow\infty}$ in $\mathcal{F}{L^{p}}_{w}$ if and only if $\mathop{\widehat{f}_{n}\longrightarrow\widehat{f}}\limits_{n\longrightarrow\infty}$ in $L^{p}(\mathbb{R}^{d},w\,d\lambda)$. Let $\varphi\in\mathcal{S}(\mathbb{R}^{d})$, then, if $\dfrac{1}{p}+\dfrac{1}{q}=1$, by definition of the Fourier Transform of a tempered distribution and Hölder’s inequality we get:

[TABLE]

[TABLE]

[TABLE]

which proves the last assertion of Proposition II.6. ∎

The following estimate for the $\mathcal{F}{L^{p}}_{w}$ norm will be useful in the sequel.

Lemma** II.7****.**

Let $1\leq p\leq 2$, then $L^{2}(\mathbb{R}^{d})\subset\mathcal{F}{L^{p}}_{w}$ and moreover, if $Q=\left[{-\dfrac{1}{4},\dfrac{1}{4}}\right)^{d}$, there exits a positive constant $C_{p\,d}$ such that for every $f\in L^{2}(\mathbb{R}^{d})$, $f=0$ a.e. in $Q^{c}$, the following inequality holds:

[TABLE]

Proof.

If $p=2$ the result is immediate. To prove the first assertion for $p\neq 2$, by Hölder’s inequality one has the following estimate

[TABLE]

For the second assertion, under these conditions we can write

[TABLE]

as in Theorem II.1 and therefore:

[TABLE]

[TABLE]

[TABLE]

since $(1+|\lambda|^{2})^{-d}\leq 2^{d}(1+|k|^{2})^{-d}(1+|\lambda-k|^{2})^{d}$ by Peetre’s inequality. If $\dfrac{1}{p}+\dfrac{1}{q}=1$, take $a_{k}(\lambda)=|\phi(\lambda-k)|^{\frac{1}{q}}$ and

[TABLE]

by Hölder’s inequality we get:

[TABLE]

finally, since there exists some positive constant $C$ such that:

[TABLE]

then equation (14) becomes

[TABLE]

[TABLE]

∎

II-B Some probability, stable laws and generalized random processes.

Let $(\Omega,\mathcal{F},\mathbf{P})$ be a probability space and $X$ a random variable variable defined on it. The distribution function of $X$ is defined, for $x\in\mathbb{R}$, as $F_{X}(x)=\mathbf{P}(X\leq x)$. If $\varphi$ is any Borel measurable real function, we will denote the expectation of $\varphi(X)$ with $\mathbf{E}(\varphi(X))$. The characteristic function of $X$ is $\Phi_{X}(\xi)=\mathbf{E}(e^{i\xi X})$. For $p\in(0,2]$, we say that a random variable $\eta$ is symmetric $p$-stable of parameter $\sigma>0$ if $\Phi_{\eta}(\xi)=e^{-\sigma^{p}|\xi|^{p}}$. A symmetric $p$-stable random variable $\eta$ will be denoted as $\eta\sim SpS$. When we write $F_{\eta_{p}}$ we shall be referring to the distribution function of such a random variable with $\sigma=1$. Note that $p=2$ corresponds to the Gaussian case and therefore $\eta\sim\mathcal{N}(0,\sigma)$. Let us review some basic properties of stable distributions, see [16], Chapter 1, p. 10, and [10], Chapter 0, p.5.

If $\eta_{1},\dots,\eta_{n}$ are independent and $\eta_{i}\sim SpS$, with parameter $\sigma_{i}$ then $\sum\limits_{i=1}^{n}\eta_{i}\sim SpS$, with $\sigma^{\prime}=\left\|(\sigma_{\eta_{i}})_{i}\right\|_{l^{p}}$. 2. 2.

Let $p<2$. If $\eta\sim SpS$ and $0<r<p$ then $(\mathbf{E}|\eta|^{r})^{1/r}=\,C_{r}\,\sigma_{\eta}$, where ${C_{r}}^{r}=\mathbf{E}|\eta_{p}|^{r}$, and $\mathbf{E}|\eta|^{r}=\infty$ for $r\geq p$.

Let $\mu$ be a non negative Borel measure on $\mathbb{R}^{d}$. We shall need a result on the a.s. convergence of random elements in $L^{r}(\mathbb{R}^{d},d\mu)$. This theorem is a particular case of a more general one in [10], Chapter 2.

Theorem** II.8****.**

Let $0<r<p<2$, $\{f_{j}\}_{j\in\mathbb{N}}\subset L^{r}(\mathbb{R}^{d},d\mu)$, and let $\{\eta_{j}\}_{j\in\mathbb{N}}\sim SpS$ be a sequence of independent and identically distributed random variables. Then the series $\sum\limits_{i=1}^{\infty}\eta_{i}f_{i}$ converges in $L^{r}(\mathbb{R}^{d},d\mu)$ a.s. if and only if

[TABLE]

Our results, are aimed at the construction of certain random variables taking values in $\mathcal{D}^{\prime}(\mathbb{R}^{d})$. In this case, every $\mathcal{D}^{\prime}(\mathbb{R}^{d})$- valued random variable, say $X$, takes the form of a random linear functional defined on $\mathcal{D}(\mathbb{R}^{d})$. Previously, we will also need to define the class of generalized random processes, of which these $\mathcal{D}^{\prime}(\mathbb{R}^{d})$- valued random variables are particular cases. Following [5], Chapter 3, p. 237, and [19], Chapter 4, p. 57, we will say that a generalized random functional is defined on $\mathcal{D}(\mathbb{R}^{d})$ if for every $\varphi\,\in\,\mathcal{D}(\mathbb{R}^{d})$ there is associated a real valued random variable $X(\varphi)=\langle{X,\varphi}\rangle$. In accordance with the usual specification of the probability distributions of a countable set of real random variables, given $n\in\mathbb{N}$, $\varphi_{1},\dots,\varphi_{n}\,\in\,\mathcal{D}(\mathbb{R}^{d})$ define the probability of the events, $\{a_{k}\leq\langle{X,\varphi_{k}}\rangle<b_{k}\},\,\,\;k=1,\dots,n\;,$ which will have to be compatible in the usual sense. On the other hand, linearity means that for any $a,b\in\mathbb{R}$, $\varphi,\,\psi\in\mathcal{D}(\mathbb{R}^{d})$: $\langle{X,a\varphi+b\psi}\rangle=a\langle{X,\varphi}\rangle+b\langle{X,\psi}\rangle\;\;\;\textrm{a.s.}$. For a comprehensive study on this topic, see [5]. In an analogous way to real valued random variables, for each $\varphi\in\mathcal{D}(\mathbb{R}^{d})$ we can calculate the characteristic function of the real random variable $\langle{X,\varphi}\rangle$, $\Phi_{\langle{X,\varphi}\rangle}(\xi)=\mathbf{E}(e^{i\xi\langle{X,\varphi}\rangle})$. In fact if $\xi=1$ and considering $\varphi$ as a variable, this gives the characteristic functional of $X$, $\Phi_{X}(\varphi)=\mathbf{E}(e^{i\langle{X,\varphi}\rangle})$, which completely determines its distributions as in the case of ordinary random processes. Finally, self-similarity for generalized random processes can be defined in the following analogous way to [19], p. 178: $X$ is self-similar if there exists a constant $\delta>0$ such that

[TABLE]

for every dilation factor $a>0$ and $\varphi\in\mathcal{D}(\mathbb{R}^{d})$. This means that $X$ is equivalent, in probability law, to $a^{r}X(\,.\,/a)$, for some appropriate constant $r$. In this context, we recall the Hausdorff dimension, see [3], Chapter 2, p. 21, of a subset $A$ of $\mathbb{R}^{d}$ denoted by $dim_{H}(A)$. Although self similarity is associated to the notion of “fractality”, the last one has not a precise meaning. However, subsets of $\mathbb{R}^{d}$ with non integer Hausdorff dimension are considered as displaying a fractal behaviour. A way for the study of the fractal behaviour of the graph of a function is the calculation of its Hausdorff dimension. Usually, the estimation of a lower bound for this value is calculated by potential methods, see [3], Chapter 2, p. 26, and [9], Chapter 10, p.132. An example is:

Lemma** II.9****.**

If $B$ is a compact subset of $\mathbb{R}^{d}$ and $\mathcal{G}\subset\mathbb{R}^{d+1}$ denotes the graph of a measurable function $f:B\longrightarrow\mathbb{R}$ and $\int\limits_{B}\int\limits_{B}(|x-x^{\prime}|^{2}+|f(x)-f(x^{\prime})|^{2})^{-\rho/2}\,dx\,dx^{\prime}<\infty$ then $dim_{H}(\mathcal{G})>\rho$.

Other related results will be introduced in the final section, for the estimation of the Hausdorff dimension of certain processes arising from the construction introduced in equation (1).

II-C Wavelets.

Let $\{\psi_{j\,k}^{e}\}_{j\in\mathbb{Z}\,k\in\mathbb{Z}^{d}\,e\in E}$, with $E=\{0,1\}^{d}$, be an orthonormal wavelet basis of $L^{2}(\mathbb{R^{d}})$, [14], Chapter 2. The Parseval identity for this case is:

[TABLE]

Therefore the norm $\left\|f\right\|^{2}_{L^{2}(\mathbb{R}^{2})}$ can be estimated from the wavelet coefficients $\langle{f,\psi_{j\,k}^{e}}\rangle$. Under some additional conditions, for example if the wavelet basis arises from a $r$-regular wavelet multirresolution approximation of $L^{2}(\mathbb{R}^{d})$, then, if $\{I_{j\,k}\}_{j\in\mathbb{Z},k\in\mathbb{Z}^{d}}$ denotes the family of dyadic cubes of $\mathbb{R}^{d}$, for some positive constants $c_{p},c_{p\,s},C_{p},C_{p\,s}$, we have the following estimations for the $L^{p}(\mathbb{R}^{d})$ and $H^{p}_{s}(\mathbb{R}^{d})$ norms respectively, [14], Chapter 6:

[TABLE]

and for $0\leq s\leq r$,

[TABLE]

In order to simplify the notation involving wavelet expansions we will sometimes omit the summation limits as in equations (17) and (18).

III Main Results.

III-A Convergence.

First, we prove an inequality involving the $l^{p}$ norm of the wavelet coefficients of a function. As a byproduct, this inequality implies one case of the Sobolev’s embeddings, see e.g. [1], Theorem 7.57.

Theorem** III.1****.**

Let $\{\psi_{j\,k}^{e}\}_{j\,k\,e}$ be an $r$-regular orthonormal wavelet basis, $1<p<2$ and $d\left({\dfrac{1}{p}-\dfrac{1}{2}}\right)<s<r$ then there exists a positive constant $C_{p\,s}$ such that:

[TABLE]

for all $f\in H^{p}_{s}(\mathbb{R}^{d})$. If $p=2$, the inequality (19) holds for $s\geq 0$.

Proof.

The case $p=2$ is immediate since $\left\|f\right\|_{L^{2}(\mathbb{R}^{d})}\leq\left\|f\right\|_{H^{2}_{s}(\mathbb{R}^{d})}$. If $1<p\leq 2$, the lower bound holds, since

[TABLE]

The upper bound is obtained splitting the sum:

[TABLE]

Then for each $e\in E$:

[TABLE]

since for fixed $j$, $I_{j\,k}\bigcap I_{j\,k^{\prime}}=\emptyset$ if $k\neq k^{\prime}$. The inner integrand can be rewritten as

[TABLE]

[TABLE]

by Hölder’s inequality with exponents $\dfrac{2}{p}$ and $\dfrac{2}{2-p}$ and since $s>d\left({\dfrac{1}{p}-\dfrac{1}{2}}\right)$. Hence

[TABLE]

[TABLE]

For the bound on the other term, we proceed similarly to the previous case:

[TABLE]

Therefore by by Hölder’s inequality with exponents $\dfrac{2}{p}$ and $\dfrac{2}{2-p}$, if

[TABLE]

we get

[TABLE]

[TABLE]

Combining equations (20) and (21) and since $E$ is finite we get the result. ∎

Now, we can prove one of the main results of this work.

Theorem** III.2****.**

Let $\{\psi_{j\,k}^{e}\}_{j\,k\,e}$ be an $r$-regular orthonormal wavelet series, with $d\left({\dfrac{1}{p}-\dfrac{1}{2}}\right)<\gamma\leq d\left({1-\dfrac{1}{p}}\right)$, $\dfrac{3}{4}\leq p\leq 2$, $\gamma<r$ and $(\eta_{j\,k\,e})_{j\,k\,e}$ a sequence of independent identically distributed random variables such that $\eta_{j\,k\,e}\sim SpS$. Then the series defined by

[TABLE]

converges a.s. in $\mathcal{D}^{\prime}(\mathbb{R}^{d})$. If $p=2$, the result remains true for $0\leq\gamma\leq\dfrac{d}{2}$.

Proof.

We shall prove the case $p<2$, the $p=2$ case is very similar using Parseval’s identity instead of Theorem III.1. Let $Q=\left[{\dfrac{-1}{4},\dfrac{1}{4}}\right)^{d}$, since $(\mathcal{I}_{\gamma}\psi_{j\,k}^{e})\mathbf{1}_{Q}\,\in\,L^{2}(\mathbb{R}^{d})$, then by lemma II.7,

[TABLE]

thus

[TABLE]

[TABLE]

But, if $e_{n}(x)=\mathbf{1}_{Q}(x)e^{i2\pi nx}$, a density argument applied to equation (6) gives:

[TABLE]

Therefore, by Theorem III.1, and taking $\gamma=s$ :

[TABLE]

[TABLE]

The last inequality holds by the Hardy-Littlewood and Sobolev Inequality with exponents $\dfrac{1}{r}-\dfrac{1}{p}=\dfrac{\gamma}{d}$. Note that the validity of this last step is granted since $\dfrac{4}{3}\leq p\leq 2$ and $d\left({\dfrac{1}{p}-\dfrac{1}{2}}\right)\leq\gamma\leq d\left({1-\dfrac{1}{p}}\right)$. Moreover $\left\|e_{n}\right\|_{L^{r}(\mathbb{R}^{d})}+\left\|e_{n}\right\|_{L^{p}(\mathbb{R}^{d})}$ is finite and constant in $n$. Thus from the definition of $\mathcal{F}{L^{p}}_{w}$ combined with equations (24), (23) and (22):

[TABLE]

[TABLE]

Taking any $1<r<p$, by Hölder’s inequality combined with equation (25):

[TABLE]

[TABLE]

then, by Theorem II.8,

[TABLE]

converges a.s. in $L^{r}(\mathbb{R}^{d},w\,d\lambda)$ and therefore $\sum\limits_{j\,k\,e}\eta_{j\,k\,e}{(\mathcal{I}_{\gamma}\psi_{j\,k}^{e})\mathbf{1}_{Q}}$ converges a.s. in $\mathcal{F}{L^{r}}_{w}$ and in $\mathcal{S}^{\prime}(\mathbb{R}^{d})$. With slight modifications, the same argument works with any translate of $Q$. Finally, to verify that $\sum\limits_{j\,k\,e}\eta_{j\,k\,e}{\mathcal{I}_{\gamma}\psi_{j\,k}^{e}}$ converges a.s. in $\mathcal{D}^{\prime}(\mathbb{R}^{d})$, take $\mathcal{Q}=\left\{{Q=\left[{\dfrac{-1}{4},\dfrac{1}{4}}\right)^{d}+\dfrac{k}{2},\,k\in\mathbb{Z}^{d}}\right\}$ , $\Omega^{\prime}$ with $\mathbf{P}(\Omega^{\prime})=1$ defined by

[TABLE]

and $\varphi\in\mathcal{D}(\mathbb{R}^{d})$. For fixed $Q\in\mathcal{Q}$, $\omega\in\Omega^{\prime}$ and $N,M\in\mathbb{N}$ we have

[TABLE]

and then

[TABLE]

for some $Q_{i}$ such that $supp(\varphi)\subset\mathop{\bigcup}\limits_{i=1}^{m}Q_{i}$ since $\varphi$ has compact support. The result follows from the convergence of $\langle{s_{N\,M\,Q_{i}}(\omega),\varphi}\rangle$ when $N,M\longrightarrow\infty$ for each $i=1\dots m$. ∎

Alternatively, considering $\gamma>\dfrac{d}{2}$ and the operators $\mathcal{K}_{\gamma}$ instead of $\mathcal{I}_{\gamma}$ we can prove:

Theorem** III.3****.**

Let $\{\psi_{j\,k}^{e}\}_{j\,k\,e}$ be an $r$-regular orthonormal wavelet series, $\dfrac{d}{2}<\gamma\leq d\left({1-\dfrac{1}{p}}\right)+1$, $1\leq p\leq 2$, $\gamma<r$ and $(\eta_{j\,k\,e})_{j\,k\,e}$ a sequence of independent identically distributed random variables such that $\eta_{j\,k\,e}\sim SpS$. Then, for each $x\in\mathbb{R}^{d}$ the series defined by

[TABLE]

converges almost surely. Moreover, $\{Y_{\gamma}(x)\}_{x\in\mathbb{R}^{d}}$ has a measurable version. If $p=2$, the result remains true for $\dfrac{d}{2}\leq\gamma\leq\dfrac{d}{2}+1$.

Remark.

Note that the range of validity of the result depends on the dimension $d$, since the restrictions imply that $1<\dfrac{2d}{d+2}<p\leq 2$ for $d\geq 2$.

Proof.

Recall the properties of the $p$ stable random variables reviewed in Section II-B. For each $x\in\mathbb{R}^{d}$, we can prove the convergence in $r$-mean ($r<p$) of the sum defining $Y_{\gamma}(x)$. By Theorem III.1, and taking any $s$ such that $d\left({\dfrac{1}{p}-\dfrac{1}{2}}\right)<s<\gamma-d\left({1-\dfrac{1}{p}}\right)$, since $\mathcal{K}_{\gamma}\psi_{j\,k}^{e}(x)=\langle{K_{\gamma}(x,\,.\,),\psi_{j\,k}^{e}}\rangle$ for some constant $C$. we obtain:

[TABLE]

since, recalling from Section II-A the Lemma II.4, and the equivalence of norms of $H^{p}_{s}(\mathbb{R}^{d})$ given by equation (11), one obtains:

[TABLE]

[TABLE]

The sum defining $Y_{\gamma}(x)$ converges a.s. since convergence in the $r$-mean of independent random variables implies a.s. convergence. Similarly to the previous bound, if $|x-x^{\prime}|<1$, by Lemma II.4 (ii) one gets:

[TABLE]

[TABLE]

[TABLE]

From this, applying Tchebychev’s inequality, it follows the stochastic continuity of $Y_{\gamma}(x)$, and then there exists a measurable version (Theorem 1, p.157 of [6]) of $\{Y_{\gamma}(x)\}_{x\in\mathbb{R}^{d}}$. ∎

III-B Self similarity analysis

Self similarity in the sense of equation (15) is broken if $p\neq 2$. However, the following results show that, in some sense, the rescaled versions of $X_{\gamma}$ are stochastically dominated. Furthermore, we may expect some kind of fractal behavior for an integrated version of $X_{\gamma}$, as the realizations of $Y_{\gamma}$ considering a Daubechies wavelet basis suggest, see Figures 1 and 2.

Theorem** III.4****.**

Under the same hypothesis of Theorem III.2, the generalized random process $X_{\gamma}$ defined by:

[TABLE]

is self similar if $p=2$, in the sense that for every $\varphi\in\mathcal{D}(\mathbb{R}^{d})$, $a^{\frac{d}{2}+\gamma}\langle{X_{\gamma},\varphi(a\,.\,)}\rangle$ has the same distribution function as $\langle{X_{\gamma},\varphi}\rangle$, and otherwise, for every $s>\gamma$, there exists a positive constant $C_{p\,s}$ such that the following bounds hold:

[TABLE]

[TABLE]

for every $a>0$, $\varphi\in\mathcal{D}(\mathbb{R}^{d})$ and $x>0$.

Remark

Note that in the case $p=2$ it is easy to verify that the limit process is a Gaussian fractional noise with characteristic functional $\phi_{X_{\gamma}}(\varphi)=e^{-\left\|\mathcal{I}_{\gamma}\varphi\right\|^{2}_{L^{2}(\mathbb{R}^{d})}}$, and that this stationary generalized random process has a spectral measure, [5], Chapter 3, given by $d\mu_{X_{\gamma}}(\lambda)=(2\pi)^{-2\gamma}\dfrac{d\lambda}{|\lambda|^{2\gamma}}$. However, if $p\neq 2$, the analogous result for the stable case does not hold, since $\phi_{X_{\gamma}}(\varphi)\neq e^{-\left\|\mathcal{I}_{\gamma}\varphi\right\|^{p}_{L^{p}(\mathbb{R}^{d})}}$, which corresponds to the case of fractional stable noise.

Proof.

Let $p<2$ and $\varphi\in\mathcal{D}(\mathbb{R}^{d})$. To prove equation (27) it is sufficient to analyze $\Phi_{X(\varphi(a\,.\,))}$, the characteristic function of the real random variable $a^{\frac{d}{2}+\gamma}\langle{X_{\gamma},\varphi(a\,.\,)}\rangle$. From the scaling property of $\mathcal{I}_{\gamma}$:

[TABLE]

Assume $\sigma=1$ with no loss of generality. Since the $\eta_{j\,k\,e}$’s are independent and identically distributed with characteristic function $\Phi_{\eta_{j\,k\,e}}(\xi)=e^{-|\xi|^{p}}$, then the sum defining $a^{\frac{d}{2}+\gamma}\langle{X_{\gamma},\varphi(a\,.\,)}\rangle$ has characteristic function given by:

[TABLE]

which corresponds to the distribution

[TABLE]

Then, the upper bound follows combining Theorem III.1 and the fact that $F_{\eta_{p}}$ is monotone. The lower bound is obtained similarly estimating the norm

[TABLE]

Finally, the case $p=2$ is obtained in an analogous way with equality due to Parseval’s identity for the orthonormal basis $\{\psi_{j\,k}^{e}\}_{j\,k\,e}$ of $L^{2}(\mathbb{R}^{d})$. ∎

The previous result is a consequence of the bound derived from Theorem III.1:

[TABLE]

For $x\in\mathbb{R}^{d}$, and taking a sequence $\varphi_{n\,x}\in\mathcal{D}(\mathbb{R}^{d})$ such that $\mathop{\varphi_{n\,x}\longrightarrow K_{\beta}(x,\,.\,)}$ in $L^{p}(\mathbb{R}^{d})$ as $n\longrightarrow\infty$, provided that $\gamma+\beta$ are as in Theorem III.3, we can interpret $Y_{\gamma+\beta}$ as an integrated observation of $X_{\gamma}$: $Y_{\gamma+\beta}(x)=\langle{X_{\gamma},K_{\beta}(x,\,.\,)}\rangle=\int\limits_{\mathbb{R}^{d}}K_{\beta}(x,y)X_{\gamma}(y)dy$, where these equalities are only formal. In fact $Y_{\gamma+\beta}(x)$ is a well defined ordinary random variable for each $x\in\mathbb{R}^{d}$. Recalling equation (8) and Section II-B, its characteristic function is given by

[TABLE]

which is the pointwise limit of the sequence of characteristic functions

[TABLE]

This is a consequence of the following bound, which again can be derived from Theorem III.1 with $s=\gamma$:

[TABLE]

[TABLE]

[TABLE]

The Lebesgue measure in $\mathbb{R}^{d+1}$ of a measurable version of $\{Y_{\gamma}(x)\}_{x\in\mathbb{R}^{d}}$ is zero. Let us bound, from below, the Hausdorff dimension of the graph $\mathcal{G}\subset\mathbb{R}^{d+1}$ of $Y_{\gamma}(x)$. As a consequence, we shall see that for suitable parameters, the Hausdorff dimension has non integer values.

Theorem** III.5****.**

Under the same hypothesis of Theorem III.3, then $\dfrac{3d}{2}-\gamma+1\leq dim_{H}(\mathcal{G})$ a.s., where $\mathcal{G}\subset\mathbb{R}^{d+1}$ is the graph of $Y_{\gamma}(x)$.

Proof.

The lower bound is a consequence of Lemma II.9. We shall prove that

[TABLE]

if $\rho<\dfrac{3d}{2}-\gamma+1$. Let us write $\Delta(x,x^{\prime})=Y(x)-Y(x^{\prime})$, then recalling equation (29), by Lemma II.4, (i) and (ii), one gets:

[TABLE]

[TABLE]

Hence, from equation (30) :

[TABLE]

[TABLE]

and therefore, if for example without loss of generality $B=\{|x|\leq 1\}$,

[TABLE]

[TABLE]

provided that $\rho<\dfrac{3d}{2}-\gamma+1$, which concludes the proof. ∎

Acknowledgment.

The authors thank the collaboration of Alexandre Chevallier, visiting student from the École Internationale des Sciences du Traîtement de l’Information, École d’Ingénieurs Mathématiques, for the computer simulations corresponding to Figures 1 and 2.