Marianne Clausel
email: clausel@univ-paris12.fr

Gaussian fields satisfying simultaneous operator scaling relations

In this paper we define a special class of group self-similar Gaussian fields. We present an harmonizable representation of m parameter group self-similar Gaussian field by utilizing Haar measure of this group. These fields have also stationary rectangular increments according to special directions linked to co reduction of matrices of the considered m parameter group.

Introduction

Random fields are a useful tool for modelling spatial phenomenon like environmental fields, including for example, hydrology, geology, oceanography and medical images. Many times the chosen model has to include some statistical dependence structure that might be present across the scales. Thus, an usual assumption is self-similarity (see [START_REF] Lamperti | Semi-stable stochastic processes Trans[END_REF]), defined for a random field {X(x)} x∈R d on R d by {X(ax)} x∈R d (f.d.)

= {a H X(x)} x∈R d for some H ∈ R (called the Hurst index). As usual, (f.d.)

= denotes equality of all finite-dimensional marginal distributions. The most famous example of self-similar processes is Fractional Brownian Motion (FBM) {B H (x)} x∈R d , introduced in 1940 by Kolmogorov (see [START_REF] Kolmogorov | Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum C[END_REF]) and first studied in the famous paper of Mandelbrot and Van Ness (see [START_REF] Mandelbrot | Fractional Brownian Motion[END_REF]).

Moreover in many cases, random fields have an anisotropic nature in the sense that they have different geometric characteristics along different directions (see, for example, Davies and Hall ([DH99]), Bonami and Estrade ( [START_REF] Bonami | Anisotropic analysis of some Gaussian models[END_REF]) and Benson,et al.([Ben06])). The classical notion of self-similarityby construction isotropic-has then to be changed in order to fit anisotropic situations. For this reason, an increasing interest has been paid in defining a suitable concept for anisotropic self-similarity. Many authors have developed techniques to handle anisotropy in the scaling : In [START_REF] Hudson | Operator-self-similar processes in a finite dimensionnal space Trans[END_REF] Hudson and Mason introduced operator-self-similar processes {X(t)} t∈R with values in R d . Moreover in [START_REF] Schertzer | Generalised scale invariance in turbulent phenomena[END_REF][START_REF] Schertzer | Physically based rain and cloud modeling by anisotropic, multiplicative turbulent cascades[END_REF] Schertzer and Lovejoy introduced a general concept of scaling with respect to a one-parameter group (which may be a matricial one) for random fields. These authors have in mind the particular case of linear case and its application to the study to atmospheric stratification.

In [START_REF] Kamont | On the Fractional Anisotropic Wiener Field Prob[END_REF], A.Kamont introduced a first example of anisotropic selfsimilar Gaussian field : Fractional Brownian Sheet (FBS). For any (H 1 , • • • , H d ) in (0, 1) d , FBS with Hurst indices (H 1 , • • • , H d )-denoted {B H1,••• ,H d (x)} x∈R dcan be defined through its harmonizable representation (see [START_REF] Ayache | Drap Brownien Fractionnaire Pot[END_REF]) :

B H1,••• ,H d (x) = R d (e i<x1,ξ1> -1) • • • (e i<x d ,ξ d > -1) |ξ 1 | H1+ 1 2 • • • |ξ d | H d + 1 2 d W ξ1,••• ,ξ d , (1)
where dW x1,•

, • • • , a d ) ∈ (R * + ) d        {B H1,••• ,H d (a 1 x 1 , • • • , x d )} x∈R d (f.d.) = {a H1 1 B H1,••• ,H d (x)} x∈R d . . . {B H1,••• ,H d (x 1 , • • • , a d x d )} x∈R d (f.d.) = {a H d d B H1,••• ,H d (x)} x∈R d (2)
Moreover, it follows from definition (1) that FBS admits stationary rectangular increments according to the coordinate axes. For example in the bidimensional case (d = 2) if we denote

∆ h1,h2 B H1,H2 (x 1 , x 2 ) = B H1,H2 (x 1 + h 1 , x 2 + h 2 ) -B H1,H2 (x 1 + h 1 , x 2 ) -B H1,H2 (x 1 , x 2 + h 2 ) + B H1,H2 (x 1 , x 2 ) then (Proposition 2 of [ALP02]) for any (x 1 , x 2 ) ∈ R 2 : {∆ h1,h2 B H1,H2 (x 1 , x 2 )} (h1,h2)∈R 2 (f.d.) = {∆ h1,h2 B H1,H2 (0, 0)} (h1,h2)∈R 2 .
Conversely, any Gaussian field {X(x)} x∈R d of the form

X(x) = R d (e i<x1,ξ1> -1) • • • (e i<x d ,ξ d > -1)φ(ξ)d W ξ1,••• ,ξ d (3) with min(1, |ξ 1 | 2 ) • • • min(1, |ξ d | 2 )|φ(ξ)| 2 dξ < +∞,
admits stationary rectangular increments according to the coordinate axes (see Section 3). Furthermore if, as in the case of FBS,

∀(a 1 , • • • , a d ) ∈ (R * + ) d , |φ(a 1 • • • a d ξ)| 2 = a -2H1-1 1 • • • a -2H d -1 d |φ(ξ)| 2 , ( 4
)
then the Gaussian field {X(x)} x∈R d defined by (3) satisfies properties (2).

Another model of anisotropic self-similar random field is the class of Operator Scaling Random Fields (OSRF) introduced by H.Biermé, M.Meerschaert and H.P.Scheffler in [START_REF] Biermé | Operator Scaling Stable Random Fields Stoch[END_REF]. These fields satisfy the following scaling relation :

∀a > 0, {X(a E x)} x∈R d (f.d.) = {a H X(x)} x∈R d . (5)
for some matrix E (called an anisotropy of the field) whose eigenvalues have a positive real part and some H > 0 (called an Hurst index of the field).

Remind that for any real a > 0, a E denotes the matrix

a E = exp(E log(a)) = k≥0 E k log k (a) k!
. Moreover H. (Re(λ))-Gaussian fields with stationary increments satisfying (5) can be defined in the following way

X(x) = R d (e i<x,ξ> -1)ρ(ξ) -(H+ T r(E) 2 ) d W ξ ,
where ρ is a (R d , E t ) pseudo-norm (see [START_REF] Lemarié-Rieusset | Projecteurs invariants, matrices de dilatation, ondelettes et analyses multi-résolutions[END_REF]) that is a continuous function defined on R d \ {0} with positives values satisfying :

∀ξ ∈ R d \ {0}, ρ(a E t ξ) = aρ(ξ).
Then, the main difficulty to overcome is to define a suitable (R d , E t ) pseudonorm. In [START_REF] Biermé | Operator Scaling Stable Random Fields Stoch[END_REF], for any matrix E whose eigenvalues have positive real parts, is proved that

ρ(ξ) = ( S0 ∞ 0 (1 -cos(< ξ, r E θ >)) dr r 2 dµ(θ))
is a (R d , E t ) pseudo-norm (S 0 denotes the unit sphere of R d for a well-chosen norm, and µ a finite measure on S 0 ). We will generalize this result using another approach based on Haar measure of an m parameter group.

Here, our purpose is to introduce another class of anisotropic Gaussian fields satisfying given simultaneous operator scaling relations : For any

(a 1 , • • • , a m ) ∈ (R * + ) m        {X(a E1 1 x)} x∈R d (f.d.) = {a H1 1 X(x)} x∈R d , . . . {X(a Em m x)} x∈R d (f.d.) = {a Hm m X(x)} x∈R d , . (6)
where E 1 , • • • , E m are m given pairwise commuting matrices. We now illustrate through an example the potential usefulness of this model.

Example 1. Let us consider the two commuting matrices

E 1 =   1 1 0 -1 1 0 0 0 0   , E 2 =   0 1 0 -1 0 0 0 0 1   .
For any θ ∈ R, denote R θ = cos(θ) -sin(θ) sin(θ) cos(θ) and remark that for any positive numbers a 1 , a 2 :

a E1 1 = a 1 R log(a1) 0 0 1 , a E2 2 = R log(a2) 0 0 a 2 .
As a consequence of Theorem 1 (see Section 4), for all (H 1 , H 2 ) ∈ (0, 1) 2 we can define an anisotropic field {X(x

)} x∈R 3 = {X(x 1 , x 2 , t)} (x1,x2,t)∈R 3 such that :    ∀a 1 ∈ R * + , {X(a E1 1 x)} x∈R 3 (f.d.) = {a H1 1 X(x)} x∈R 3 , ∀a 2 ∈ R * + , {X(a E2 2 x)} x∈R 3 (f.d.) = {a H2 2 X(x)} x∈R 3 . (7)
Let us make some comments about these two scaling properties of field {X(x)} x∈R 3 . Assume that x 1 , x 2 denote two space variables whereas t denotes times. Then, the first scaling property means that at fixed time {X(x)} x∈R 3 is a (maybe anisotropic) operator scaling Gaussian field. The second scaling property means that anisotropy of the field evolves throughout time. In fact, we defined a fixed time anisotropic Gaussian field whose anisotropy rotates with time.

Our objective is now to define such Gaussian fields. In the two following Sections, we present our approach. We first consider the special case of Gaussian fields satisfying simultaneous uncoupled relations.

A first example of field satisfying simultaneous operator scaling properties

We first consider a particular case.

Let (d 1 , • • • , d m ) ∈ N m such that d 1 + • • •+d m = d, (E 1 , • • • , E m ) m given matrices in (M d1 (R), • • • , M dm (R)) whose eigenvalues have positive real parts and (H 1 , • • • , H m ) in m =1 (0, ρ min (E )).
Combining the model of FBS and this of OSRF, one can easily define a Gaussian field

{X(x 1 , • • • , x m )} (x1,••• ,xm)∈R d 1 ו••×R dm satisfying the following simultaneous operator scaling relations : For any (a 1 , • • • , a m ) ∈ (R * + ) m        {X(a E1 1 x 1 , • • • , x m )} x∈R d (f.d.) = {a H1 1 X(x 1 , • • • , x m )} x∈R d . . . {X(x 1 , • • • , a Em m x m )} x∈R d (f.d.) = {a Hm m X(x 1 , • • • , x m )} x∈R d (8)
Indeed, the Gaussian field {X(x)} x∈R d can be defined through its harmonizable representation

X(x) = Re( R d 1 ו••×R dm m =1 ((e i<x ,ξ > -1)(e -i<y ,ξ > -1)|φ (ξ )| 2 )d W ξ ) (9)
where for any , φ denotes a (R d \ {0}, E t ) pseudo-norm. By construction, the Gaussian field {X(x)} x∈R d satisfies simultaneously the m operator scaling relationships (8). As FBS, this field does not admit stationary increments but satisfies a weaker stationarity property : It admits rectangular stationary increments according to some special directions. Before giving a precise statement about stationarity properties of the field {X(x)} x∈R d , we first recall the notion of Gaussian field with rectangular stationary increments. Let us begin by defining the notion of rectangular increments of a function :

Definition 1. Let M 1 , • • • , M m m subspaces of R d in direct sum, f a function defined on R d . For any x ∈ R d and any (h 1 , • • • , h m ) ∈ M 1 × • • • × M m define ∆ h1,••• ,hm f (x) = m =0 1≤i1<•••<i ≤m (-1) m-f (x + h i1 + • • • + h i ). with for = 0, f (x + h i1 + • • • + h i ) = f (x). Example 2. In the case m = 1, M 1 = R d ∆ h f (x) = f (x + h) -f (x). In the case m = d = 2, M 1 = R × {0}, M 2 = {0} × R, if (h, k) ∈ M 1 × M 2 ∆ h,k f (x 1 , x 2 ) = f (x 1 + h, x 2 + k) -f (x 1 + h, x 2 ) -f (x 1 , x 2 + k) + f (x 1 , x 2 ). Definition 2. Let M 1 , • • • , M m m subspaces of R d in direct sum. A Gaussian field {X(x)} x∈R d admits stationary rectangular increments according to the directions M 1 , • • • , M m if for any x ∈ R d {∆ h1,••• ,hm X(x)} (h1,••• ,hm)∈M1ו••×Mm (L) = {∆ h1,••• ,hm X(0)} (h1,••• ,hm)∈M1ו••×Mm .
Example 3. In the case m = 1 we recover the classical notion of a random field with stationary increments. Bidimensional FBS admits stationary rectangular increments according to the directions

M 1 = R × {0} and M 2 = {0} × R.
Here, in the example above, the Gaussian field defined by (9) admits stationary rectangular increments according to

M 1 = R d1 × • • • × {0 R dm }, • • • , M m = {0 R d 1 } × • • • × R dm .
We now use the special case introduced in this section in order to present a general approach to define Gaussian fields satisfying the m given scaling properties (6).

A general approach

In the general case, we will formulate the problem as in Section 2 and define a Gaussian field {X(x)} x∈R d which admits stationary rectangular increments according to some special directions (M 1 , • • • , M m ) in direct sum with m ≥ m. These special directions follow from the simultaneous reduction of matrices E 1 , • • • , E m (see Section 6.2 above) and then are invariant through these matrices, that is : For any

j in {1, • • • , m}, for any in {1, • • • , m } E j M ⊂ M . Subspaces (M 1 , • • • , M m )
will be called renormalization directions of Gaussian field {X(x)} x∈R d and have to be defined.

After definition of these renormalization directions, the Gaussian field {X(x)} x∈R d will be defined as follows. We are given a function φ with positive values satisfying the m following simultaneous properties : For any

a 1 , • • • , a m , for almost any ξ in R d        φ(a -E1 1 ξ) = a H1+ T r(E 1 ) 2 1 φ(ξ) . . . φ(a -Em m ξ) = a Hm+ T r(Em) 2 m φ(ξ) (10)
Furthermore, in order that integral (12) exists, we assume that

∀x ∈ R d , R d |φ(ξ)| 2 (min(1, | < ξ, x > | 2 ))dξ < +∞. (11)
Function |φ(ξ)| 2 is then called a spectral density of Gaussian field {X(x)} x∈R d . Thereafter, for any

x = (x 1 , • • • , x m ) ∈ M 1 × • • • × M m we set X(x) = Re   R d m =1 (e i<x ,ξ> -1)φ(ξ)d W ξ   . ( 12
)
The following proposition proves that the Gaussian field {X(x)} x∈R d satisfies the required properties Proof Indeed, since condition (11) is satisfied, {X(x)} x∈R d defined by (12) exists for any x ∈ R d . Denote

Proposition 1. Let (E 1 , • • • , E m ) m pairwise commuting matrices. Assume that there exist (M 1 , • • • , M m ) m subspaces of R d in direct sum, invariant through the action of matrices (E 1 , • • • , E m )
X 1 = Re R d h 1 (ξ)d W ξ , X 2 = Re R d h 2 (ξ)d W ξ ,
and remark that using corollary 6.3.2 of [START_REF] Samorodnitsky | Stable Non-Gaussian Random Processes Chapman Hall[END_REF], in the special case of Gaussian random variables

E(X 1 X 2 ) = Re R d h 1 (ξ)h 2 (ξ)dξ .
It is then sufficient to prove that {Y (x)} x∈R d defined as

Y (x) = R d m =1 (e i<x ,ξ> -1)φ(ξ)d W ξ ,
satisfies the required properties.

Then remark that the homogeneity properties (10) satisfied by φ imply that {Y (x)} x∈R d satisfies (6). Moreover, {Y (x)} x∈R d admits stationary rectangular increments according to directions

M 1 , • • • , M m . Indeed for any x ∈ R d , (h 1 , • • • , h m ) ∈ M 1 × • • • × M m , (k 1 , • • • , k m ) ∈ M 1 × • • • × M m E(∆ h1,••• ,h m Y (x)∆ k1,••• ,k m Y (x)) = m =1
(e i<h ,ξ> -1)(e -i<k ,ξ> -1)|φ(ξ)| 2 dξ.

This expression does not depend on x, then the result follows.

Let us illustrate this proposition by giving an explicit construction of a Gaussian field satisfying the two simultaneous operator scaling properties introduced in example 1 :

Example 4. The notations are those of example 1. Our objective is to define a Gaussian field satisfying the two simultaneous scaling properties (7). Having Proposition 1 in mind, we consider

X(x 1 , x 2 , t) = R 3
(e i(x1ξ 1 space +x2ξ 2 space ) -1)(e itξtime -1)φ(ξ 1 space , ξ 2 space , ξ time )d W ξ .

It will be required that function φ satisfies (10) and (11). Let us set

φ(ξ 1 space , ξ 2 space , ξ time ) = ξ 1 space cos(log(|ξ space | • |ξ time |)) -ξ 2 space cos(log(|ξ space | • |ξ time |)) |ξ space | H1+1 |ξ time | H2+ 1 2
One can easily check that φ fullfills the assumptions (10) and (11) of Proposition 1 and thus the Gaussian field {X(x)} x∈R 3 satisfies the two simultaneous scaling properties (7).

We now state the existence results proved in this paper.

Existence results

In order to state our existence results some hypotheses are needed about matrices (E 1 , • • • , E m ). We first recall what is real diagonalizable part of a matrix.

Real diagonalizable part of a matrix

As usual (see [START_REF] Hudson | Operator-self-similar processes in a finite dimensionnal space Trans[END_REF] ou [START_REF] Meerschaert | Limit Distributions for Sums of Independent Random Vectors Wiley Series in Probability ans Statistics[END_REF]), our main tool will be complete additive Jordan decomposition of a matrix. We refer to lemma 7.1, chap 9 of [Helg78] :

Proposition 2. Any matrix M of M d (R) can be decomposed into a sum of commuting real matrices M = D+S +N where D is a diagonalizable matrix in M d (R), S is a diagonalizable matrix in M d (C) with zero or imaginary complex eigenvalues, and N is a nilpotent matrix. Matrix D is called the real diagonalizable part of M .

Below, we give examples of real diagonalizable part of a matrix E.

1. If E =    λ 0 . . . 0 λ    or E =       λ 1 0 . . . . . . . . . 1 0 λ      
then E admits λ as unique eigenvalue and as real diagonalizable part D = λId where Id denotes the identity matrix.

If

E =    ∆ 0 . . . 0 ∆    or E =       ∆ I 2 0 . . . . . . . . . I 2 0 ∆       with ∆ = α -β β α
, then E admits exactly two conjugate complex eigenvalues

λ 1 = α + iβ, λ 2 = α -iβ. It implies that D = αId.
Recall that in general, a square real matrix is similar to a block diagonal matrix where each block is a square matrix of the form above. Thus, from the previous examples we can deduce the complete additive Jordan decomposition of any square real matrix. We now state our assumptions on matrices

E 1 , • • • , E m .

Assumptions on matrices

E 1 , • • • , E m Denote D 1 , • • • , D m the real diagonalizable parts of matrices E 1 , • • • , E m .
Hypotheses 4.1 We assume that

1. Matrices E 1 , • • • , E m are pairwise commuting matrices. 2. Matrices (D 1 , • • • , D m ) are linearly independent matrices in M d (R).
Under these assumptions we now state our main result :

Theorem 1. Assume that Hypotheses (4.1) are satisfied. Then for some C 0 ∈ (0, 1) depending only on matrices

E 1 , • • • , E m ,
∀(a 1 , • • • , a m ) ∈ (R * + ) m ,        {X(a E1 1 x)} x∈R d (f.d.) = {a H1 1 X(x)} x∈R d . . . {X(a Em m x)} x∈R d (f.d.) = {a Hm m X(x)} x∈R d (13)
Before any proof of this result, we need first to reformulate our problem in terms of group self-similarity. It will then allow us to use the concept of Haar measure in order to define a spectral density of Gaussian field {X(x)} x∈R d .

Reformulation of the problem in terms of group self-similarity

In [START_REF] Kolodynski | Group Self-Similar Stable Processes in R d[END_REF], S.Kolodynski and J.Rosinski defined the notion of group selfsimilar random field. We adapt this definition to our setting Definition 3. Let A a subgroup of Gl d (R) and χ a continuous mapping from

A into R * + . The random field {X(x)} x∈R d is A-self-similar with coefficient χ if ∀A ∈ A, {X(Ax)} x∈R d (f.d.) = {χ(A)X(x)} x∈R d . (14)
Remark 1. Remark that (see [START_REF] Kolodynski | Group Self-Similar Stable Processes in R d[END_REF]) the self-similarity coefficient of a Gaussian field is necessary an homomorphism from A into R * + .

Our problem can now be reformulated in terms of group self-similarity. Let us define the following m parameter group :

A = {a E1 1 • • • a Em m , (a 1 , • • • , a m ) ∈ (R * + ) m }. (15)
Proposition 3. The two problems are equivalent :

1. Find sufficient conditions on H 1 , • • • , H m for the existence of a Gaussian field {X(x)} x∈R d satisfying the simultaneous scaling properties (6).

2. Find sufficient conditions on homomorphism χ for the existence of a Gaussian field {X(x)} x∈R d A-self-similar with coefficient χ where A is the m parameter group defined by (15).

The proof of Proposition 3 is based on the complete description of the continuous homomorphisms from A into R * + . More precisely, one can easily check that (a more detailed proof may be found in [START_REF] Clausel | Etude de quelques notions d'irrégularité : le point de vue ondelettes[END_REF]): Proposition 4. If Hypotheses (4.1) are satisfied, the mapping

(a 1 , • • • , a m ) → a E1 1 • • • a Em m is a bicontinous isomorphism from (R * + ) m into A.
Since the homomorphisms from (R * + ) m into R * + are well-known, if Hypotheses (4.1) are satisfied, Proposition 4 directly implies that (a more detailed proof may be found in [START_REF] Clausel | Etude de quelques notions d'irrégularité : le point de vue ondelettes[END_REF]) Proposition 5. Let A a subgroup of Gl d (R) of the form Assume that Hypotheses (4.1) are satisfied. Then for any continuous homomorphism χ from A into R * + , there exists a unique

(H 1 , • • • , H m ) ∈ (R * + ) m such that ∀(a 1 , • • • , a m ) ∈ (R * + ) m , χ(a E1 1 • • • a Em m ) = a H1 1 • • • a Hm m .
Then any A-self-similar Gaussian field with coefficient χ, where χ is a continuous mapping from A into R * + , necessary satisfies

∀(a 1 , • • • , a m ) ∈ (R * + ) m , {X(a E1 1 • • • a Em m x)} x∈R d (L) = {a H1 1 • • • a Hm m X(x)} x∈R d for some (H 1 , • • • , H m ) ∈ (R * + ) m .
We then proved Proposition 3. Example 5. We now give examples of A-group self-similar Gaussian fields with coefficient χ.

• FBM is group self-similar with A = {aId} a∈R * + and χ(aId) = a H .

• FBS is group self-similar with A being the group of diagonal matrices with positive coefficients and χ(

   λ 1 0 . . . 0 λ n   ) = λ H1 1 • • • λ Hn n .
• Any OSRF is group self-similar with A = {a E , a > 0}, χ(a E ) = a H .

• The Gaussian field defined in Section 2 above is also group self-similar

with A = {    a E1 1 0 . . . 0 a Em m    , ∀i a i > 0} , χ(    a E1 1 0 . . . 0 a Em m   ) = a H .

Definition of the desired Gaussian field

As announced in Section 3, we now define the desired Gaussian field in two steps. We first define a suitable spectral density (see Section 6.1). Thereafter using simultaneous reduction of matrices E 1 , • • • , E m we define renormalization directions M 1 , • • • , M m (see Section 6.2). Finally in Section 6.3, we will use Proposition 1 to prove that the Gaussian field we just defined satisfies the required properties.

Definition of a suitable spectral density

We reformulate the properties required on spectral density using group A defined by (15) Proposition 6. Let φ a function defined on R d with positive values. The two properties are equivalent :

1. Function |φ(ξ)| 2 satisfies the m simultaneous relations (10).

Function |φ(ξ)|

2 is A homogeneous with coefficient χ 2 (•)| det(•)| that is : ∀A ∈ A, a.e.ξ ∈ R n , |φ((A -1 ) t ξ)| 2 = |χ(A)| 2 | det(A)||φ(ξ)| 2 , ( 16
)
with χ(a E1 1 • • • a Em m ) = a H1 1 • • • a Hm m .
We now define a suitable spectral density using a Haar measure µ A of group

A. Let |φ(ξ)| 2 = A χ(A) 2 | det(A)|| ψ(-A t ξ)| 2 dµ A (A), (17)
ψ ∈ H m+1 (R d ). As usual

H m+1 (R d ) = {f ∈ L 2 (R d ), |ξ| m+1 f ∈ L 2 (R d )}.
Then, the invariance property of any Haar measure of group A implies that : Function |φ(ξ)| 2 defined by (17) then satisfies the required properties of a spectral density. We now give some details about the definition of renormalization directions.

Definition of renormalization directions

In this Section, our purpose is to define m special subspaces of

R d M 1 , • • • , M m in direct sum, invariant through matrices E 1 , • • • , E m such that hypotheses
of Proposition 1 are fullfilled. These subspaces will be called renormalization directions and are invariant through group A. Proposition 1 then ensures the existence of a Gaussian field satisfying the required properties. Moreover (see Section 2) if

A = {diag(a E1 1 , . . . , a Em m ), (a 1 , • • • , a m ) ∈ (R * + ) m }, (18) with
(E 1 , • • • , E m ) ∈ M d1 (R) × • • • × M dm (R)
, whose eigenvalues have positive real parts (see Section 2), we can choose as renormalization directions

M 1 = R d1 × • • • × {0}, • • • , M m = {0} × • • • × R dm .
Now we want to extend the approach of Section 2 to the general case. In order to define renormalization directions, we simultaneously diagonalize matrices

A D = {a D1 1 • • • a Dm m , (a 1 , • • • , a m ) ∈ (R * + ) m }.
We can reduce simultaneously matrices of group A D :

Proposition 9. Assume that Hypotheses 4.1 are fullfilled. There exists an invertible matrix P such that Remark 3. In the proof of the existence of the desired Gaussian field, one can then replace A by A P , χ by χ P and ψ by ψ P .

A D = {P × diag(a ∆1 1 , . . . , a ∆m m , a D m+1 1 1 ..a D m+1 m m ) × P -1 , (a 1 , • • • , a m ) ∈ (R * + ) m } where
Proof To prove this result, we perform the changes of variable ξ = P t ζ, A P = P AP -1 . This Lemma leads us to consider a special case : Proposition 11. Notations are those of proposition 10. Assume that for any i ∈ {1, • • • , m}, -ρ min (∆ - i ) < H i < ρ min (∆ + i ), with ρ min (∆ - i ) = 0 if matrix ∆ - i does not exist. Then condition 20 is satisfied for χ P defined from

A P into R * + as χ(a F1 1 • • • a Fm m ) = a H 1 1 • • • a H m m .
Proof Proof detailed in [START_REF] Clausel | Etude de quelques notions d'irrégularité : le point de vue ondelettes[END_REF]. Lemma 1 and Proposition 11 then implies Theorem 1 (see [START_REF] Clausel | Etude de quelques notions d'irrégularité : le point de vue ondelettes[END_REF] for more details).

  and φ a function with positive values satisfying properties (10) and condition (11). Then {X(x)} x∈R d defined by (12) is with stationary rectangular increments according to directions M 1 , • • • , M m and satisfies the m simultaneous operator scaling relations (6).

  for any ∈ {1, • • • , m} and any 0 < H < C 0 ρ min (E ) there exists m ≥ m, m subspaces M 1 , • • • , M m of R d in direct sum and a Gaussian field {X(x)} x∈R d with rectangular stationary increments according to the directions M 1 , • • • , M m satisfying the m simultaneous operator scaling properties :

Proposition 7 .

 7 Function |φ(ξ)| 2 defined by (17), is an A t homogeneous function with coefficient χ 2 (•)| det(•)|. Remark 2. Proposition 7 does not prove that function |φ(ξ)| 2 defined by (17) is finite almost everywhere. The finitness of function |φ(ξ)| 2 comes from the existence of covariance function of the required Gaussian field.

  D 1 , • • • , D m using the following Proposition : Proposition 8. Let E 1 , • • • , E m be m pairwise commuting square matrices. Denote D 1 , • • • , D m their real diagonalizable parts. Then 1. Matrices D 1 , • • • , D m are pairwise commuting and then simultaneously diagonalizable. 2. Matrices E 1 , • • • , E m are all commuting with matrices D 1 , • • • , m . Definition of renormalization directions will follow from this simultaneous reduction of matrices D 1 , • • • , D m . The following notation will be needed : Notation 2 Let A the group defined by (15). Then denote

1.

  For any k ∈ {1, • • • , m}, ∆ k = any k ∈ {1, • • • , m}, matrices ∆ + k ( resp ∆ - k , D m+1 ) are diagonal matrices with positive coefficients (resp negative, unspecified).3. Matrices ∆ +k always exists for any k ∈ {1, • • • , m} whereas matrices ∆ - k , D m+1 can possibly not exist for some values of k or .Proof Proof detailed in in[START_REF] Clausel | Etude de quelques notions d'irrégularité : le point de vue ondelettes[END_REF]. Let us illustrate Proposition 9 through an example.Example 6. Let us consider the following groupA = {a E1 1 a E2 2 , (a 1 , a 2 ) ∈ (R * + ) 2 }, with E 1 = D 1 = A = A D , matrices D 1 et D 2 are diagonal and6.3 Proof of Theorem 1Consider function |φ(ξ)| 2 defined by (17) and the renormalization directions M 1 , • • • , M m defined in Section 6.2. We want to find sufficient conditions on χ in order that condition (11) of Proposition 1 is fullfilled. Let us first remark that :Lemma 1. Let A P the following m parameter groupA P = P -1 AP = {a F1 1 • • • a Fm m , (a 1 , • • • , a m ) ∈ (R * + ) m },and χ P defined as χ P (A P ) = χ(P A P P -1 ) for any A P ∈ A P . Condition (11) is satisfied iff for any x ∈ R d the following integralR d (min(1, | < x W + l , ζ > | 2 ) min(1, | < x W - l , ζ > | 2 ))|φ P t (ζ)| 2 dζ,(20) is finite with |φ P t (•)| 2 = |φ(P t •)| 2 = χ 2 P (A P )| det(A P )|| ψ P (-A t P •)| 2 dµ A P (A P ) and ψ P (•) = ψ(P •).

V ect < D 1 , D 2 >= V ect < ∆ 0 0 5 , 2∆ 0 0 1 >, with ∆ = 1 0 0 -2 .

To prove that matrices (D 1 , D 2 ) are linearly independent matrices remark that ( 1 5 , 2 1 ) are linearly independent vectors. Since

Hence we deduce that

Thus we recover the result of Proposition 9 with

Proposition 9 implies the following description of group A : Proposition 10. We use notations of Proposition 9. Group A is of the form

where for any , matrix F admits as real diagonalizable part the diagonal matrix diag(0,

Proof Proof detailed in [START_REF] Clausel | Etude de quelques notions d'irrégularité : le point de vue ondelettes[END_REF]. Proposition 10 will allow us to define renormalization directions. Let us define for any ∈ {1, • • • , m}, W

We then choose as renormalization directions

These subspaces are all invariant through matrices E 1 , • • • , E m . The sets V - l , V m+1 can be possibly equal to {0}. Notation 3 Denote M 1 , • • • , M m the m non zero sets within the spaces

are called non trivial renormalization directions. Example 7. In example 6 above, the renormalization directions are

In the following Section, we prove that this construction method is effective.


M Clausel
email: clausel@univ-paris12.fr

Lacunary Fractional Brownian Motion. 6

Keywords: Lacunary Gaussian Fields, Non uniqueness of the tangent field, Uniform irregularity, wavelets. 2000 MSC: 42C40 26B35

In this paper, a new class of Gaussian field is introduced called Lacunary Fractional Brownian Motion. Surprisingly we show that usually their tangent fields are not unique at every point. We also investigate the smoothness of the sample paths of Lacunary Fractional Brownian Motion using wavelet analysis.

Introduction

In [Falc02] and [START_REF] Falconer | The local structure of random processes[END_REF], Falconer generalized the concept of tangent field previously defined in [START_REF] Benassi | Elliptic Gaussian random processes[END_REF] by Benassi, Jaffard and Roux to study the local structure of Multifractional Brownian Motion. A field Y is a weak tangent field of the random field X at x 0 , if there exists two decreasing sequences of non negative real numbers converging to zero ρ n,1 and ρ n,2 such that,

{ X(x 0 + ρ n,1 t) -X(x 0 ) ρ n,2 } → {Y (t)}
where convergence is defined in an appropriate sense.

In many specific cases, such as Fractional Brownian Motion (FBM) or Multifractional Brownian Motion, the tangent field is essentially unique. More precisely, there exists a field Y such that the collection of all tangent fields of X at x 0 , called the tangent space of X at x 0 , equals < Y >, the family of non negative scalar multiple of Y . In that specific case, Y is called the tangent field of X at x 0 . Furthermore, under the assumption of the uniqueness of the tangent field, Falconer proves that tangent fields must be self-similar at the origin and must have stationary increments almost everywhere. In the special case of Gaussian fields for which tangent field is unique at every point, it implies that almost everywhere either the tangent field is "smooth" or it is locally "fractal" like a FBM. Natural questions appear : Can we define a Gaussian field for which there is no uniqueness of the tangent field at any point? What is the structure of the tangent space of such a field at every point? The Lacunary Fractional Brownian Motion (LBFM) provides a simple answer to these questions. At every point, LFBM admits several tangent fields. In addition, although LBFM is a Gaussian field, we can give two explicit examples of tangent fields unlike a FBM at every point. This last property results from asymptotic self-similarity properties thoroughly described in the following pages. Additionally, the sample paths of LFBM satisfy some specific regularity properties that we study using wavelets techniques.

Our plan will be as follows : In Section 2, we first briefly recall some definitions concerning FBM and Multi-scale FBM. Thereafter, we introduce two new classes of Gaussian fields : Infinite Scale Fractional Brownian Motion (ISFBM in short) and a special case of ISFBM : Lacunary Fractional Brownian Motion which satisfies paradoxical self-similarity properties. In Section 4 we review some well-known facts about tangent fields. Then we prove that LFBM is a Gaussian field admitting several tangent fields at every point. In this case, our main theorem describes the structure of tangent space at every point and the related asymptotic self-similarity properties according to the different scales. In Section 5 we investigate the sample paths properties of ISFBM. More precisely, we will see that different situations can appear : The sample paths regularity associated with a sequence of scales can be all the same for every sequence or on the opposite, can depend on the chosen sequence. In the example of LFBM, this field presents different regularity properties of the sample paths according to the different sequences of scales. Section 6 is devoted to the proofs of the results stated in Section 4 and Section 5.

Some multi-scale Gaussian models

The aim of this Section (see Section 2.1) is firstly to recall basic facts about an already known multi-scale Gaussian model : The Multi-scale Fractional Brownian Motion. This field was recently introduced in [START_REF] Bardet | Definition, properties and Wavelets analysis of Multiscale Fractional Brownian Motion[END_REF] by Bardet and Bertrand and generalizes in some sense FBM. In Section 2.2, we introduce two new classes of multi-scale Gaussian fields derived from Multiscale Fractional Brownian Motion : Infinite Scale Fractional Brownian Motion and Lacunary Fractional Brownian Motion whose definition is the main purpose of this paper.

The Multi-scale Fractional Brownian Motion

The Multi-scale Fractional Brownian Motion is one of the numerous generalizations of Fractional Brownian Motion. Let us recall that Fractional Brownian Motion, introduced by Kolmogorov in [START_REF] Kolmogorov | Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum[END_REF] and studied by B.Mandelbrot and J.Van Ness in [START_REF] Mandelbrot | Fractional Brownian Motion, fractional noises and applications[END_REF], is the continuous mean-zero Gaussian process {B H (t)} t∈R depending on a parameter H ∈ (0, 1), called the Hurst index, with covariance kernel

E(B H (t)B H (s)) = |s| 2H + |t| 2H -|s -t| 2H .
This process has important applications in modelling (see [START_REF] Beran | Statistics for Long-Memory processes[END_REF]), for example in hydrology or in finance. Its main properties are all related to its Hurst index H. More precisely, since FBM is self-similar, the sample paths regularity (related to the high frequency behavior) and the long memory properties of the increments (related to the low frequency behavior) are all driven by the same Hurst index H.

In some applications, one needs more flexibility and two Hurst indices are required in order to describe the low and the high frequency behavior of the process. Let us quote an example in finance given by Bardet and Bertrand in in [START_REF] Bardet | Definition, properties and Wavelets analysis of Multiscale Fractional Brownian Motion[END_REF]. If we are interested in the price process, statistical studies show that this process satisfies long memory properties (see [START_REF] Willinger | Stock market price and long-range dependence[END_REF]). Thus, at low frequencies, the Hurst index should be larger than 1/2. But financial theory assumes it is not possible to make profit without any risk. This implies that the price process is a semi-martingale and that the Hurst index should be H = 1/2 at high frequencies. So in that case one must introduce two different Hurst indices to describe the behavior of the process at high and low frequencies. Others examples can also be found in turbulence or in biomechanics (see [START_REF] Frisch | Turbulence, the legacy of A[END_REF] or [START_REF] Collins | Open loop and closed loop control of posture : a random walk analysis of center of pressure trajectories Exp[END_REF]).

For this reason, new models of Gaussian fields with a Hurst index varying as a piecewise function of frequencies have been introduced implicitely by Collins and al. [START_REF] Collins | Open loop and closed loop control of posture : a random walk analysis of center of pressure trajectories Exp[END_REF]) or explicitely by Benassi and Deguy (see [START_REF] Benassi | Multi-scale Fractional Motion : definition and identification Preprint LAIC[END_REF]). Both Collins and al. and Benassi and Deguy defined a process with two Hurst indices : One related to high frequencies, another related to low frequencies.

To study precisely the probabilistic properties of this class of processes, in [START_REF] Bardet | Definition, properties and Wavelets analysis of Multiscale Fractional Brownian Motion[END_REF] Bardet and Bertrand introduced a much more general model called Multi-scale Fractional Brownian Motion where the Hurst index is a piecewise function of frequency. They pick a finite number of frequencies,

ω 0 = 0 < • • • < ω J+1 = +∞, a finite number of Hurst indices H 0 , • • • , H J+1 , and a finite number of ampli- tudes σ 0 , • • • , σ J+1 . The Multi-scale Fractional Brownian motion {B ω,H,σ (t)} t∈R d is then defined by its harmonisable representation B ω,H,σ (t) = J ℓ=0 σ ℓ ω ℓ <|ξ|<ω ℓ+1 e it.ξ -1 |ξ| H ℓ + d 2 d W (ξ).
Then, they study the main properties of Multi-scale Fractional Brownian Motion : Regularity of the sample paths and long memory properties.

Let us be more precise about the distinction between the model proposed by Benassi and Deguy in [START_REF] Benassi | Multi-scale Fractional Motion : definition and identification Preprint LAIC[END_REF] and this proposed by Bardet and Bertrand in [START_REF] Bardet | Definition, properties and Wavelets analysis of Multiscale Fractional Brownian Motion[END_REF]. Following the literature on change point detection (see for e.g. [START_REF] Csörgö | Non parametric method for change point problems in Handbook of statistics[END_REF] or [START_REF] Basseville | Detection of abrupt changes-Theory and applications[END_REF]), the model proposed by Benassi and Deguy is At Most One Change (AMOC), whether this of Bardet and Bertrand admits a spectral density with More Than One Change. It is well known in the statistical community working on abrupt change detection that there is a gap between AMOC and More Than One Change problems.

The main goal of this paper is to generalize the Multi-scale model of Bardet and Bertrand in order to recover specific properties as non uniqueness of the tangent field. To this end we introduce a new model called Infinite Scale Fractional Brownian Motion (ISFBM). Thereafter, we focus on a special case of ISFBM : Lacunary Fractional Brownian Motion (LFBM) which satisfy the required property of non-uniqueness of tangent fields at any point.

Infinite Scale Fractional Brownian Motion and Lacunary Fractional Brow-

nian Motion Our first model, Infinite Scale Fractional Brownian Motion, is a refinement of Multi-scale Fractional Brownian Motion. Our aim is to define a field presenting different behaviors according to different family of scales. For this reason, we need to consider not a finite number of band of frequencies but an infinite one. This leads us to introduce ISFBM : Definition 2.1. Let H = (H j ) j be a sequence of reals numbers satisfying simultaneously the two following conditions :

   0 < lim inf j→+∞ H j , lim sup j→+∞ H j < 1, (2.1)
and σ = (σ j ) j be a bounded sequence of positive real numbers. The Infinity Scale Fractional Brownian Motion {B H,σ (t)} t∈R d with Hurst indices H = (H j ) j and amplitudes σ = (σ j ) j is the real valued Gaussian field with stationary increments whose harmonizable representation is given by

B H,σ (t) = +∞ ℓ=0 σ ℓ D ℓ e itξ -1 |ξ| H ℓ + d 2 d W (ξ), (2.2)
where D 0 = B d (0, 1) = {ξ, |ξ| < 1} and for all ℓ ≥ 1,

D ℓ = {ξ, 2 ℓ-1 < |ξ| < 2 ℓ }.
Remark In some specific cases, the ISFBM coincides with already known Gaussian fields. We give some examples below :

Y H,λ (t) = 0≤|ξ|<1 e itξ -1 |ξ| h ℓ (t)+ d 2 d W (ξ) + +∞ ℓ=0 λ ℓ ≤|ξ|<λ ℓ+1 e itξ -1 |ξ| h ℓ (t)+ d 2 d W (ξ).
If λ = 2 and for all ℓ, h ℓ (•) ≡ H ℓ , we recover the Infinity Scale Fractional Brownian Motion {B H,σ (t)} t∈R d with Hurst indices H = (H j ) j and amplitudes σ ≡ 1.

In the following, we will not focus on the case σ ≡ 1. On the opposite, we will be especially interested in another special case : The LFBM. In that case, we allow some values of the amplitude sequence σ = (σ j ) j to vanish. Thus we recover some very special properties, as non unicity of tangent fields, reflecting that the behavior of LFBM is different according to the different families of scales. Our aim is to introduce a Gaussian field admitting different behaviors according two different families of scales related to two different Hurst indices. We are given a first Hurst index H ∈ (0, 1). Our approach consists in introducing a first family of frequencies (denoted 2 ℓn in the following) and a first Gaussian field

X 1 (x) = +∞ n=0 D ℓn e ixξ -1 |ξ| H+ d 2 d W ξ .
At frequencies 2 ℓn , the behavior of this field is clearly driven by H. If the sequence 2 ℓn is sufficiently lacunary, we will prove that there exists a sequence of intermediary frequencies 2 jn -depending both on the lacunarity degree τ of the sequence 2 ℓn and on H-where the influence of frequencies located at the scale 2 ℓn is minimal. We thus can add to the previous Gaussian field another Gaussian part :

X 2 (x) = +∞ n=0 Dj n e ixξ -1 |ξ| H+ d 2 d W ξ .
The Hurst index H is well-chosen to ensure that X 2 is always negligible with respect to X 1 except at scales 2 -jn and is related to H, τ . We then define a new model-Lacunary Fractional Brownian Motion-which satisfies all the required properties. Let us be more precise :

Definition 2.2. Let τ > 1 and H ∈ (0, 1). Define H as H = τ H 1 -H + τ H . (2.4)
Define the two following sequences (j n ) n∈N et (ℓ n ) n∈N by j 0 = 1 and for all integer n ≥ 1 as

ℓ n = H H j n + τ log(j n ), j n+1 = 1 -H 1 -H ℓ n + τ log(ℓ n ).
(2.5)

The Lacunary Fractional Brownian Motion with lower Hurst index H and lacunary index τ is the real valued Gaussian field with stationary increments whose harmonizable representation is given by :

B H,τ (x) = +∞ n=0 Dj n e ixξ -1 |ξ| H+ d 2 d W ξ + +∞ n=0 D ℓn e ixξ -1 |ξ| H+ d 2 d W ξ . (2.6)

Numerical simulations

We now illustrate through numerical simulations the multiscale behavior of LFBM. To this end, let X be a LFBM with lower Hurst index H = 0.3 and 3) for N = 2000, ∆ N = 1/2000 using the explicit expression of the covariance and the circulant matrix embedding method . Recall that this method consists of embedding the covariance matrix of the Gaussian vector (X(∆ N ),

lacunary index τ = 2. A discretized trajectory (X(∆ N ), • • • , X(N ∆ N )) is nu- merically obtained (see Figure
• • • , X(N ∆ N )) in a non-negative definite matrix R of size M ≥ 2(N -1)
which is a circulant one (see Dietrich and Newsam [DN97], Wood and Chan [START_REF] Wood | Simulation of stationary Gaussian processes in [0; 1] d[END_REF], Beran [START_REF] Beran | Statistics for Long-Memory processes[END_REF], Davies and Harte [START_REF] Davies | Tests for Hurst effect[END_REF]). We also refer to the survey of Bardet, Lang, Oppenheim, Phillipe and Taqqu (see [START_REF] Bardet | Taqqu Generators of long-range dependent processes: A survey In Doukhan Oppenheim Taqqu[END_REF]) for an overview about numerical simulations of Gaussian processes. Our code is a slight adaptation of the generator of Multi-scale Fractional Brownian Motion furnished to us by Jean-Marc Bardet.

: A discretized path (X(∆ N ), • • • , X(N ∆ N )) with H = 0.3 and τ = 2
In Figure 4, in order to vizualize the two different behaviors related to the two characteristic family of scales 2 -jn and 2 -ℓn involved in the definition of the LFBM X, we represent for different values of h, a discretized trajectory

(X(h∆ N ), • • • , X(hN ∆ N ) -X(h(N -1)∆ N )
) of the increments of the Gaussian process X.

In this case, N = 2000, H = 0.5, τ = 1.5 and then H = 0.6. The increments h take the values 2 -j1 , 2 -ℓ1 , 2 -j2 , 2 -ℓ2 and are linked to the two families of characteristic scales. We see that two regimes begin to appear (compare the first and the third figure and then the second and fourth figure) related to the two different characteristic families of scales and the different Hurst indices H and H. In this paper, our aim is to state and to prove in a proper way the special self-similarity and sample paths properties satisfied by LFBM. In next Section we investigate the self-similarity properties of LFBM.

Self-similarity properties of Lacunary Fractional Brownian Motion

The aim of this Section is to state the main result of this paper : LFBM admits several tangent fields at any point (see Section 4.2). To this end we need some prerequisites about tangent fields and tangent spaces of a Gaussian field (see Section 4.1).

Tangent spaces and tangent fields

Here, we provide the reader with some material about tangent fields and tangent spaces of a Gaussian field. All that follows can be found with more details in [Falc02,[START_REF] Falconer | The local structure of random processes[END_REF]. The notion of tangent field was initially introduced in [BJR97] by Benassi, Jaffard, Roux related to this of local asymptotic selfsimilarity. In [Falc02] and [START_REF] Falconer | The local structure of random processes[END_REF]

) n such that lim n→+∞ X(x 0 + ρ 1,n t) -X(x 0 ) ρ 2,n t∈R d = {Y (t)} t∈R d . (4.1)
where lim means the convergence in distribution on the space of continuous functions endowed with the topology of the uniform convergence on compact sets.

In the special case where ρ 2,n = ρ H 1,n for H ∈ (0, 1), we recover a weak notion of local asymptotic self-similarity : Definition 4.2. Let (ρ n ) n∈N be a decreasing sequence of non negative real numbers converging to zero. A random field {X(x)} x∈R is locally asymptotically self-similar with exponent H at x 0 according the family of scales

(ρ n ) n∈N if the random field X(x 0 + ρ n u) -X(x 0 ) ρ H n u∈R
has a non trivial limit when n tends to +∞.

To define formally the concept of uniqueness of tangent field, Falconer needs to use the notion of tangent space. The tangent space of a random field {X(t)} t∈R d , written T an(X, x 0 ), is the aggregate of all the tangent fields of {X(t)} t∈R d at x 0 .

One can remark that if λ is a positive constant and if {Y (t)} t∈R d is a tangent field to the random field {X(t)} t∈R d then {λY (t)} t∈R d is also a tangent field to the random field {X(t)} t∈R d . The tangent field to X at x 0 is said to be unique if

T an(X, x 0 ) = {λY ′ , Y ′ (L) = Y },
for some random field Y .

One can also say that {Y (t)} t∈R d is the unique tangent field to the random field {X(t)} t∈R d . The random field X admits several tangent fields at x 0 if there is no uniqueness of the tangent field to X at x 0 .

In the special case of Gaussian fields, Falconer proved the following result (Corollary 3.10 of [START_REF] Falconer | The local structure of random processes[END_REF]) which enables to describe the local structure of the tangent space at almost every point : Theorem 4.1. Let X be a Gaussian field with continuous sample paths. For almost all x 0 in R d at which X has a unique tangent field Y x0 then the following alternative holds :

1. Either there exists a d-dimensional Gaussian random variable Z x0 such that almost surely, Y x0 (t) = t.Z x0 for all t in R d .
2. Or there exists H ∈ (0, 1) such that for all t = 0, {Y x0 (rt)} r∈R is a constant multiple of index H Fractional Brownian Motion.

In short, in the case of a Gaussian field admitting at each point a unique tangent field, this tangent field is either 'smooth' or 'fractal' like a FBM. This result comes from two properties of tangent spaces proved in [Falc02, Falc03] : the tangent space of a random field is a.e. shift invariant and always scaling invariant.

One can wonder what kind of behavior can be expected if these assumptions are not satisfied. Using LFBM defined above, we will prove in next section that we can define a Gaussian field admitting several tangent fields at each point and that in that case the tangent fields could be unlike a FBM. Further, we also prove-since we exhibit two tangent fields of LFBM which are not self similar-that in the general case tangent fields are not self-similar.

Tangent fields of Lacunary Fractional Brownian Motion

Now, we state the two central results of this paper concerning the non uniqueness at any point of the tangent field of the LFBM defined by Equation (2.6).

Proposition 4.2.

1. The LFBM {B H,τ (x)} x∈R d defined by Equation (2.6) is asymptotically locally self-similar with exponent H at every point x 0 according to the family of scales (ρ n,1 ) n∈N = (2 -ℓn ) n . Moreover, for all

x 0 ∈ R d lim n→+∞ B H,τ (x 0 + ρ n,1 t) -B H,τ (x 0 ) ρ H n,1 t∈R d = { B H (t)} t∈R d ,
where lim means the convergence in distribution on the space of continuous functions endowed with the topology of the uniform convergence on compact sets and

B H (t) = 1 2 <|ξ|<1 e itξ -1 |ξ| H+ d 2 d W (ξ). ( 4
) n∈N = (2 -jn ) n . Moreover, for all x 0 ∈ R d lim n→+∞ B H,τ (x 0 + ρ n,2 t) -B H,τ (x 0 ) ρ H n,2 t∈R d = { B H (t)} t∈R d ,
where lim means the convergence in distribution on the space of continuous functions endowed with the topology of the uniform convergence on compact sets and Let us make some comments about these self-similarity properties of LFBM. Roughly speaking, LFBM 'looks like' the Gaussian field { B H (t)} t∈R d when being considered at the scales (ρ n,1 ) n∈N = (2 -ℓn ) n whereas it 'looks like' the Gaussian field { B H (t)} t∈R d when being considered at the scales (ρ n,2 ) n∈N = (2 -jn ) n . From the regularity point of view, this non uniform behavior of LFBM according to the different scales is reflected in its sample paths properties studied in next Section.

B H (t) = 1 2 <|ξ|<1 e itξ -

Sample paths properties of Lacunary Fractional Brownian Motion

Hereafter we investigate the sample paths properties of LFBM. In Section 5.1, we first give a result about uniform Hölderian regularity of a more general model : ISFBM. Then, in Section 5.2 we focus on the special case of LFBM. We illustrate the non-uniformity in scales of sample paths of LFBM stating our results in terms of uniform irregularity.

Regularity of the sample paths of the Infinite Scale Fractional Brownian Motion

We first study the regularity of the sample paths of LFBM. In fact, our result can be stated for a more general model than LFBM : ISFBM. But before an accurate study of uniform regularity of the sample paths of ISFBM, we first precise the regularity concept we used. This is done in Section 5.1.1.

Uniform Hölderian regularity

The usual concept for uniform regularity is uniform Hölderian regularity. Uniform Hölderian regularity of a function can be characterized by its belongness to classical Hölder spaces C α (R d ) :

Definition 5.1. A bounded function f belongs to C α (R d ) with α ∈ (0, 1) if there exists a real positive number r 0 and a positive constant C 0 such that for every scale 0 < r ≤ r 0 , sup

|h|≤r sup x∈R d |f (x + h) -f (x)| ≤ C 0 r α .
A bounded function f is said to be uniformly Hölder if for some

ε 0 > 0, f ∈ C ε0 (R d ).
In the following, we will be interested in local sample paths properties of ISFBM.

To state our result, we need a notion of locally uniform Hölder regularity.

Definition 5.2. The locally bounded function f belongs to C α loc (R d ) with α ∈ (0, 1), if for any ϕ ∈ D(R d ), the function ϕf belongs to C α (R d ). One can define the local uniform Hölder exponent of function f as :

H loc (f ) = sup{α > 0, f ∈ C α loc (R d )}.
Our result of local uniform Hölder regularity of the sample paths of ISFBM is based on wavelets technics. Indeed, discrete wavelet transform and multiresolution analysis are particularly efficient tools to study the uniform Hölderian regularity of a function (see e.g. [START_REF] Meyer | Ondelettes et opérateurs Hermann[END_REF]). Hereafter, we review a result binding the regularity of a function and its wavelet coefficients. We first review some definitions and notations about wavelets (see for more details for example [START_REF] Mallat | A wavelet tour of signal processing[END_REF][START_REF] Meyer | Ondelettes et opérateurs Hermann[END_REF]).

Under some general assumptions, there exists 2 d -1 functions (ψ

(i) ) 1≤i<2 d , called wavelets, such that {ψ (i) (2 j x -j) : 1 ≤ i < 2 d , k ∈ Z d , j ∈ Z} is an orthogonal basis of L 2 (R d ). Any function f ∈ L 2 (R d ) can be decomposed as follows f (x) = j∈Z k∈Z d 1≤i<2 d c (i) j,k ψ (i) (2 j x -k),
where c

(i) j,k = 2 dj R d f (x)ψ (i) (2 j x -k) dx.
Let us remark that we do not choose the L 2 normalization for the wavelets, but rather an L ∞ normalization, which is better fitted to the study of Hölder regularity. If (ψ (i) ) i ∈ C r (R d ) and if (∂ α ψ (i) ) i (|α| ≤ r) have fast decay, the multiresolution analysis is said to be r-smooth. We always suppose that the multiresolution analysis is r-smooth with r > α.

A dyadic cube of scale j is a cube of the form

λ = [ k 1 2 j , k 1 + 1 2 j ) × • • • × [ k d 2 j , k d + 1 2 j ),
where k = (k 1 , . . . , k d ) ∈ Z d . From now on, wavelets and wavelet coefficients will be indexed with dyadic cubes. Since i takes 2 d -1 values, we can assume that it takes values in {0, 1} d -(0, . . . , 0). In the sequel we use the following notations

• λ = λ(i, j, k) = k 2 j + i 2 j+1 + [0, 1 2 j+1 ) d , • c λ = c (i) j,k , • ψ λ = ψ (i) (2 j -k).
We state the usual wavelet criterium for uniform Hölderian regularity (see [START_REF] Meyer | Ondelettes et opérateurs Hermann[END_REF]).

Theorem 5.1. Let α ∈ (0, 1), f ∈ C α (R d ), if and only if there exists C 0 > 0 such that ∀j ≥ 0, sup

|λ|=2 -j |c λ | ≤ C 0 2 -αj .
(5.1)

We now take an interest in the sample paths properties of ISFBM. Next Section is devoted to the statement of our regularity result.

Sample paths Hölderian regularity of Infinite Scale Fractional Brownian

Motion Now, we consider the local Hölderian regularity properties of the sample paths of ISFBM. When σ ≡ 1, ISFBM is a special case of GMBM. Therefore we could expect that, as in the case of GMBM (see [START_REF] Ayache | Generalized Multifractional Brownian Motion : definition and preliminary results[END_REF]), the local properties of the sample paths are given by the high frequencies of the harmonizable representation. The following proposition shows that this property is satisfied.

Proposition 5.2. For any integer j, let τ j = -log 2 (σ j ) j with τ j = +∞ if

σ j = 0. Set H = lim inf j→+∞ (H j + τ j ).
The local uniform Hölder exponent H loc of the Infinite Scale Fractional Brownian Motion {B H,σ (x)} x∈R d with Hurst indices H = (H j ) j and amplitudes σ = (σ j ) j satisfies almost surely

H loc (B H,σ ) = H. (5.2)
This Proposition is proved in Section 6.2 using wavelets technics and the wavelet criterion stated in Theorem 5.1 above. Now we focus on the special case of LFBM. Our aim is to understand how the non-uniform behavior of LFBM according to the different scales is reflected in its sample paths properties. To this end we need to complete our study and investigate the irregularity properties of LFBM.

Study of the uniform irregularity of the sample paths in the special case of

Lacunary Fractional Brownian Motion Here our aim is to illustrate in a relevant way the specific sample paths properties of LFBM using adapted concepts of uniform regularity. In a classical way, we are interested in the behavior of the so-called modulus of continuity of LFBM r → sup on any compact K.

Remark that α coincides with the uniform exponent of the LFBM on the compact K and has already been estimated in Section 5.1. The exponent α is related to another concept of uniform regularity than the classical one : Weak uniform Hölderian regularity defined in Section 5.2.1. The estimation of α (see Section 5.2.2) uses the wavelet criterion for weak uniform irregularity stated in Section 5.2.1.

Alternative definitions of uniform regularity

We refer to [START_REF] Clausel | More about uniform irregularity : the wavelet point of view[END_REF] for all the material of this Section. We first introduce the concept of uniform irregularity : Definition 5.3. A bounded function f is said to be uniformly irregular with exponent β if there exists a positive constant C 0 and a positive real number r 0 such that for every scale r in (0, r 0 )

sup |h|≤r sup x∈R d |f (x + h) -f (x)| ≥ C 0 r β .
A weak notion for uniform regularity can be associated with the concept of uniform irregularity : Definition 5.4. A bounded function f is said to be weakly uniformly Hölder of exponent β in (0, 1)

written f ∈ C β (R d ) if f ∈ I β (R d ) i.e. if
x∈R d |f (x + h) -f (x)| ≤ Cr β n .
The uniform irregular exponent of function f is defined as

I(f ) = inf{β > 0, f ∈ I β (R d )} = sup{β > 0, f ∈ C β (R d )}.
Remark then that

I(f ) = lim sup r→0 log( sup |h|≤r sup x∈R d (|f (x + h) -f (x)|) log(r) .
whereas

H(f ) = lim inf r→0 log( sup |h|≤r sup x∈R d (|f (x + h) -f (x)|) log(r) .
The simultaneous knowledge of the uniform Hölder exponent and the uniform irregular exponent of a bounded function f gives us a lower and an upper bound at r = 0 of the modulus of continuity of f , ω f , defined as

ω f (r) = sup |h|≤r sup x∈R d (|f (x + h) -f (x)|).
Since we are interested in the local behavior of sample paths, we need the following definitions : Definition 5.5. The locally bounded function f is said to be locally weak uniformly Hölder with exponent β ∈ (0, 1),

written f ∈ C β loc (R d ), if for any ϕ ∈ D(R d ), the function ϕf is C β (R d ).
The function f is said to be locally uniformly irregular with exponent exponent β ∈ (0, 1),

written f ∈ I β loc (R d ), if f ∈ C β loc (R d ).

The local irregularity exponent is defined as

I loc (f ) = sup{β, f ∈ C β loc (R d )}.
To study the irregularity properties of the sample paths of LFBM, we need to use a wavelet criterion for weak uniform Hölder regularity. In [START_REF] Clausel | More about uniform irregularity : the wavelet point of view[END_REF] is proved the following result Theorem 5.3. Let β ∈ (0, 1).

1. If f ∈ C β (R d ) then :
For every C > 0, there exists a strictly increasing sequence of integers (j n ) n ∈ N such that :

∀ n ≥ 0, ∀ j ∈ {j n , • • • , j n+1 -1}, sup |λ|=2 -j (|c λ |) ≤ C inf(2 -jn β , 2 (1-β)jn+1 2 -j ), (5.3) 2.
Conversely, if f is uniformly Hölder and if for any C > 0, there exists a strictly increasing sequence of integers (j n ) n ∈ N such that (5.3) holds, then

∀ n ≥ 0, ∀ j ∈ {j n , • • • , j n+1 -1}, sup x∈R d sup |h|≤2 -j (|f (x + h) -f (x)|) ≤ C 0 C inf(2 -jn β j γ n , 2 (1-β)jn+1 2 -j j γ ),
(5.4) where C 0 > 0 depends only on multi resolution analysis and (j n ) n denotes the sequence given by (5.3). In particular, f ∈ C

β | log | γ (R d ) for all γ > 1. Remark 5.1. Remark that, if ∀ j ∈ N, sup |λ|=2 -j (|c λ |) ≥ C 0 2 -jβ ,
(5.5) then (5.3) cannot hold and f belongs to I β (R d ). Of course (5.5) is sufficient to prove the uniform irregularity of f but not necessary.

By contraposition of Theorem 5.3, we get a wavelet criterium for uniform irregularity :

Theorem 5.4. Let β ∈ (0, 1) and f be a bounded function on R d .

1. Assume there exists C 0 > 0 such that for any integer j ≥ 0 :

max( sup |λ|≤2 -j (|c λ |), 2 -j sup |λ|≥2 -j ( |c λ | |λ| )) ≥ C 0 2 -jβ , (5.6) then f ∈ I β (R d ).
2. If for any γ > 1, f is uniformly Hölder and belongs to

I β | log | γ (R d
) then there exists C 0 > 0 such that for any integer j ≥ 0 then (5.6) holds. Now, we will apply these two wavelet criteria to the study of the uniform irregularity of the sample paths of LFBM defined above.

Smoothness of the sample paths of LFBM

Let us consider the LFBM defined by Equation (2.6). The study of the weak uniform regularity properties will be based on the fact that in short, the large wavelets coefficients of LFBM are located on scales 2 -ℓn and on the scales 2 -jn and that, at these scales, the wavelets coefficients satisfy sup

|λ|=2 -ℓn |c λ | ≈ 2 -ℓnH , sup |λ|=2 -jn |c λ | ≈ 2 -jnH .
Thus in Section 6.3 we will prove that,

I loc (B H,τ ) = H,
showing that we are nearly in the case of a lacunary wavelet series of the form

n∈N |λ|=2 -ℓn 2 -ℓnH ψ λ + n∈N |λ|=2 -jn 2 -jnH ψ λ .
The proof is detailed in Section 6.3. Here, we give some heuristic arguments to explain what happens and state our main result about the weak uniform irregularity of LFBM.

The example of LFBM is complicated because there is no lower bound of the wavelets coefficients at every scale of the form ∀j ≥ 0, sup

|λ|=2 -j |c λ | ≥ C 0 2 -jα .
To estimate the uniform irregularity exponent of LFBM, the wavelets criteria recalled in Theorems 5.3 and 5.4 will be needed. Contrary on the uniform Hölder exponent, the uniform irregularity exponent of LFBM will depend on the lacunarity index of the field.

Let us first consider the wavelet series n∈N |λ|=2 -ℓn 2 -ℓnH ψ λ . Indeed, the wavelets criterium for weak uniform regularity, asserts that the wavelets coefficients at scale (2 -ℓn ) n∈N have an effect on scales below and above the scale 2 -ℓn . This influence is decreasing as far as we get from the scale 2 -ℓn . This implies that when the band of intermediary scales between the scale 2 -ℓn is wide, there exists a sequence of intermediary scale (2 -tn ) n for which the wavelet coefficients located at the scale 2 -ℓn have little weight. Thus, the higher the lacunarity index is, the higher the uniform irregularity exponent of this wavelet series and thus of LFBM is.

To prove this, there will be two main difficulties to overcome. Firstly, the sequence (2 -tn ) n of intermediate scales has to be defined. This sequence will be defined such that for all n, the competitive influence of the large wavelets coefficients located at the scale 2 -ℓn-1 and at the scale 2 -ℓn are balanced exactly at the scale 2 -tn . Furthermore, the index of uniform irregularity α has to be estimated.

We want the uniform irregularity exponent to be larger than an index α to estimate. If we suppose that for any n, t n ≤ ℓ n ≤ t n+1 -1, we will require that, 2 -ℓnH ≤ inf(2 -tnα , 2 tn+1(1-α)-ℓn ).

In the limit case, the relationship above becomes

2 -ℓnH = 2 -tnα = 2 tn+1(1-α)-ℓn .
This means that the influence of wavelets coefficients located at the scale 2 -ℓn is exactly the same at the intermediary scale just below 2 -tn and just above 2 -tn+1 . Thus, this leads us to define the following sequence of integer (t n ) n∈N by,

t n = [ H α ℓ n ] + 1.
(5.7)

Now, we can express α as a function of the lacunarity index τ . By definition of the sequence

(t n ) n∈N lim n ℓ n t n = α H and lim n ℓ n t n+1 = 1 -α 1 -H . Thus lim n ℓ n+1 ℓ n = α H × 1 -H 1 -α . Since lim n (ℓ n+1 /ℓ n ) = τ , it implies that α = τ H 1 -H + τ H .
We then obtain that α = H and thus the sequence (t n ) n∈N coincide more or less with the sequence (j n ) n∈N involved in the definition of LFBM. The uniform irregularity exponent of the wavelet series |λ|=2 -ℓn c λ ψ λ should be equal to H.

Moreover, the sequence (j n ) n∈N has been chosen such that the additional wavelet series

|ℓ n | 1 2 +ε ≤ sup {(x,y)∈K 2 ,|x-y|≤2 -ℓn } |B H,τ (x) -B H,τ (y)| ≤ C 2 2 -Hℓn |ℓ n | τ + 1 2 .
(5.8) 2. Almost surely

I loc (B H,τ ) = H.
Furthermore there exists C 3 , C 4 almost surely positive such that for any n ∈ N

C 3 2 -jnH j 1 2 +ε n ≤ sup {(x,y)∈K 2 ,|x-y|≤2 -jn } |B H,τ (x) -B H,τ (y)| ≤ C 4 2 -jnH j τ + 1 2 n .
(5.9)

Next Section is devoted to the proofs of the results stated in the previous Sections.

Proofs

Proof of Proposition 4.2

The proof of Proposition 4.2 relies on the following Lemma : Lemma 6.1. For all x, h in R d ,

lim n0→+∞ E(|B H,τ (x + 2 -jn 0 h) -B H,τ (x)| 2 ) 2 -2jn 0 H = 1 2 <|ζ|<1 |e ihζ -1| 2 |ζ| 2H+1 dζ = E(| B H (h)| 2 ),
(6.1) and

lim n0→+∞ E(|B H,τ (x + 2 -jn 0 ,1 h) -B H,τ (x)| 2 ) 2 -2jn 0 H = 1 2 <|ξ|<1 |e ihζ -1| 2 |ζ| 2H+1 dζ = E(| B H (h)| 2 ). (6.2)
Proof. Since the field {B H,τ (x)} x∈R d has stationary increments

E(|B H,τ (x + 2 -jn 0 h) -B H,τ (x)| 2 ) -2 -2jn 0 H 1 2 <|ξ|<1 |e ihζ -1| 2 |ζ| 2H+1 dζ = E(|B H,τ (2 -jn 0 h)| 2 ) -2 -2jn 0 H 1 2 <|ξ|<1 |e ihζ -1| 2 |ζ| 2H+1 dζ
Moreover, performing the change of variables ζ = 2 -jn 0 ξ in equation (2.6) leads to

E(|B H,τ (x + 2 -jn 0 h) -B H,τ (x)| 2 ) -2 -2jn 0 H 1 2 <|ξ|<1 |e ihζ -1| 2 |ζ| 2H+1 dζ = +∞ n=0 2 -2jn 0 H 2 -jn 0 D ℓn |e ihζ -1| 2 |ζ| 2H+1 dζ + +∞ n=0 2 -2jn 0 H 2 -jn 0 Dj n |e ihζ -1| 2 |ζ| 2H+1 dζ - 1 2 <|ξ|<1 |e ihζ -1| 2 |ζ| 2H+1 dζ Hence E(|B H,τ (x + 2 -jn 0 h) -B H,τ (x)| 2 ) -2 -2jn 0 H = n0-1 n=0 2 -2jn 0 H 2 -jn 0 D ℓn |e ihζ -1| 2 |ζ| 2H+1 dζ + +∞ n=n0 2 -2jn 0 H 2 -jn 0 D ℓn |e ihζ -1| 2 |ζ| 2H+1 dζ + n0-1 n=0 2 -2jn 0 H 2 -jn 0 Dj n |e ihζ -1| 2 |ζ| 2H+1 dζ + +∞ n=n0+1 2 -2jn 0 H 2 -jn 0 Dj n |e ihζ -1| 2 |ζ| 2H+1 dζ
We will give upper bounds for each sum above. We bound the first sum in the following way :

n0-1 n=0 2 -jn 0 H 2 -jn 0 D ℓn |e ihζ -1| 2 |ζ| 2H+1 dζ ≤ n0-1 n=0 2 -2jn 0 H 2 -jn 0 D ℓn |h| 2 |ζ| 1-2H dζ ≤ |h| 2 2 -2jn 0 H n0-1 n=0 2 2(ℓn-jn 0 )(1-H) ≤ |h| 2 2 -2jn 0 H 2 2(jn 0 -1,1 -jn 0 )(1-H) ≤ |h| 2 2 -2jn 0 2 2jn 0 (1-H)-α1 log(jn 0 ) ≤ |h| 2 2 -2jn 0 (1-H) 2 -2Hjn 0 j α1 n0
for some α 1 > 0 depending on γ and H.

We then bound the second sum :

+∞ n=n0 2 -2jn 0 H 2 -jn 0 D ℓn |e ihζ -1| 2 |ζ| 2H+1 dζ ≤ 2 -2jn 0 H +∞ n0 2 -jn 0 D ℓn dζ |ζ| 2H+1 ≤ 2 -2jn 0 H +∞ n=n0 2 -jn 0 D ℓn dζ |ζ| 2H+1 ≤ 2 -2jn 0 H +∞ n=n0 2 -2(ℓn-jn 0 )H ≤ 2 -2ℓn 0 H = 2 -2jn 0 Hλ j Hγ n0
using the definition of ℓ n0 .

The third sum can be bounded in the same way as the first one

n0-1 n=0 2 -2jn 0 H 2 -jn 0 Dj n |e ihζ -1| 2 |ζ| 2H+1 dζ ≤ 2 -2jn 0 H n0-1 n=0 2 2(jn-jn 0 )(1-H) ≤ 2 -2jn 0 H 2 2(jn 0 -1-jn 0 )(1-H) ≤ 2 -2jn 0 H j 2(1-H) n0 2 -2jn 0 (1-1 β )(1-H)
using (2.5).

The fourth sum can be bounded in the same way as the second one,

+∞ n=n0+1 2 -2jn 0 H 2 -jn 0 Dj n |e ihζ -1| 2 |ζ| 2H+1 dζ ≤ 2 -2jn 0 H +∞ n=n0+1 2 -2(jn-jn 0 )H ≤ 2 -2jn 0 H 2 -2(jn 0 +1-jn 0 )H ≤ 2 -2jn 0 H j γ n0 2 -2jn 0 (β-1
)H Thus we proved (6.1). The proof of (6.2) is similar. The above lemma yields the convergence of the finite dimensional distribution of the field

B H,τ (x + 2 -jn u) -B H,τ (x) 2 -jnH u∈R d to those of { B H (t)} t∈R d .
We also proved the convergence of the finite dimensional distribution of the field

B H,τ (x + 2 -ℓn u) -B H,τ (x) 2 -ℓnH u∈R d to those of { B H (t)} t∈R d .
In order to prove the convergence in distribution for the topology of the uniform convergence on compact sets the following tightness result is required (see for e.g. the Kolmogorov criterium recalled in [START_REF] Karatzas | Brownian Motion and stochastic calculus[END_REF]) :

Proposition 6.1. Let (Z n ) n∈N a sequence of continuous random fields on R d vanishing at the origin. Suppose that for all T > 0,

∀(s, t) ∈ [-T, T ] d , sup n∈N E(|Z n (t) -Z n (s)| α ) ≤ C 0 (T )|t -s| 1+β ,
for some positive constant C 0 (T ) which may depend on T and α, β which are universal positive constants. If, moreover, for any compact K the finite dimensional distributions of sequence (Z n ) n∈N converge on K then the convergence is in distribution on K.

Using Proposition 6.1 above and a method similar to the prove of Lemma 6.1, we apply this tightness criterium to the family,

Z 1 n (u) u∈R d = B H,τ (x + 2 -jn u) -B H,τ (x) 2 -jnH u∈R d
and to the family

Z 2 n (u) u∈R d = B H,τ (x + 2 -ℓn u) -B H,τ (x) 2 -ℓnH u∈R d
So we can conclude.

Proof of Proposition 5.2

The proof uses several technics introduced in [CKR93]; (5.2) will be obtained through a serie expansion of B H,σ in the orthonormal basis of L 2 (R d ) given by the Lemarié-Meyer wavelets (see [START_REF] Meyer | Ondelettes et opérateurs Hermann[END_REF]). The proof of Proposition 5.2 relies on a key lemma that we state and prove in next section.

Estimates on the wavelet coefficients of the field

Lemma 6.2. Let j ≥ 1, (k, k ′ ) ∈ (Z d ) 2 .
1. The following estimates of the covariance of the wavelets coefficients of the field {B H,σ (x)} x∈R d hold

∃C 0 > 0, |E(c j,k c j,k ′ )| ≤ C 0 max(σ 2 j 2 -jHj , σ 2 j+1 2 -jHj+1 , σ 2 j+2 2 -jHj+2 ) 1 + |k -k ′ | . 2. If k = k ′ then ∃C 1 , C 2 > 0 C 1 max ℓ∈{j,j+1,j+2} (σ 2 ℓ 2 -2jH ℓ ) ≤ E(c 2 j,k ) ≤ C 2 max ℓ∈{j,j+1,j+2} (σ 2 ℓ 2 -2jH ℓ ).
Proof.

1. We use the Meyer wavelet basis which satisfies for any i, supp( ψ

(i) ) ⊂ [-4π/3, -2π/3] ∪ [2π/3, 4π/3].
The proof is based on the harmonisable representation of wavelet coefficients of ISFBM. For any Gaussian field with stationary increments admitting a spectral density f , one has

∀j ≥ 1, ∀k ∈ Z d , c i j,k = R d e i2 -j kξ ψ (i) (2 -j ξ)f (ξ) 1/2 d W (ξ).
Hence, in the special case of ISFBM, one deduces that

∀j ≥ 1, ∀k ∈ Z d , c i j,k = +∞ ℓ=0 σ ℓ D ℓ e ik2 -j ξ ψ (i) (2 -j ξ) |ξ| H ℓ +d/2 d W (ξ). Then for any i ∈ {1, • • • , 2 d -1}, j ≥ 1 and any (k, k ′ ) ∈ (Z d ) 2 E(c i j,k c i j,k ′ ) = +∞ ℓ=0 σ 2 ℓ D ℓ e i2 -j (k-k ′ )ξ | ψ (i) (2 -j ξ)| 2 |ξ| 2H ℓ +d dξ Perform the change of variable ζ = 2 -j ξ : E(c i j,k c i j,k ′ ) = σ 2 j 2 -2jHj 1<|ζ|<2 e i(k-k ′ )ζ | ψ (i) (ζ)| 2 |ζ| 2Hj +d dζ +σ 2 j+1 2 -2jHj+1 2<|ζ|<4 e i(k-k ′ )ζ | ψ (i) (ζ)| 2 |ζ| 2Hj+1+d dζ +σ 2 j+2 2 -2jHj+2 4<|ζ|<8 e i(k-k ′ )ζ | ψ (i) (ζ)| 2 |ζ| 2Hj+2+d dζ.
Let j a given integer and p ∈ {1, 2, 3}, we define F p j as

F p j (x) = 2 p <|ζ|<2 p+1 e ixζ | ψ (i) (ζ)| 2 |ζ| 2Hj +d dζ,
We want to bound sup

x∈R d ((1 + |x|)|F p j (x)|).
Remark that

F p j L ∞ (R d ) ≤ 2 p <|ζ|<2 p+1 | ψ(ζ)| 2 |ζ| 2Hj +d dζ ≤ 1<|ζ|<8 | ψ(ζ)| 2 |ζ| 2H+d dζ = C(H).
Furthermore, for all ℓ and p (perform an integral by parts) :

x ℓ F p j = e ix ℓ 2 p+1 | ψ(2 p+1 )| 2 |2 p+1 | 2Hj +d -e ix ℓ 2 p | ψ(2 p )| 2 |2 p | 2Hj +d - 2 p <|ζ|<2 p+1 e ixζ ∂ ℓ (| ψ (i) (ζ)| 2 /|ζ| 2Hj +d ) ∂ζ ℓ dζ
which can be bounded in a similar way than F p j L ∞ (R d ) since 0 ∈ supp( ψ). If we set x = k -k ′ , we thus obtain the required result.

If

k = k ′ then, E(|c i j,k | 2 ) = σ 2 j 2 -2jHj 1<|ζ|<2 | ψ(ζ)| 2 |ζ| 2Hj +d dζ + σ 2 j+1 2 -2jHj+1 2<|ζ|<4 | ψ(ζ)| 2 |ζ| 2Hj+1+d dζ +σ 2 j+2 2 -2jHj+2 4<|ζ|<8 | ψ(ζ)| 2 |ζ| 2Hj+2+d dζ ≥ σ 2 j 2 -2jHj 1<|ζ|<2 | ψ(ζ)| 2 |ζ| 2H+d dζ + σ 2 j+1 2 -2jHj+1 2<|ζ|<4 | ψ(ζ)| 2 |ζ| 2H+d dζ +σ 2 j+2 2 -2jHj+2 4<|ζ|<8 | ψ(ζ)| 2 |ζ| 2H+d dζ.
The result follows.

Determination of the local uniform Hölder exponent

Let B 1 H,σ (x) = j≥0 |k|<j2 jd c i j,k (ω)ψ i j,k (x), and
B 2 H,σ (x) = k∈Z d C k (ω)φ k (x) + j≥0 |k|>j2 jd c i j,k (ω)ψ i j,k (x)
.

We will focus on the estimate of the uniform regularity exponent of field

{B 1 H,σ (x)} x∈R d . We first prove that H ≥ H loc (B 1 H,σ ). (6.3)
For any integer j, let us set τ j = -log 2 (σ j ) j with τ j = +∞ if σ j = 0.

By assumption, for any positive real ε, there exists an integer j 0 such that,

∀j ≥ j 0 , H j + τ j ≥ H -ε.
Then, by a classical lemma (see lemma 3 of [START_REF] Meyer | Wavelets, generalized white noise and fractional integration : the synthesis of fractional brownian motion[END_REF]) and lemma 6.2 we deduce that there exists a almost surely positive constant C 0 (ω) and an integer j 0 (ω) such that for any j ≥ j 0 (ω) and any k ∈ Z d |c i j,k | ≤ C 0 log(k) log(j) max(2 -j(Hj +τj ) , 2 -j(Hj+1+τj+1) , 2 -j(Hj+2+τj+2) ) ≤ C 0 j log(j) max(2 -j(Hj +τj ) , 2 -j(Hj+1+τj+1) , 2 -j(Hj+2+τj+2) ) ≤ C 0 j log(j)2 -j(H-ε) . (6.4) Hence, if |k| ≤ j2 jd |c i j,k | ≤ C 0 jd log(j) max(2 -j(Hj +τj ) , 2 -j(Hj+1+τj+1) , 2 -j(Hj+2+τj+2) ) ≤ C 0 j log(j)2 -j(H-ε) .

Thus, by the usual wavelet criterium for uniform Hölder regularity (see Theorem 5.1 above) we deduce the required upper bound of uniform Hölder exponent of the random field B 1 H,σ .

Theorem II.7 of [START_REF] Ciesielski | Quelques espaces fonctionnels associés à des processus gaussiens Stud[END_REF] Proof. Let 0 < H < H ′ < 1, ε > 0 and ϕ ∈ D(R d ). We may assume that supp(ϕ) ⊂ B d (0, 1) = {x, |x| ≤ 1} and 0 ≤ ϕ ≤ 1. We denote by Y the random field ϕB 2 H,σ . We want to give an upper bound of |Y (x + h) -Y (x)| for any given x, h in B d (0, 1). Here Ψ i j,k , Φ k will denote respectively ϕψ i j,k and ϕφ k . Remark that if

f j,k = Ψ j,k (2 -j •) then |f j,k | ≤ ψ(• -k), (6.6)
and Ψ j,k (x + h) -Ψ j,k (x) = f j,k (2 j x + 2 j h) -f j,k (2 j x).

Using (6.4) The lemma follows if we choose M suficiently large.

|Y (x + h) -Y (x)| = | +∞ j=1 |k|>j2 jd c i j,k (f j,k (2 j x + 2 j h) -f j,k (2 j x)) + k∈Z d C k (Φ k (x + h) -Φ k (x

Proof of Proposition 5.5

We will use the wavelets criterium recalled in section 5. By definition of the two sequences (j n ) n , (ℓ n ) n and by Lemma 6.2 : For any integer n, and all j ∈ {j n ,

• • • , j n+1 -1}      max j∈{ℓn-2,••• ,ℓn+2} (E(|c i j,k | 2 )) ≤ C 0 2 -2ℓnH , max j∈{jn,jn+1,jn+2} (E(|c i j,k | 2 )) ≤ C 0 2 -2jnH , max j∈{jn+1-2,jn+1-1} (E(|c i j,k | 2 )) ≤ C 0 2 -2jn+1H
and if j ∈ {ℓ n -2, • • • , ℓ n+2 } ∪ {j n , j n + 1, j n + 2} ∪ {j n+1 -2, j n+1 -1}, E(|c i j,k | 2 ) = 0.

Then, for any integer n, and all j ∈ {j n , • • • , j n+1 -1}

E(|c i j,k | 2 ) ≤ C 2 0 inf(2 -jnH , 2 jn+1(1-H) , 2 -j ) 2 ,
and by the same method as above it follows that for any γ > 1 sup

x∈R d sup |h|≤2 -j
|f (x + h) -f (x)| ≤ C 0 inf(2 -jnH , 2 jn+1(1-H) , 2 -j )j γ+ 1 2 .

We then deduce almost surely the two upper bounds in equations (5.8), (5.9) and H ≥ I loc (B 1 H,σ ) = I loc (B H,σ ). We now want to bound the local uniform irregularity exponent of LFBM.

By the same proof as in the previous section we obtain that almost surely there exists a constant almost surely positive C 0 (ω) and an integer n 0 (ω) such that ∀n ≥ n 0 (ω), max Then using the wavelet criterium, we can conclude. Moreover, using for any integer j,

|c i j,k | ≤ C ψ,

Figure 1 :

 1 Figure 1: An example of spectral density of ISFBM as function of the frequencies

Figure 2

 2 Figure 2 just below represents an example of the spectral density of an LFBM with Hurst index H = 0.5 and lacunarity index τ = 1.6.

Figure 2 :

 2 Figure 2: An example of spectral density of LFBM as function of the frequencies

  Figure 3: A discretized path (X(∆ N ), • • • , X(N ∆ N )) with H = 0.3 and τ = 2

Figure 4 :

 4 Figure 4: A discretized path of the Gaussian process x → X(x + h) -X(x) for different values of h

  |h|≤r sup x∈K |B H,τ (x + h) -B H,τ (x)| near r = 0 on any compact K. So we are lead to estimate the two following exponents α = lim inf r→0 log( sup |h|≤r sup x∈K |B H,τ (x + h) -B H,τ (x)τ (x + h) -B H,τ (x)|) log(r).

|λ|=2 -jn c λ ψ λ has

  not enough large wavelet coefficients to have an influence on the uniform irregularity properties of LFBM whereas it has enough large wavelet coefficients to influence the self-similarity properties of LFBM. This heuristic argument leads us to state the Proposition below proved in Section 6.3 : Proposition 5.5. The sample paths of LFBM defined by Equation (2.6) satisfy the following properties :1. Almost surely H loc (B H,τ ) = H.Furthermore there exists C 1 , C 2 almost surely positive such that for any n ∈ N C 1 2 -Hℓn

(

  j x,2 j x+2 j h] |γ|=1 |k|>j2 jd 2 -j(H-ε) log(2 + |k|) log(2 + j)|2 j h| (1 + |y -k|) M +|h| sup y∈[x,x+h] k∈Z d ( log(2 + |k|) (1 + |y -k| M ))| using (6.6) and the fast decay of φ and ψ. Then|Y (x + h) -Y (x)| ≤ |h|( +∞ j=1 C2 j(1-H-ε) j log(2 + j) (1 + j2 j ) M ) + C)

•

  |k|≤ℓn2 ℓnd |c i ℓn,k | ≥ C 0 (ω)2 -ℓnH . (6.7)Hence : For any integer n and anyj ∈ {j n , • • • , ℓ n }, max(sup ℓ≥j |c i ℓ,k |, 2 -j sup ℓ≤j 2 ℓ |c i ℓ,k |) ≥ C 0 (ω)2 -ℓnH ≥ C 0 (ω)2 -jH .• For any integer n and any j ∈ {ℓ n + 1,• • • , j n+1 -1} max(sup ℓ≥j |c i ℓ,k |, 2 -j sup ℓ≤j 2 ℓ |c i ℓ,k |) ≥ C 0 (ω)2 -j 2 ℓn(1-H) ≥ C 0 (ω)2 -jH .

  If there exists an integer J such that, for all j ≥ J, σ j = σ J and H j = H J then the Infinity Scale Fractional Brownian Motion {B H,σ (t)} t∈R d with Hurst indices H = (H j ) j and amplitudes σ = (σ j ) j is a Multi-scale Fractional Brownian Motion as defined in[START_REF] Bardet | Definition, properties and Wavelets analysis of Multiscale Fractional Brownian Motion[END_REF].

	Example 2.1. The Infinity Scale Fractional Brownian Motion {B H,σ (t)} t∈R d
	with Hurst indices H ≡ H and amplitudes σ ≡ 1 is the classical Fractional
	Brownian Motion.
	Example 2.2. Example 2.3. If for all integer j, σ j = 1, ISFBM is a special case of the
	Generalized Multifractional Brownian Motion introduced by Ayache et Lévy-
	Véhel in [ALV99]. Let us recall that, if λ > 1 is an arbitrary fixed real and
	h = (h ℓ (•)) ℓ is a sequence of Hölder functions with values in [a, b] ⊂ (0, 1), then
	the GMBM with parameters λ and H is the continuous Gaussian process Y H,λ
	defined as

  , Falconer generalizes this concept and gives the following general definition : Definition 4.1. Let {X(t)} t∈R d be a random field with continuous sample paths. The random field {Y (t)} t∈R d vanishing almost surely at t = 0 is a tangent field at x 0 to the random field {X(t)} t∈R d if there exist two decreasing sequences of non negative real numbers converging to 0, (ρ 1,n ) n and (ρ 2,n

  .2) 2. The LFBM {B H,τ (x)} x∈R d defined by Equation (2.6) is asymptotically locally self-similar with exponent H at every point x 0 according to the family of scales (ρ n,2

  1

	2 |ξ| H+ d	d W (ξ).	(4.3)

Remark 4.1. Remark that the two Gaussian fields B H and B H are not selfsimilar. The uniqueness assumption is thus necessary to recover the results of Falconer about self-similarity of tangent fields. Since the LFBM admits stationary increments, its tangent fields also admits stationary increments. Hence our model does not yield an example of Gaussian fields with unstationary tangent fields. In

[START_REF] Clausel | More about uniform irregularity : the wavelet point of view[END_REF]

the case of a modified version of Generalized Fractional Brownian Motion has been studied. Unfortunately, in this case the tangent fields remains with stationary increments. The question of defining a Gaussian field with unstationary tangent fields remains still open.

We thus obtain our main result about the structure of tangent space of the LFBM defined by Equation (2.6) :

Theorem 4.3. For all x 0 ∈ R d ,

the two Gaussian fields { B H (t)} t∈R d and { B H (t)} t∈R d defined by Equation (4.2) and (4.3) are two tangent fields at x 0 to the LFBM {B H,τ (x)} x∈R d defined by Equation (2.6). Hence, {B H,τ (x)} x∈R d admits at every point several tangent fields unlike a FBM.

  applied to the sequence (g n,k,i ) = Lemma 6.3. Almost surely, the sample paths of the field{B 2 H,σ (x)} x∈R d are C H ′ loc (R d ) for any 0 < H < H ′ < 1.Then, the uniform regularity and irregularity exponents of the sample paths of ISFBM are those of the field {B 1 H,σ (x)} x∈R d .

		c i ℓn,k
		E(|c i ℓn,k | 2 )	1 2
	yields the converse inequality
	H ≤ H loc (B 1 H,σ ),	(6.5)
	by using the first point of lemma 6.2. The conclusion is then straightforward
	using the following lemma :

  ω sup |B H,τ (x + h) -B H,τ (x)|and (6.7) we deduce the lower bound in (5.9). Since for any integer n, Hℓ n = Hj n and

			sup
		x∈R d	|h|≤2 -j
	sup	sup		sup
	x∈R d	|h|≤2 -ℓn	x∈R d	|h|≤2 -jn

|B H,τ (x + h) -B H,τ (x)| ≤ sup |B H,τ (x + h) -B H,τ (x);

the lower bound in equation (5.8) follows.

Acknowledgements

The author would like to thank the anonymous referees whose valuable comments greatly improve the earlier version of this work.