Stationary increments harmonizable stable ﬁelds: upper estimates on path behaviour

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Stationary increments harmonizable stable fields:

upper estimates on path behaviour

Antoine Ayache and Geoffrey Boutard

UMR CNRS 8524 Laboratoire Paul Painlev´e Universit´e Lille 1

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Introduction and motivations

Organization of the talk

1 Introduction and motivations

2 Wavelet type random series representation

3 Results on path behaviour

A. Ayache and G. Boutard (Universit´e Lille 1) Behaviour of harmonizable stable fields Workshop Fractality and Fractionality 2 / 30

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Stable random variables and stable stochastic integrals

LetZ be a real-valued random variable, andχ_Z its characteristic function defined as: ∀λ∈R,χ_Z(λ) :=E(e^iλZ). Z is said to have a symmetric stable distribution of stability parameterα∈(0,2] and scale parameterσ∈R+, if:

∀λ∈R, χZ(λ) = exp(−σ^α|λ|^α). (1.1)

→whenα= 2,Z reduces to a centered Gaussian random variable of variance 2σ².

→The situation is very different whenα∈(0,2) andσ >0; the distribution ofZ becomes heavy-tailed:

P(|Z|>z)∼c(α)σ^αz^−α, when z →+∞. (1.2) This, in particular, implies that:

E(|Z|^γ)<+∞ whenγ < α, and E(|Z|^γ) = +∞ whenγ≥α. (1.3)

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We denote byMeα a complex-valued rotationally invariantα-stable random measure onR^d with Lebesgue control measure.

The related stable stochastic integral is denoted byR

R^d ·

dMeα. It is a linear map on the Lebesgue spaceL^α(R^d) such that, for any deterministic function g ∈L^α(R^d), the real partRe R

R^dg(ξ)dMeα(ξ) is a real-valued symmetric α-stable random variable with a scale parameter satisfying

σ Re

Z

R^d

g(ξ)dMeα(ξ) ^α

= Z

R^d

g(ξ)

αdξ. (1.4)

The equality (1.4) is reminiscent of the classical isometry property of Wiener integrals; in particular, it implies thatRe R

R^dgn(ξ)dMeα(ξ) converges to Re R

R^dg(ξ)dMe_α(ξ) in probability, when a sequence (g_n)_nconverges tog in L^α(R^d). This will be useful for us.

A classical reference on stable distributions and related topics, including stable random measures and their associated stochastic integrals, is the book of Samorodnitsky and Taqqu (1994).

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Stationary increments harmonizable stable fields

Let us now focus on the definition of

X(t),t ∈R^d , the class of the real-valued stationary increments harmonizable stable fields we intend to study.

The main ingredient of the definiton isf, an arbitrary real-valued Lebesgue measurable even function onR^d satisfying the condition:

Z

R^d

min 1,||ξ||^α f(ξ)

αdξ <+∞, (1.5) where||·||denotes the Euclidian norm onR^d. Notice that, by analogy with the Gaussian case (see Bonami and Estrade (2003), for instance), the function|f|^α is calledthe spectral density of the field X.

Thanks to (1.5), for anyt ∈R^d, the functionξ7→ e^it·ξ−1

f(ξ) belongs to L^α(R^d), and thus it is integrable with respect toMe_α. The field

X(t),t ∈R^d is defined, for allt∈R^d, as

X(t) =X[f](t) :=Re Z

R^d

e^it·ξ−1

f(ξ)dMe_α(ξ)

, (1.6)

wheret·ξdenotes the usual inner product oft andξ.

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Harmonizable and Linear Fractional Stable Motions

A classical particular case: harmonizable fractional stable motion (hfsm)of Hurst parameterH∈(0,1), denoted by{X^hfsm(t),t ∈R}and defined as: ∀t ∈R,

X^hfsm(t) :=Re Z

R

e^itξ−1

|ξ|^−H−1/αdMe_α(ξ)

. (1.7)

X^hfsm is one of the two most classical extensions of the well-knownfractional Brownian motion (fBm)to the setting of the heavy-tailed stable distributions.

The other classical extension of fBm to this setting is calledlinear fractional stable motion (lfsm)and denoted by

Y^lfsm(t),t ∈R^d . In contrast withX^hfsm, the processY^lfsmis not defined through an integral in the frequency domain but through an integral in the time domain:

Y^lfsm(t) :=

Z

R

|t+s|^H−1/α− |s|^H−1/α

dM_α(s), (1.8) whereMα is anα-stable real-valued random measure onR.

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In spite of the fact that the stable processesX^hfsm andY^lfsmextend the same Gaussian process (the fBm) there are hudge differences between these two stable processes. For instance:

→Sample paths ofY^lfsm are continuous functions only whenH>1/α; in the latter case their critical H¨older regularity isH−1/α. This result can be derived from Kolmogorov’s H¨older continuity Theorem, since, for each fixedγ∈(0, α), one has:

∀t₁,t₂∈R, E

Y^lfsm(t₁)−Y^lfsm(t₂)

γ

=c_α,γ t₁−t₂

1+γ(H−1/γ)

. (1.9)

→Sample paths ofX^hfsmare always continuous functions and their H¨older regularity isH−η, for anyη >0. This result can not be derived from

Kolmogorov’s H¨older continuity Theorem, even ifX^hfsmalso satisfies (1.9). It was obtained by Kˆono and Maejima (1991) thanks to a LePage type series

representation ofX^hfsm.

Generally speaking there are hudge differences between stable stochastic fields defined through stochastic integrals in the frequency domain, and those defined through stochastic integrals in the time domain.

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Our two motivations

In the continuous case, wavelet methods (see Ayache, Roueff and Xiao (2009)) have turned out to be efficient in the fine study of global and directional sample path behaviour of linear fractional stable sheet; that is the extension toR^d of the processY^lfsm. Can this methodology be adapted to the general stationary

increments stable fieldX? This issue is the main motivation of our talk.

Also we mention that the study of global and directional sample path behaviour of X may have an impact on future development of new applications related with modelling of anisotropic materials in frames of heavy-tailed stable distributions. It is worthwhile to note that in Gaussian frames such a modelling has already proved to be useful, in particular for detecting osteoporosis in human bones through the analysis of their radiographic images (see for instance Bonami and Estrade (2003) or Bierm´e, Richard, Rachidi and Benhamou (2009)).

Typically,X is an anisotropic model when the rate of vanishing at infinity of the corresponding spectral density|f|^αchanges from one axis of R^d to another;

therefore, we focus on the classAof the so-called admissible functionsf, defined in the following way.

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We setp_∗:= max

2,b1/αc+ 1 .

A functionf belongs toAwhen it satisfies (H₁), (H₂) and (H₃).

(H₁) For all multi-indexp:= (p₁,p₂, . . . ,p_d)∈

0,1,2, . . . ,p_∗ ^d, the partial derivative function

∂^pf := ∂^p¹∂^p². . . ∂^p^d

(∂ξ₁)^p¹(∂ξ₂)^p². . .(∂ξ_d)^p^d f (with the convention that∂⁰f :=f) is well-defined and continuous on the open set R\ {0}^d

; that is the Cartesian product ofR\ {0} with itselfd times.

(H2) There are a positive constantc⁰ and an exponent a⁰∈(0,1) such that, for eachp∈

0,1,2, . . . ,p_∗ ^d, andξ∈ R\ {0}^d ,

||ξ|| ≤1 =⇒

∂^pf(ξ)

≤c⁰||ξ||^−a⁰^{−d/α−l(p)}, (1.10) wherel(p) :=p1+p2+· · ·+pd is the length of the multi-indexp.

(H3) There exist a positive constantc andd positive exponentsa1, . . . ,ad such that for everyp∈

0,1,2, . . . ,p_∗ ^d, andξ∈ R\ {0}d

,

||ξ|| ≥1 =⇒

∂^pf(ξ) ≤c

d

Y

l=1

(1 +|ξl|)^−a^l^−1/α−p^l. (1.11)

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Wavelet type random series representation

Organization of the talk

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The Gaussian case α = 2

We denote by

ψJ,K : (J,K)∈Z^d×Z^d the orthonormal basis ofL²(R^d) defined in the following way: for all (J,K) := (j₁, . . . ,j_d,k₁, . . . ,k_d)∈Z^d×Z^d and x:= (x₁, . . . ,x_d)∈R^d

ψJ,K(x) :=

d

Y

l=1

2^j^l^/2ψ¹(2^j^lxl−kl), (2.1) whereψ¹denotes an usual 1D Lemari´e-Meyer mother wavelet. We refer to the book of Meyer (1990) and to that of Daubechies (1992) for a complete

description of the wavelet tools used in the present section. It is worthwhile noting thatψ¹is a real-valued function belonging to the Schwartz classS(R), and that its Fourier transformψc¹ is a compactly supportedC^∞ function onR, such that

suppψc¹⊆

λ∈R: 2π

3 ≤ |λ| ≤ 8π 3

. (2.2)

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The fact that the Fourier transform map is an isometry fromL²(R^d) into itself implies that ”the Fourier transform of the basis

ψJ,K : (J,K)∈Z^d×Z^d ”,that is

ψb_J,K : (J,K)∈Z^d×Z^d ,is also an orthonormal basis of L²(R^d). Thus, for any fixedt ∈R^d, the kernel functionξ7→ e^it·ξ−1

f(ξ), associated withX(t), can be expressed as:

e^it·ξ−1

f(ξ) = X

(J,K)∈Z^d×Z^d

sJ,K(t)ψbJ,K(ξ) (inL²(R^d)). (2.3)

The coefficientss_J,K(t) are given by s_J,K(t) :=

Z

R^d

e^it·ξ−1

f(ξ)ψb_J,K(ξ)dξ= Ψ_J 2^Jt−K

−Ψ_J(−K), (2.4) where, for allx ∈R^d,

ΨJ(x) := 2^(j¹^+···+j^d^)/2 Z

R^d

e^ix·ξf 2^Jξ

ψb0,0(ξ)dξ, (2.5) with the convention that 2^Jξ:= (2^j¹ξ1, . . . ,2^j^dξd).

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Therefore, we get that

X(t) =Re





 Z

R^d

X

(J,K)∈Z^d×Z^d

Ψ_J 2^Jt−K

−Ψ_J(−K) ψb_J_,K(ξ)

dMe₂(ξ)





 .

(2.6) Finally, in view of the isometry property of the stochastic integralR

R^d · dMe2, it turns out that one can interchange in (2.6) the integration and the summation.

Thus, we obtain that

X(t) = X

(J,K)∈Z^d×Z^d

Ψ_J 2^Jt−K

−Ψ_J(−K)

_J,K, (2.7)

where the series converges inL²(Ω), and theJ,K’s are the independentN(0,1) Gaussian random variables

_J,K :=Re Z

R^d

ψb_J,K(ξ)dMe₂(ξ)

. (2.8)

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The general case α ∈ (0, 2)

The arguments, we have used in the ”convenient” framework of the Hilbert space L²(R^d), have to be adapted to the ”more hostile” framework of the spaceL^α R^d . The main difficulty comes from the fact that

ψbJ,K: (J,K)∈Z^d×Z^d is no longer a basis ofL^α R^d

.

The functionψ_α,J,K = 2^(j¹^+···+j^d^{)(1/2−1/α)}ψ_J_,K denotes the renormalized version of the functionψJ,K so thatkψbα,J,Kk_Lα(R^d) does not depend on (J,K).

Thus setting Ψα,J = 2^(j¹^+···+j^d^{)(1/α−1/2)}ΨJ, it follows that for every (J,K)∈Z^d×Z^d and (t, ξ)∈R^d×R^d, one has

s_J,K(t)ψb_J,K(ξ) = Ψ_J 2^Jt−K

−Ψ_J(−K)

ψb_J_,K(ξ) (2.9)

= Ψ_α,J 2^Jt−K

−Ψ_α,J(−K)

ψb_α,J_,K(ξ).

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Recall thatL^α R^d

is a complete metric space for the distance

Dα(g1,g2) :=







kg1−g2k_Lα(R^d)=R

R^d|g1(ξ)−g2(ξ)|^α dξ^1/α

, ifα≥1, kg1−g2k^α_Lα(R^d)=R

R^d|g1(ξ)−g2(ξ)|^αdξ, else.

(2.10) Also, noticeD_α(g₁,g₂) =D_α(g₁−g₂,0).

The proof of the fact that, for any fixedt∈R^d, the kernel function ξ7→ e^it·ξ−1

f(ξ), associated withX(t), can be expressed as:

e^it·ξ−1

f(ξ) = X

(J,K)∈Z^d×Z^d

s_J,K(t)ψb_J,K(ξ) (inL^α(R^d)), (2.11)

is divided in two steps.

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Step 1. We show that

X

(J,K)∈Z^d×Z^d

Dα

sJ,K(t)ψbα,J,K(·),0

<+∞. (2.12)

Notice that, in view of the completeness ofL^α R^d

, (2.12) implies that there existsF(t,·) inL^α R^d

such that F(t, ξ) = X

(J,K)∈Z^d×Z^d

sJ,K(t)ψbJ,K(ξ) (inL^α(R^d)). (2.13)

Step 2. We show that, for allt∈R^d and almost allξ∈R^d, F(t, ξ) = e^it·ξ−1

f(ξ). (2.14)

Basically the Step 2 is derived from the fact that, for any fixed arbitrarily small η >0, the functionξ7→ e^it·ξ−1

f(ξ)Qd

l=11_{|ξ_l_|≥η} belongs toL²(R^d).

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Basically, the Step 1 is derived from the following proposition.

Proposition 2.1

Ψα,J is infinitely differentiable onR^d. Moreover, all the functions∂^bΨα,J, b∈Z^d+, are well-localized:

(i) There is a positive constant c, such that for all J∈Z^d+, and x= (x1, . . . ,xd)∈R^d,

∂^bΨ_α,−J(x)

≤c 2^−j¹+· · ·+ 2^−j^d−a⁰−d/αQd

l=12^−j^l^/α Qd

l=1(1 +|xl|)^p^∗ , (2.15) (ii) For eachζ= (ζ1, . . . , ζd)∈ {0,1}^d\ {(0, . . . ,0)}, there exists a positive

constant c, such that for every J∈Qd

l=1Zζ_l (Z1=NandZ0=Z−) and x= (x1, . . . ,xd)∈R^d,

∂^bΨα,J(x) ≤c

d

Y

l=1

2^(1−ζ^l^)j^l^/α2^−j^l^ζ^l^a^l

(1 +|xl|)^p^∗ . (2.16) Recall that p_∗:= max

2,b1/αc+ 1 .

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Next, using arguments rather similar to those in the Gaussian case, we can show that:

Proposition 2.2 (Wavelet representation ofX)

The field{X(t),t∈R^d}can be expressed, for each fixed t ∈R^d, as

X(t) = X

(J,K)∈Z^d×Z^d

Ψ_α,J 2^Jt−K

−Ψ_α,J(−K)

_α,J,K, (2.17)

where the series converges in probability, and theα,J,K’s are the identically distributed symmetricα-stable random variables

α,J,K :=Re Z

R^d

ψbα,J,K(ξ)dMeα(ξ)

. (2.18)

→The convergence of the series in (2.17) can be strenghtened to almost sure uniform convergence int belonging to any compact subset ofR^d.

→In contrast with the Gaussian case, the_α,J,K’s are not independent, they even have a complicated dependence structure.

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Results on path behaviour

Organization of the talk

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Basically, path behaviour ofX is determined by asymptotic behaviour of the sequence

_α,J_,K : (J,K)∈Z^d×Z^d . Let us state three crucial lemmas on this latter one.

Lemma 3.1 (the caseα∈(0,1))

There exists an eventΩ^∗ of probability 1 which depends onαand satisfies the following property: for all fixedδ∈(0,+∞)andω∈Ω^∗, there is a finite constant C(ω)>0(depending onα,δandω), such that, for every J = (j1, . . . ,jd)∈Z^d and K ∈Z^d, one has

|α,J,K(ω)| ≤C(ω)

d

Y

l=1

(1 +|jl|)^1/α+δ. (3.1) A rather surprising fact is that|α,J,K(ω)|can be bounded independently onK whenα∈(0,1).

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The previous lemma and the next one are obtained by using a LePage series representation of the complex-valuedα-stable process

Z

R^d

ψb_α,J,K(ξ)dMe_α(ξ) : (J,K)∈Z^d×Z^d

.

Lemma 3.2 (the caseα∈[1,2))

There exists an eventΩ^∗ of probability 1 which depends onαand satisfies the following property: for each fixedδ∈(0,+∞)andω∈Ω^∗, there is a finite constant C(ω)>0(depending on α,δ andω), such that for all

(J,K) = (j₁, . . . ,j_d,k₁, . . . ,k_d)∈Z^d×Z^d,

|α,J,K(ω)| ≤C(ω) v u u

tlog 3 +

d

X

l=1

|jl|+|kl|

! _d Y

l=1

(1 +|jl|)^1/α+δ. (3.2)

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In contrast with the previous two lemmas, the following one is a rather classical result.

Lemma 3.3 (the Gaussian caseα= 2)

There exists an eventΩ^∗ of probability 1 satisfying the following property: for every fixedω∈Ω^∗, there is a finite constant C(ω)>0 (depending onω), such that for each(J,K) = (j₁, . . . ,j_d,k₁, . . . ,k_d)∈Z^d×Z^d,

|J,K(ω)|:=|2,J,K(ω)| ≤C(ω) v u

utlog 3 +

d

X

l=1

|jl|+|kl|

!

. (3.3)

Notice that in the three crucial lemmas, we have just stated, the events of full probability Ω^∗ are universal in the sense that they do not depend on the particular choice of the functionf associated with the fieldX. The results, we will obtain, on path behaviour ofX are valid on these universal events.

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Sketch of the proof of Lemma 3.1: Let{κ^m:m∈N},{Γm:m∈N}, and {gm:m∈N} be three arbitrary mutually independent sequences of random variables having the following three properties.

1 Theκ^m’s,m∈N, areR^d-valued, independent, identically distributed and absolutely continuous, with a probability density function, denoted byφ, such that the measureφ(ξ)dξis equivalent to the Lebesgue measuredξonR^d.

2 The Γm’s,m∈N, are Poisson arrival times with unit rate.

3 Thegm’s,m∈N, are complex-valued, independent, identically distributed, rotationally invariant and satisfyE[|Re(gm)|^α] = 1.

LePage representation: there is a deterministic constanta(α)>0 such that (Z

R^d

ψbα,J,K(ξ)dMeα(ξ) : (J,K)∈Z^d×Z^d )

has the same distribution as (

a(α)

+∞

X

m=1

g_mΓ^−1/α_m φ(κ^m)^−1/αψb_α,J,K(κ^m) : (J,K)∈Z^d×Z^d )

.

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From now on, these two processes are identified, also, we assume that theg_m’s, m∈N, are complex-valued centred Gaussian random variables, and that the probability density functionφis such that, for allξ= (ξ₁, . . . , ξ_d)∈R^d\ {0},

φ(ξ) :=η 4

^dY^d

l=1

|ξl|⁻¹(1 +|log|ξl||)^−1−η, (3.4) whereη >0 is arbitrary fixed.

→Using the fact that, for all (J,K)∈Z^d×Z^d andξ∈R^d, ψb_α,J,K(ξ) =

d

Y

l=1

2^−j^l^/αe⁻ⁱ²⁻^jl^k^l^ξ^lψc¹(2^−j^lξ_l), (3.5) we get, for some deterministic constantc1, not depending on (J,K) and m, that

φ(κ^m)^−1/α

ψb_α,J,K(κ^m)

≤η 4

^−d/αY^d

l=1

2^−j^lκ^m_l

1/α 1 +|jl|+ log

2^−j^lκ^m_l

^(1+η)/α

ψc¹(2^−j^lκ^m_l )

≤c₁

d

Y

l=1

(1 +|j_l|)^(1+η)/α. (3.6)

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→In view of the Gaussianity assumption on theg_m’s,m∈N, it can be derived from the Borel-Cantelli’s Lemma that, almost surely, for allm∈N, one has

|g_m| ≤C₂p

log (3 +m), (3.7)

whereC2is a finite random variable not depending on (J,K) and m.

→It results from the strong law of large number, that almost surely, for any m∈N, the Poisson arrival time Γ_msatisfies

C3m≤Γm≤C4m, (3.8)

whereC3andC4 are two positive finite random variables not depending on (J,K) andm.

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Finally, it follows from (3.6) to (3.8) that, almost surely, for all (J,K)∈Z^d×Z^d, one has

Z

R^d

ψbα,J,K(ξ)dMeα(ξ)

≤ a(α)

+∞

X

m=1

|gm|Γ^−1/α_m φ(κ^m)^−1/α

ψbα,J,K(κ^m)

≤ C5 d

Y

l=1

(1 +|jl|)^(1+η)/α, (3.9)

where the random variable

C₅:=a(α)c₁C₂C₃^−1/α

+∞

X

m=1

m^−1/αp

log (3 +m) (3.10)

is almost surely finite sinceα∈(0,1).

Let us now turn to the statements of our results on path behaviour ofX. First, we mention that we will consider directional increments ofX in a generalized sense, more precisely:

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For every fixedk∈ {1, . . . ,d}, andhk ∈R, we denote by ∆^k_h

k, the operator from the space of the real-valued functions onR^d, into itself; so that, when g is such a function, ∆^k_h

kg is then the function defined, for allx∈R^d as,

∆^k_h

kg

(x) =g(x+h_ke_k)−g(x), (3.11) ek being the vector ofR^d whosek-th coordinate equals 1 and the others vanish.

Notice that the operators ∆^k_h

k are commutative, in the sense that, for all (k,k⁰)∈ {1, . . . ,d}²and (hk,h_k⁰0)∈R², one has,

∆^k_h⁰0 k0 ◦∆^k_h

k = ∆^k_h

k ◦∆^k_h⁰0 k0,

where the symbole ”◦” denotes the usual composition of operators. For every h= (h1, . . . ,hd)∈R^d and multi-index B= (b1, . . . ,bd)∈Z^d+, we denote by

∆^B_(h), the operator from the space of the real-valued functions onR^d, into itself, defined by

∆^B_(h):= ∆^1,b_h ¹

1 ◦ · · · ◦∆^d,b_h ^d

d , (3.12)

where, for allk ∈ {1, . . . ,d}, ∆^k,b_h ^k

k is ∆^k_h

k composed with itselfbk times, with the convention that ∆^k_h^,0

k is the identity.

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Definition 3.1 (it concerns powers of logarithmic factors in the next theorem) (i) We denote byL2 the function defined, for each(a,b)∈R²+, as

L2(a,b) := 1/2 1_{b≥a}+ 1_{b=a}. (3.13) More precisely, one has: L2(a,b) = 0 if a>b,L2(a,b) = 3/2if a=b, and L₂(a,b) = 1/2 if a<b.

(ii) For any fixedα∈(0,2), we denote byLαthe function defined, for each (a,b, δ)∈R³+, as

Lα(a,b, δ) := 1/α+bαc/2 +δ

1_{b≥a}+ 1_{b=a}, (3.14) wherebαcis the integer part ofα. More precisely,

whenα∈(0,1), one has: Lα(a,b, δ) = 0if a>b,Lα(a,b, δ) = 1/α+ 1 +δ if a=b, andLα(a,b, δ) = 1/α+δif a<b;

whenα∈[1,2), one has: Lα(a,b, δ) = 0if a>b,Lα(a,b, δ) = 1/α+ 3/2 +δ if a=b, andLα(a,b, δ) = 1/α+ 1/2 +δif a<b.

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Theorem 3.1 (directional behaviour ofX)

Let a1, . . . ,ad be the exponents governing the asymptotic behaviour at infinity of f along the axes ofR^d. Moreover we assume that B∈Z^d+, T ∈(0,+∞)and ω∈Ω^∗ are arbitrary an fixed.

(i) Whenα= 2, one has

sup

h∈[−T,T]^d











∆^B_(h)X(·, ω) _T,∞

d

Y

l=1

|h_l|^min(b^l^,a^l⁾ log

3 +|h_l|⁻¹L2(al,bl)











<+∞. (3.15)

(ii) Whenα∈(0,2), for all arbitrarily small positive real numberδ, one has

sup

h∈[−T,T]^d











∆^B_(h)X(·, ω) T,∞

d

Y

l=1

|hl|^min(b^l^,a^l⁾ log

3 +|hl|⁻¹Lα(a_l,b_l,δ)











<+∞. (3.16)

(30)

Theorem 3.2 (behaviour ofX at infinity)

Let a⁰ ∈(0,1)be the exponent governing the behaviour of f in the vicinity of zero. Moreover we assumeδ∈(0,+∞)andω∈Ω^∗ are arbitrary and fixed.

1 Whenα∈(0,1)one has sup

||t||≥1

n||t||^−a⁰ log 3 +||t||^−1/α−δ

|X(t, ω)|o

<+∞. (3.17)

2 Whenα∈[1,2) one has sup

||t||≥1

n||t||^−a⁰ log 3 +||t||−1/α−δ

|X(t, ω)|o

<+∞. (3.18)

3 Whenα= 2one has sup

||t||≥1

n||t||^−a⁰ log log 3 +||t||−1/2

|X(t, ω)|o

<+∞. (3.19)