Erratic random functions: the concept of local nondeterminism and the study of Harmonizable Fractional Stable Field

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Erratic random functions: the concept of local nondeterminism and the study of Harmonizable

Fractional Stable Field

Antoine Ayache

Universit´e Lille 1 - Laboratoire Paul Painlev´e

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Organization of the talk

1 Introduction: the intuitive notion of erratic function

2 The deterministic notion of local time and the Berman’s principle

3 Stochastic fields and random local times

4 Joint continuity of random local times and local nondeterminism

5 Harmonizable Fractional Stable Field is locally nondeterministic

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In all this talk,g always denotes an arbitrary continuous and deterministic function fromR^N toR.

Definition 1.1

Letτ∈R^N be an arbitrary fixed point. Let B(τ, ρ)be the closed ball centered at τ and of arbitrary radiusρ >0. The oscillation of g on B(τ, ρ), is denoted by oscg,τ(ρ), and defined as:

oscg,τ(ρ) := sup

t¹,t²∈B(τ,ρ)

|g(t¹)−g(t²)|= max

t∈B(τ,ρ)g(t)− min

t∈B(τ,ρ)g(t). (1.1) The continuity ofg atτ clearly implies that

osc_g,τ(ρ)−−−−→

ρ→0+

0₊. (1.2)

The rate of convergence depends on the degree of smoothness/roughness ofg at τ. When it is differentiable atτ, then the rate isρ; when it is rough atτ, then the rate can be much slower.

The functiong is said to be erratic when the convergence in (1.2) holds slowly (slower thanρ) for all the pointsτ inR^N, or at least for most of them.

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0 2 4 6 8 10 12 14 x 10⁴

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1 1.2

Figure: Graph of an erratic function

Finely studying the local behavior of an erratic functiong consists in finding, for anyτ∈R^N and smallρ,an optimalexplicit ”nice” upper boundM_g,τ(ρ) for oscg,τ(ρ).

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Usually, one looks for an optimal upper boundM_g,τ(ρ) of the form:

M_g,τ(ρ) = (power function in ρ)×(logarithmic factor in ρ).

In this issue, the most difficult task is to show the optimality ofM_g,τ(ρ), that is:

lim sup

ρ→0+

osc_g,τ(ρ)

Mg,τ(ρ) >0. (1.3)

A closely related, but less general, issue is whether or notg is a nowhere differentiable function.

In such kind of problems, loosely speaking, one of the major difficulties comes from the fact that in Real Analysis, usually, more effort is required in order to derive lower bounds (even in the sense of some sequence) than for deriving upper bounds. One reason is that in the case of lower bounds, the use of the triangle inequality is, in general, ”less natural” than in that of upper bounds.

How can one overcome this difficulty?

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Organization of the talk

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The article by Geman and Horowitz ”Occupation densities” (inAnnals of

Probability 1980) is a classical and excellent survey on the notions to be presented in this section and the next one. Throughout the present section, we denote byg an arbitrary continuous function fromR^N toR, depending on the

multidimensional variablet ∈R^N, which is nevertheless viewed asa time variable.

Definition 2.1 (occupation measure)

Let T be an arbitrary compact subset ofR^N. The occupation measureassociated with g on T , is the deterministic positive finite measureµg,T on B(R), the Borel σ-field overR, defined as

µ_g,T(A) :=λ_N {t∈T :g(t)∈A}

= Z

T

1_A g(t)

dt, for all A∈ B(R), (2.1) whereλ_N denotes the Lebesgue measure on R^N. Observe that the quantity µg,T(A)can be viewed as a measure of ”the amout of time t spent by the function g in the Borel set A during the time period T ”.

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Remarks 2.2

1 The finite measureµg,T is supported on the compact g(T) :=

g(t) :t ∈T .

2 Assume that f is either a Borel measurable non-negative function onR, or a Borel measurable bounded complex-valued function onR. Then

Z

R

f(x)dµg,T(x) = Z

T

f g(t)

dt. (2.2)

3 The occupation measureµg,T is completely determined by its Fourier transformbµg,T defined as

µbg,T(ξ) :=

Z

R

e^−iξxdµg,T(x), for eachξ∈R; (2.3) also, in view of (2.2), one has

µb_g,T(ξ) = Z

T

e^−iξg(t)dt. (2.4)

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The notion of local time ”mesure du voisinage” was first introduced by Paul L´evy ('1940) in some his pioneering works on Brownian Motion.

Definition 2.3 (local time)

For having the existence ofLg(•,T), the local time onT of the functiong, the occupation measureµg,T(•) needs to be absolutely continuous with respect toλ, the Lebesgue measure onR. When this condition is fulfilled, then Lg(•,T)is defined as the Radon-Nikod´ym derivative of µg,T(•)with respect toλ. In other words, L_g(•,T)is defined as the unique (up to a Lebesgue negligible set) non-negative function in the Lebesgue spaceL¹(R), such that the equality

Z

R

f(x)dµg,T(x) = Z

R

f(x)Lg(x,T)dx, (2.5) holds for any f , which is either a Borel measurable non-negative function onR, or a Borel measurable bounded complex-valued function onR.

Notice that the fact thatµ_g,T is supported ong(T) implies thatL_g(•,T) is supported ong(T) as well.

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Proposition 2.4

A sufficient condition for the local time L_g(•,T)to exist is that the Fourier transformbµ_g,T belongs to the Lebesgue Hilbert spaceL²(R). Notice that, under this condition, one also has L_g(•,T)∈L²(R).

Proof: We denote byF(·) the Fourier transform map. It is a bijection from the Schwartz classS(R) into itself; also it is a bijective ”isometry” (Plancherel’s formula) fromL²(R) into itself. Letϕ∈S(R) be arbitrary; we have, for allx∈R, ϕ(x) =_2π¹ R

Re^iξxϕ(ξ)b dξ. Thus, using Fubini-Tonelli’s theorem, we get that Z

R

ϕ(x)dµg,T(x) = 1 2π

Z

R

ϕ(ξ)b Z

R

e^iξxdµg,T(x) dξ

= 1

2π Z

R

ϕ(ξ)b bµg,T(ξ)dξ

= Z

R

ϕ(x)F⁻¹(bµ_g,T)(x)dx (Plancherel’s formula) which means thatF⁻¹(µbg,T) is the local time we are looking for.

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Notice that whenL_g(•,T) exists thenL_g(•,S) also exists for any compact S ⊆T; this a consequence of the Radon-Nikod´ym Theorem (see for instance the well-known Rudin’s book). Thus,L_g can be viewed as a function of 2 variables:

the space variablex∈Randthe set variable (or time variable)S ⊆T.

The Berman’s principle ('1970): The more regular (smooth) is the local time L_g the more irregular (rough) is the associated functiong. This principle is somehow clarified by the following lemma.

Lemma 2.5

Letτ∈R^N be a fixed point such that, for someρ >0small enough (or

equivalently for allρsmall enough), the local time of g on the closed ball B(τ, ρ), that is Lg(•,B(τ, ρ)), exists. One sets

L^∗_g(B(τ, ρ)) := sup

x∈R

Lg(x,B(τ, ρ)) = sup

x∈g(B(τ,ρ))

Lg(x,B(τ, ρ)). (2.6) Then, assuming that the constant c>0is the Lebesgue measure of the unit ball ofR^N, one has

cρ^N≤oscg,τ(ρ)L^∗_g(B(τ, ρ)). (2.7)

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Thus, it turns out that in order to show thatMg,τ(ρ) is ”a lower bound” for the oscillationoscg,τ(ρ), it is enough to prove that the ratioρ^N/Mg,τ(ρ) is ”an upper bound” forL^∗_g(B(τ, ρ)), more precisely:

lim sup

ρ→0+

ρ^N

Mg,τ(ρ)L^∗_g(B(τ, ρ)) >0 =⇒lim sup

ρ→0+

osc_g,τ(ρ) Mg,τ(ρ) >0.

Proof of Lemma 2.5: On one hand, one has µ_g,B_(τ,ρ)(R) :=λN {t∈B(τ, ρ) :g(t)∈R}

=λN B(τ, ρ)

=cρ^N. (2.8) On the other hand, one has

µg,B(τ,ρ)(R) = Z

g(B(τ,ρ))

Lg(x,B(τ, ρ))dx

≤ λ g(B(τ, ρ) sup

x∈g(B(τ,ρ))

Lg(x,B(τ, ρ))

= oscg,τ(ρ)L^∗_g(B(τ, ρ)). (2.9)

Combining (2.8) and (2.9) one gets the lemma.

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Yet, Lemma 2.5 is of no interest whenL^∗_g(B(τ, ρ)) = +∞! In fact, it becomes to be interesting, when one has, at least, that

ρ→0lim+

L^∗_g(B(τ, ρ)) = 0. (2.10)

Notice that ifN= 1 and (2.10) is satisfied, then one has

limρ→0₊ρ⁻¹oscg,τ(ρ) = +∞, which entails thatg is not differentiable atτ.

Let us now provide a sufficient condition on the local timeLg under which (2.10) is valid. To this end,we assume, for a while, thatN= 1, thus the ballB(τ, ρ) reduces to the compact interval [τ−ρ, τ+ρ]⊂R.

Condition (joint continuity (JC)):There existsρ₀>0, such that the function from R×[τ−ρ₀, τ+ρ₀] intoR+, (x,s)7→L_g(x,[τ−ρ₀,s]) is continuous. The continuity is thenuniformsince the function is compactly supported.

The condition (JC) implies that (2.10) holds; since one has, for all 0< ρ≤ρ0, L^∗_g(B(τ, ρ)) = sup

x∈R

Lg(x,[τ−ρ0, τ+ρ])−Lg(x,[τ−ρ0, τ−ρ])

. (2.11)

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This argumentation can be extended to the general case whereN≥1 is arbitrary;

the condition (JC) is then defined as follows:

Definition 2.6 (jointly continuous local time)

The function g is said to have a jointly continuous local time on a fixed compact rectangleQN

l=1[al,bl]ofR^N, if Lg(•,QN

l=1[al,bl])exists, and the function from R×QN

l=1[a_l,b_l] intoR+,(x,s₁, . . . ,s_N)7→L_g(x,QN

l=1[a_l,s_l])is continuous.

Studying local behavior of erratic deterministic functions via their local times is very unusual. This strategy is much more adapted to the random framework of stochastic processes and fields.

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Organization of the talk

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SαS random variables

Let(Ω,A,P)be aprobability space; the integralR

Ω(·)dP, with respect to the probability measureP, is calledthe expectation operatorand denoted byE(·).

A real-valuedrandom variable Z on Ω is a measurable function from (Ω,A) into (R,B(R)). Z is (almost) completely determined by its characteristic function:

ΦZ(ξ) :=E(e^iξZ), for allξ∈R. (3.1) Letα∈(0,2] be a parameter (called the stability parameter), one says thatZ is Symmetricα-Stable (SαS), if there existsσ(Z)≥0 (called the scale parameter ofZ), such that

ΦZ(ξ) = exp − |σ(Z)ξ|^α), for anyξ∈R. (3.2)

→Whenα= 2,Z reduces to a centered Gaussian random variablewith Var(Z) =E(Z²) = 2σ(Z)²; observe thatE(|Z|^γ)<+∞, for everyγ∈R+.

→The situation is different whenα∈(0,2);Z has a heavy-tailed distribution, which, in particular implies thatE(|Z|^γ) = +∞, as soon asγ≥α.

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Local times of sample paths

Astochastic fieldY on Ω is a collection{Y(t),t ∈R^N}of real-valued random variables on Ω; whenN= 1, thenY is called astochastic process.

For all fixedω∈Ω,the functionY(·, ω), fromR^N toR,t 7→Y(t, ω) is called a sample pathof the fieldY. In the setting of this talk, any sample path is always a continuous onR^N. Each Y(·, ω) plays the same role as the functiong in the previous section. The associated occupation measure, on an arbitrary compact T ⊂R^N,is denoted byµ_Y_,T(•, ω), instead ofµ_Y_(·,ω),T(•). Thus,when it exists, the corresponding local time is denoted byLY(•,T, ω).

→One of the main messages, we would like to deliver, is the following:

”Basically, it is a less difficult problem to obtain a generic result onP-almost all local timesLY(•,T, ω) (that is an almost sure result onLY(•,T)), than a result on a unique specified local timeLg(•,T)”. We mention that the Fubini-Tonelli’s theorem plays an important role in this issue.

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Existence and integrability of local times of S αS fields

One says that{Y(t),t∈R^N}is aSαS field, iffany linear combination of the random variablesY(t)’s is aSαS random variable: ∀m∈N,∀a1, . . . ,am∈R and∀t¹, . . . ,t^m∈R^N, the random variablePm

l=1a_lY(t^l) isSαS.

σ Pm

l=1a_lY(t^l)

denotes the scale parameter ofPm

l=1a_lY(t^l).

Theorem 3.1

Let{Y(t),t∈R^N}be aSαS field and T a compact ofR^N. A ”simple” sufficient condition for the local time LY(•,T)to exist almost surely is that

Z

T²

σ Y(t¹)−Y(t²)−1

dt¹dt²<+∞. (3.3) Notice that, under this condition, one also has(x, ω)7→L_Y(x,T, ω)∈L²(R×Ω).

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Proof: For each ω∈Ω, the Fourier transform of the occupation measure µY,T(•, ω), is given by

µbY,T(ξ, ω) = Z

T

e^−iξY^(t,ω)dt, for allξ∈R. (3.4) Therefore

Z

Ω

Z

R

µbY,T(ξ, ω)

2dξdP(ω) = Z

Ω

Z

R

Z

T²

e^iξ(Y^(t¹^,ω)−(Y^(t²^,ω))dt¹dt²dξdP(ω)

= Z

T²

Z

R

Z

Ω

e^iξ(Y^(t¹^,ω)−(Y^(t²^,ω))dP(ω)dξdt¹dt² (Fubini-Tonelli)

= Z

T²

Z

R

Φ_Y_(t1)−Y(t²)(ξ)dξdt¹dt²

= Z

T²

Z

R

exp − |σ(Y(t¹)−Y(t²))ξ|^α)dξdt¹dt²(setη=σ(Y(t¹)−Y(t²))ξ)

=Z

R

e^−|η|^αdηZ

T²

σ Y(t¹)−Y(t²)−1

dt¹dt² .

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Remark 3.2

For all f ∈L^α(R^N), the stochastic integralR

R^Nf(ξ)Me_α(dξ)is a complex-valued random variable whose real part isSαS with a scale parameter equals to the (quasi) normkfk_Lα(R) :=

R

R^N

f(ξ)

αdξ1/α

.

Definition 3.3 (Harmonizable Fractional Stable Field (HFSF))

The HFSF, of Hurst parameter H∈(0,1) and of stability parameterα∈(0,2], is theSαS stochastic field denoted by X ={X(t),t ∈R^N}and given by

X(t) :=Re Z

R^N

e^it·ξ−1

|ξ|^H+N/αMeα(dξ), for all t∈R^N, (3.5) where|ξ|is the Euclidian norm ofξ, and t·ξthe classical inner product of t andξ.

HFSF generalizes classical processes and fields:

1 Brownian Motion(α= 2,N= 1, andH= 1/2);

2 Fractional Brownian Motion(α= 2,N= 1, andH arbitrary);

3 Fractional Brownian Field(α= 2 andNandH arbitrary).

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Proposition 3.4

Let X be a HFSF of arbitrary parameters, and T a compact ofR^N. Then the local time L_X(•,T)exists almost surely, moreover the function

(x, ω)7→L_X(x,T, ω)belongs toL²(R×Ω).

Proof: using the definition ofX and elementary properties of the stochastic integral in it, one can show that, for some constantc >0, one for has

σ X(t¹)−X(t²)

=c|t¹−t²|^H, for allt¹,t²∈R^N. (3.6) Therefore

Z

T²

σ X(t¹)−X(t²)−1

dt¹dt²<+∞. (3.7)

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Organization of the talk

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Theorem 4.1 (Kolmogorov’s continuity theorem)

Let Q∈N, J a compact rectangle ofR^Q, and{Z(θ), θ∈J} a stochastic field. A sufficient condition for its sample paths to be, almost surely, continuous functions on J is the following: there exist 3 positive constantsβ, c, andν such that

E |Z(θ⁰)−Z(θ⁰⁰)|^β

≤c|θ⁰−θ⁰⁰|^Q+ν, for allθ⁰, θ⁰⁰∈J. (4.1) LetI :=QN

l=1[a_l,b_l] be a compact rectangle ofR^N; for anys= (s₁, . . . ,s_N)∈I we setI(s) :=QN

l=1[al,sl]⊆I. Let{Y(t),t ∈R^N}be a stochastic field such that the local timeLY(•,I) exists almost surely, and belongs toL²(R×Ω). To say thatLY is almost surely jointly continuous onI means that, for any fixedr>0, the stochastic field

LY(x,I(s)), (x,s)∈[−r,r]×I has, almost surely,

continuous paths. To derive such a result, the classical strategy consists in trying to use Theorem 4.1, whenQ=N+ 1,J = [−r,r]×I,θ= (x,s), and

Z(θ) =LY(x,I(s)). Thus, one has to find, for some fixed well-chosen even integer k =β ≥2, an upper bound, of the same form as in (4.1), for the quantity

∆k(x⁰,x⁰⁰;s⁰,s⁰⁰) :=E

LY(x⁰,I(s⁰))−LY(x⁰⁰,I(s⁰⁰))

k

, (4.2)

where (x⁰,x⁰⁰)∈[−r,r]² and (s⁰,s⁰⁰)∈I²are arbitrary.

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Let us ”separate” the increment ins from that inx. One has

∆k(x⁰,x⁰⁰;s⁰,s⁰⁰)≤2^k⁻¹ ∆˜k(x⁰;s⁰,s⁰⁰) + ˇ∆k(x⁰,x⁰⁰;s⁰⁰)

. (4.3)

Notice that

∆˜k(x⁰;s⁰,s⁰⁰) :=E

LY(x⁰,I(s⁰))−LY(x⁰,I(s⁰⁰))

k

≤E

LY x⁰,I(s⁰)\I(s⁰⁰)k +E

LY x⁰,I(s⁰⁰)\I(s⁰)k

≤2 card(S)k−1X

j∈S

E LY(x⁰,Bj)^k

, (4.4)

whereS is a finite set such thatcard(S) |s⁰−s⁰⁰|^1−N, and eachB_j is a cube included inI satisfyingdiam(B_j) |s⁰−s⁰⁰|.

Also, notice that

∆ˇk(x⁰,x⁰⁰;s⁰⁰) :=E

LY(x⁰,I(s⁰⁰))−LY(x⁰⁰,I(s⁰⁰))k

. (4.5)

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Thus, it is useful to obtain convenient upper estimates for:

1 thekth moment of the local timeUk(x,B) :=E LY(x,B)^k

, wherex ∈R and the cubeB ⊆I are abitrary;

2 thekth moment of its increments Wk(x,y,R) :=E

LY(x,R)−LY(y,R)^k

, where (x,y)∈R²and the rectangleR⊆I are abitrary.

We only studyUk(x,B) sinceWk(x,y,R) can be studied in a rather similar way.

Using the heuristic computations LY(x,B) =F⁻¹(µbY,B)(x) = 1

2π Z

R

e^−ixξµbY,B(ξ)dξ= 1 2π

Z

R

e^−ixξ Z

B

e^iξY^(t)dtdξ and Fubini-Tonelli’s theorem, we get (see Geman and Horowitz, 1980), that

Uk(x,B) = 1 (2π)^k

Z

B^k

Z

R^k

exp

−ix

k

X

l=1

ξl

E

exp

i

k

X

l=1

ξlY(t^l)

dξd t, (4.6)

whereξ= (ξ1, . . . , ξk) andt = (t¹, . . . ,t^k). The quantityE

eⁱ^P^k^l=1^ξ^l^Y^(t^l⁾ is the value at 1 of the characteristic function of the random variablePk

l=1ξlY(t^l).

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Thus, whenY is a centered Gaussian field, one has that E

exp

i

k

X

l=1

ξlY(t^l)

= exp

−2⁻¹VarX^k

l=1

ξlY(t^l)

(4.7) and consequently that

Z

R^k

E

exp i

k

X

l=1

ξlY(t^l) dξ=

s (2π)^k

det CovMat(Y(t¹), . . . ,Y(t^k)). (4.8) Thus, the problem of finding a convenient upper estimate for thekth moment of the local timeUk(x,B), reduces to that of finding a convenient lower estimate for det CovMat(Y(t¹), . . . ,Y(t^k))

. This is in fact a Gram determinant. Therefore

det CovMat(Y(t¹), . . . ,Y(t^k))

=Var Y(t¹)

k

Y

m=2

Var Y(t^m)|Y(t¹), . . . ,Y(t^m−1 ,

(4.9) where

q

Var Y(t^m)|Y(t¹), . . . ,Y(t^m−1

:=dist_L2(Ω)

Y(t^m),span

Y(t¹), . . . ,Y(t^m−1) . (4.10)

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The concept of local nondeterminism (LND) was first introduced by Berman (1973) in the framework of Gaussian processes. Pitt (1978) extended it to the framework of Gaussian fields, and Nolan (1989) to that ofSαS fields.

Definition 4.2 (LND for Gaussian stochastic fields)

A centered Gaussian field{Y(t),t∈R^N}is said to be LND on a compact rectangle I ⊂R^N, if for any fixed integer m≥2, and for all points t¹, . . . ,t^m∈I sufficiently close together, one has

Var Y(t^m)|Y(t¹), . . . ,Y(t^m−1

≥c min

1≤q<mVar Y(t^m)−Y(t^q)

, (4.11) where c>0 is a constant which may only depend on I and m.

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Remark 4.3

When the centered Gaussian field{Y(t),t∈R^N}has a stochastic integral representation of the form: for all t∈R^N,

Y(t) =Re Z

R^N

K(t, ξ)Me2(dξ), whereK(t,·)∈L²_ξ(R^N). (4.12) Then, thanks to the isometry property (fromL²_ξ(R^N)intoL²(Ω)) of this

stochastic integral, the inequality Var Y(t^m)|Y(t¹), . . . ,Y(t^m−1

≥c min

1≤q<mVar Y(t^m)−Y(t^q) ,

can be expressed in terms of the deterministic kernel functionK:

dist_L2 ξ(R^N)

K(t^m,·),span

K(t¹,·), . . . ,K(t^m−1,·)

(4.13)

≥√ c min

1≤q<m

K(t^m,·)− K(t^q,·) _L₂

ξ(R^N).

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Definition 4.4 (LND forSαS stochastic fields)

Letα∈(0,2]and{Y(t),t ∈R^N}a SαS field having a stochastic integral representation of the form: for all t∈R^N,

Y(t) =Re Z

R^N

K(t, ξ)Meα(dξ), where K(t,·)∈L^α_ξ(R^N). (4.14) Such a field is said to be LND on a compact rectangle I ⊂R^N, if for any fixed integer m≥2, and for all points t¹, . . . ,t^m∈I sufficiently close together, one has

dist_L^α

ξ(R^N)

K(t^m,·),span

K(t¹,·), . . . ,K(t^m−1,·)

(4.15)

≥c min

1≤q<m

K(t^m,·)− K(t^q,·) _L_α

ξ(R^N), where c>0 is a constant which may only depend on I and m.

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Definition 4.5 (Nolan (1989), locally appoximately independent increments) Let{Y(t),t∈R^N}be aSαS field which has the same stochastic integral representation, through a kernel functionK, as in Definition 4.4. Such a field is said to have locally appoximately independent increments on a compact rectangle I ⊂R^N, if for any fixed integer m≥2, for every real numbers b1, . . . ,bm, and for all points t¹, . . . ,t^m∈I sufficiently close together, one has

c⁻¹

b1K(t¹,·) _Lα

ξ(R^N)+

m

X

j=2

bj K(t^j,·)− K(t^j−1,·) _Lα

ξ(R^N)

≤

b1K(t¹,·) +

m

X

j=2

bj K(t^j,·)− K(t^j−1,·) _Lα

ξ(R^N)

(4.16)

≤c

b1K(t¹,·) _L_α

ξ(R^N)+

m

X

j=2

bj K(t^j,·)− K(t^j−1,·) _L_α

ξ(R^N)

,

where c>0 is a constant which may only depend on I and m.

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Theorem 4.6 (Nolan (1989))

Let{Y(t),t∈R^N}be aSαS field which has the same stochastic integral representation as in Definition 4.4. Then this field is LND on a compact rectangle I ⊂R^N if and only if it has locally appoximately independent increments on I . For proving this theorem Nolan made use of arguments from Linear Algebra relying on a generalization of the notion of Gram determinant.

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Organization of the talk

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Recall that the Harmonizable Fractional Stable Field (HFSF), of Hurst parameter H∈(0,1) and of stability parameterα∈(0,2], is theSαS stochastic field denoted by{X(t),t∈R^N}and given by

X(t) :=Re Z

R^N

e^it·ξ−1

|ξ|^H+N/αMeα(dξ), for allt∈R^N, (5.1) where|ξ|is the Euclidian norm ofξ, andt·ξthe classical inner product oft andξ.

Theorem 5.1

Let I be a compact rectangle ofR^N such that0∈/ I . Then the HFSF {X(t),t ∈R^N} is LND on I .

This theorem was obtained by:

Pitt (1978) in the Gaussian caseα= 2;

Nolan (1989) in theSαS case, whereα∈[1,2);

Ayache and Xiao (recently) in theSαS case, whereα∈(0,1).

Notice that, in the Gaussian case, an alternative proof for Theorem 5.1 was proposed by Kahane. Actually, the method used by Ayache and Xiao, in their SαS setting, is, to a certain extent, inspired by this Kahane’s proof.

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Proof whenα= 2 (Kahane): The integerm≥2 is arbitrary an fixed. One has to show that for every real numbersa₁, . . . ,a_m−1, and for all pointst¹, . . . ,t^m∈I (sufficiently close together), the following inequality holds

Z

R^N

e^it^m^·ξ−1−

m−1

X

l=1

a_l e^it^l^·ξ−1

2 dξ

|ξ|^2H+N ≥c min

1≤q<m|t^m−t^q|^2H, (5.2)

wherec>0 is a constant which may only depend onI andm.

We suppose thatI := [ε,1]^N, ε∈(0,1) being arbitrary. We set r := min

1≤q<m|t^m−t^q| and ρ=ρ(r,|t^m|) := min r,|t^m|

. (5.3)

The inequality|t^m| ≥ε√

N implies that one has, for some constantc1>0, only depending onε,

ρ(r,|t^m|)≥c1r. (5.4)

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Next, letϕbe an arbitrary fixed infinitely differentiable function fromR^N into the compact interval [0,1] of the real line, such that

ϕ(0) = 1 andSuppϕ⊆

x ∈R^N: |x| ≤1 . (5.5) We denote byϕ_ρ the function fromR^N into the compact interval [0,1], defined as

ϕ_ρ(s) =ρ^−Nϕ ρ⁻¹s

for alls∈R^N. (5.6)

Observe that (5.5) and (5.6) entail that ϕρ(0) =ρ^−N andSuppϕρ ⊆

s∈R^N : |s| ≤ρ . (5.7) Next, let Λ = Λ a₁, . . . ,a_m−1;t¹, . . . ,t^m

be the integral defined as Λ :=

Z

R^N

e^−it^m^·ξϕbρ(ξ)

e^it^m^·ξ−1−

m−1

X

l=1

al e^it^l^·ξ−1

dξ, (5.8) whereϕbρ denotes the Fourier transform ofϕρ.

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Using (5.8) and the equality ϕρ(s) := (2π)^−N

Z

R^N

e^is·ξϕbρ(ξ)ds, for alls∈R^N, one gets that

Λ = Z

R^N

ϕb_ρ(ξ)dξ+

−1 +

m−1

X

l=1

a_lZ

R^N

e^−it^m^·ξϕb_ρ(ξ)dξ

−^m−1X

l=1

al

Z

R^N

e^−i(t^l^−t^m^)·ξϕbρ(ξ)dξ

= (2π)^−Nϕ_ρ(0) + (2π)^−N/2

−1 +

m−1

X

l=1

a_l

ϕ_ρ(−t^m) + (2π)^−Nϕ_ρ t^m−t^l .

Therefore, the equalityϕρ(0) =ρ^−N and the inequalities|t^m| ≥ρand min_1≤l<m|t^m−t^l| ≥ρimply that Λ = (2π)^−Nρ^−N.

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On the other hand, using Cauchy-Schwarz inequality, one obtains that

|Λ|² = Z

R^N

e^−it^m^·ξϕbρ(ξ)|ξ|^H+N/2

e^it^m^·ξ−1−

m−1

X

l=1

al e^it^l^·ξ−1 dξ

|ξ|^H+N/2

2

≤ Z

R^N

ϕb_ρ(ξ)

2|ξ|^2H+Ndξ× Z

R^N

e^it^m^·ξ−1−

m−1

X

l=1

a_l e^it^l^·ξ−1

2 dξ

|ξ|^2H+N.

Therefore,

(2π)^−2Nρ^−2N

≤ Z

R^N

ϕ(ρξ)b

2|ξ|^2H+Ndξ× Z

R^N

e^it^m^·ξ−1−

m−1

X

l=1

a_l e^it^l^·ξ−1

2 dξ

|ξ|^2H+N

=ρ^−2(H+N) Z

R^N

ϕ(η)b

2|η|^2H+Ndη

× Z

R^N

e^it^m^·ξ−1−

m−1

X

l=1

al e^it^l^·ξ−1

2 dξ

|ξ|^2H+N.

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Finally, one gets that Z

R^N

e^it^m^·ξ−1−

m−1

X

l=1

a_l e^it^l^·ξ−1

2 dξ

|ξ|^2H+N ≥c₂ρ^2H ≥c₃r^2H, (5.9) wherec2>0 andc3>0 are two constants only depending onH andN.