Harmonizable Fractional Stable Fields: Local Nondeterminism and Joint Continuity of the Local Times

Harmonizable Fractional Stable Fields: Local

Nondeterminism and Joint Continuity of the Local Times

Antoine Ayache and Yimin Xiao

Universit´e Lille 1 (USTL) - Laboratoire Paul Painlev´e and

Michigan State University (MSU) - Department of Statistics and Probability

Organization of the talk

1 Introduction

2 Recalls on LND for stable fields

3 Our main result and its proof

4 Joint continuity of the local times

Introduction

Harmonizable Fractional Stable Field (HFSF)

The real-valued HFSF, denoted byX ={X(t),t ∈R^N}, is one of the most classical extensions of the well-known Fractional Brownian Field (FBF), to the setting of heavy-tailed stable distributions. It is parameterized byα∈(0,2) and H∈(0,1), thestabilityandHurstparameters; moreover it is defined as:

X(t) :=cRe Z

R^N

e^it·ξ−1

|ξ|^H+N/αMe_α(dξ), ∀t ∈R^N; (1.1) Meαdenotes a complex-valued rotationally invariant α-stable random measure with Lebesgue control measure, thus,

Re Z

R^N

f(ξ)Meα(dξ)

α

α

= Z

R^N

f(ξ)

αdξ, ∀f ∈L^α(R^N); (1.2) X isH-self-similar with stationary and isotropic increments;

c=c(α,H,N)>0 denotes the normalizing constant chosen such that kX(t)k_α=|t|^H, ∀t ∈R^N. (1.3)

Several authors have studied various path properties of HFSF and its local times, for example:

using LePage series, Kˆono and Maejima (1991), showed that, on any compact rectangle ofR^N, the paths are, almost surely, H¨older continuous functions of an arbitrary orderγ <H; later, Xiao (2010) obtained a more precise uniform modulus of continuity for them;

In the case whereα∈[1,2), Nolan (1989) investigated the regularity of the local times of anR^d-valued HFSF (its components are i.i.d. real-valued HFSF); Kˆono and Shieh (1993) as well as Shieh (1993), studied existence and joint continuity of the intersection local times.

The concept oflocal nondeterminism (LND)is the keystone of most of the works on regularity of local times of Gaussian and stable fields/processes. Roughly speaking,it is a kind of generalization of the notion of independent increments.

Introduction

The concept of 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 of stable fields.

For HFSF, the property of LND was proved by Nolan (1989) for 1≤α <2.

However, the problem for the case 0< α <1 had remained open. The main objective of this talk is to resolve this problem.

Organization of the talk

1 Introduction

2 Recalls on LND for stable fields

3 Our main result and its proof

4 Joint continuity of the local times

Recalls on LND for stable fields

LetY ={Y(t),t ∈R^N} be an arbitrary real-valued stable field such that, Y(t) =Re

Z

R^N

h(t, ξ)Se_α(dξ), ∀t ∈R^N; (2.1)

Se_αis a complex-valued rotationally invariantα-stable random measure onR^N with an arbitrary control measure ∆;

h(t,·) :R^N →C(t ∈R^N) is a family of functions belonging toL^α(∆) i.e.

h(t,·)

α L^α(∆):=

Z

R^N

|h(t, ξ)|^α∆(dξ)<∞, ∀t ∈R^N. (2.2) Let the random vectorYt¹,...,t^m := Y(t¹), . . . ,Y(t^m)

wheret¹, . . . ,t^m∈R^N are arbitrary; its characteristic function Υ_Y

t1,...,tm defined as, Υ_Yt1,...,tm(d1, . . . ,dm) :=Eexp

i

m

X

n=1

dnY(tⁿ)

, ∀(d1, . . . ,dm)∈R^m, (2.3) can be expressed as,

Υ_Yt1,...,tm(d1, . . . ,dm) = exp

−

m

X

n=1

dnh(tⁿ,·)

α L^α(∆)

. (2.4)

Hence, the scale parameter of the symmetricα-stable random variable Pm

n=1dnY(tⁿ) is given by

m

X

n=1

dnY(tⁿ) _α

=

m

X

n=1

dnh(tⁿ,·) _Lα(∆)

. (2.5)

Whenm≥2, theL^α(∆)-distance from h(t^m,·) to the subspace ofL^α(∆) spanned by

h(t¹,·), . . . ,h(t^m−1,·) is defined as, inf

(

h(t^m,·)−

m−1

X

n=1

b_nh(tⁿ,·) _Lα(∆)

: ∀b₁, . . . ,b_m−1∈R )

; (2.6)

notice that the infimum is attained. Because of the analogy with the Gaussian case (α= 2), this infimum can be viewed as theL^α-error of predictingY(t^m), givenY(t¹), . . . ,Y(t^m−1); therefore we denote it by,

Y(t^m)

Y(t¹), . . . ,Y(t^m−1) _α.

Recalls on LND for stable fields

To state the property of LND in the sense of Nolan (1989), we will make use of the following partial order inR^N: for any m≥2 distinct pointst¹, . . . ,t^m∈R^N,

t¹4t²4· · ·4t^m⇐⇒ ∀1≤i<j≤m, |t^j−t^j−1| ≤ |t^j−tⁱ|; (2.7) note that the partial order defined by (2.7) is not unique (including the case N= 1); one reason is that the choice of the point to be labeledt^m can be done in mdifferent ways.

Definition 2.1 (Nolan (1989), local nondeterminism (LND))

Let I ⊂R^N be a compact rectangle. The stable field Y is said to be LND on I , if:

(a) Y(t)

_α>0, for every t ∈I ; (b)

Y(s)−Y(t)

_α>0for all distinct s,t∈I sufficiently close together;

(c) for each fixed integer m≥2, lim inf

t¹4t²4···4t^m,|t^m−t¹|→0

Y(t^m)

Y(t¹), . . . ,Y(t^m−1) _α kY(t^m)−Y(t^m−1)kα

>0. (2.8)

Conditions (a) and (b) rule out degeneracy; condition (c) is the important one.

The ratio in (2.8) is a relative linear prediction error and is always between 0 and 1. If the ratio is bounded away from 0 as|t^m−t¹| →0, then we can approximate Y(t^m) ink · kα(quasi)-norm, by the ”most recent” valueY(t^m−1) alone, with the same order of error as with all the ”past”Y(t¹), . . . ,Y(t^m−1).

Recalls on LND for stable fields

Definition 2.2 (Nolan (1989), locally appoximately independent increments)

Y is said to have locally appoximately independent increments on I , if:

(a) Y(t)

_α>0, for every t ∈I ; (b)

Y(s)−Y(t)

_α>0for all distinct s,t∈I sufficiently close together;

(c⁰) for each fixed integer m≥2, there is a constant c_m>0, such that the inequalities,

c_m⁻¹

b₁Y(t¹) _α +

m

X

j=2

b_j Y(t^j)−Y(t^j−1) _α

≤

b1Y(t¹) +

m

X

j=2

bj Y(t^j)−Y(t^j−1) _α

(2.9)

≤ cm

b1Y(t¹) _α+

m

X

j=2

bj Y(t^j)−Y(t^j−1) _α

,

hold, for every b1, . . . ,bm∈Rand all t¹4t²4· · ·4t^m sufficiently close.

Theorem 2.3 (Nolan (1989))

Y is LND on I if and only if Y has locally appoximately independent increments on I .

For proving this theorem Nolan made use of arguments from Linear Algebra relying on generalized Grammian:

g(V¹, . . . ,V^m) :=

V¹ ×

V²−span{V¹} × V³−span{V¹,V²}

×. . .×

V^m−span{V¹, . . . ,V^m−1} ,

which, generally speaking, provides a measure of dependence between arbitrary vectorsV¹, . . . ,V^min an (quasi)-normed linear space.

Our main result and its proof

Organization of the talk

1 Introduction

2 Recalls on LND for stable fields

3 Our main result and its proof

4 Joint continuity of the local times

Theorem 3.1 (Ayache and Xiao (2013))

Let I be a compact rectangle ofR^N such that0∈/ I . Then, for all the possible values of its parametersαand H, the HFSF X ={X(t),t∈R^N}satisfies the following property: for any fixed integer m≥2, there exists a constant c1=c1(m)>0, depending on m,α, H, N and I only, such that for all t¹, . . . ,t^m∈I , one has,

X(t^m)

X(t¹), . . . ,X(t^m−1)

_α≥c1min

|t^m−tⁿ|^H : 1≤n<m . (3.1) Theorem 3.1 entails thatX is LND onI, in the sense of Nolan (1989). Indeed, the equality

kX(s)−X(t)kα=|s−t|^H, ∀s,t ∈R^N, (3.2) implies thatkX(s)−X(t)kα>0 whens6=t; moreover, combining (3.2) with X(0) = 0 and 0∈/ I, one gets that inft∈IkX(t)kα>0. On the other hand, assuming thatt¹4t²4· · ·4t^m and using (3.2), the minimum in (3.1), reduces tokX(t^m)−X(t^m−1)kα; thus it results from (3.1) that,

inf

X(t^m)

X(t¹), . . . ,X(t^m−1) _α

≥c1>0. (3.3)

Our main result and its proof

From now on, we suppose thatI := [ε,1]^N,ε∈(0,1) being arbitrary. Recall that X(t) :=cRe

Z

R^N

K(t, ξ)Meα(dξ), where K(t, ξ) := e^it·ξ−1

|ξ|^H+N/α; (3.4) thus, for proving the previous theorem, one has to show that, the following inequality holds, for allt¹, . . . ,t^m∈I andb1, . . . ,bm∈R,

X(t^m)−

m−1

X

n=1

bnX(tⁿ)

α α

(3.5)

= Z

R^N

K(t^m, ξ)−

m−1

X

n=1

bnK(tⁿ, ξ)

α

dξ≥c₁^αmin

|t^m−tⁿ|^αH : 1≤n<m .

Lemma 3.2

It is sufficient to establish the inequality in (3.5), only whenmax_1≤n<m|bn| ≤1.

Sketch of the Proof of Lemma 3.2: the proof can be done by induction onm, here we only give it at the initial stepm= 2. By the assumption of the lemma, one knows that, for some constantc1>0, the inequality,

X(t²)−b1X(t¹)

α

α≥c₁^α|t²−t¹|^αH, (3.6) holds, for allt¹,t²∈I andb₁∈Rsatisfying|b1| ≤1. Let us show that it remains true whenb₁is replaced by a real numbera₁such that|a1|>1. One clearly has,

X(t²)−a₁X(t¹)

α

α = |a₁|^α

X(t¹)−a₁⁻¹X(t²)

α α

≥

X(t¹)−a⁻¹₁ X(t²)

α

α; (3.7)

then using (3.6), in which,t¹andt²are interchanged, andb₁=a⁻¹₁ , one gets that

X(t²)−a₁X(t¹)

α

α≥c₁^α|t²−t¹|^αH. (3.8)

Our main result and its proof

From now on, we assume that max_1≤n<m|bn| ≤1. We set

ρ:= 2⁻¹εN^−1/2min{|t^m−tⁿ|: 1≤n<m}; (3.9) thus,ρ∈[0,2⁻¹ε], moreover, there is no restriction to suppose thatρ6= 0. Let c_∗∈ 0,(16N)^−1/2

be an arbitrary constant, later we will impose to it to be very small. Denote byj0≥1, the unique integer such that

2^−j0−1<c_∗ρ≤2^−j0. (3.10) Thus, the fact thatK(t, ξ) := _|ξ|^eit·ξH+N/α⁻¹ , and the change of variable η= 2^−j0ξ, show that, the inequality

Z

R^N

K(t^m, ξ)−

m−1

X

n=1

b_nK(tⁿ, ξ)

α

dξ≥c₁^αρ^αH, (3.11) we want to prove, is equivalent to the inequality

Z

R^N

K(2^j0t^m, η)−

m−1

X

n=1

bnK(2^j0tⁿ, η)

α

dη ≥c2>0, (3.12) wherec2>0 is some constant non depending on thetⁿ’s and thebn’s.

In order to establish (3.12), we will make use of the ”Fourier transform” of a Lemari´e-Meyer orthonormal wavelet basis ofL²(R^N). We denote such a basis, by

n

2^jN/2ψl(2^j· −k) : 1≤l≤2^N−1;j ∈Z,k ∈Z^N o

;

a specificity of it, is that the mother waveletsψ_l, belong to the Schwartz class S(R^N), and have compactly supportedC^∞ Fourier transforms, satisfying,

suppψb_l⊆∆ :=h

−8π 3 ,8π

3 i^N

\

−2π 3 ,2π

3 ^N

. (3.13)

In view of (3.13), let us considerbg :R^N →[0,1], an even compactly supported C^∞ function, such that,

suppbg ⊆∆ and bg(4π/3,0, . . . ,0) = 1. (3.14) Then, for each fixedt ∈R^N, K(t,e ·) :=K(t,·)bg(·), is aC^∞ function with a compact support included in ∆.

Our main result and its proof

In view of the fact thatgbis with values in [0,1], for proving (3.12), it is sufficient to show that

A_α:=

Z

R^N

K(2e ^j0t^m, η)−

m−1

X

n=1

b_nK(2e ^j0tⁿ, η)

α

dη≥c₂>0. (3.15)

Observe that, sup

(t,ξ)∈R^N×R^N

eK(t, ξ)

≤ sup

(t,ξ)∈R^N×∆

|e^it·ξ−1|

|ξ|^H+N/α ≤2 3 2π

H+N/α

:=c₃; (3.16) therefore, using max1≤n<m|bn| ≤1 andα∈(0,2), one gets,

A_α = m^αc₃^α Z

R^N

m^−αc₃^−α

K(2e ^j0t^m, η)−

m−1

X

n=1

b_nK(2e ^j0tⁿ, η)

α

dη

≥ m^αc₃^α Z

R^N

m⁻²c₃⁻²

K(2e ^j0t^m, η)−

m−1

X

n=1

b_nK(2e ^j0tⁿ, η)

2

dη

≥ m^α−2c₃^α−2A2. (3.17)

A2is in fact the square of theL²(R^N)-norm of the function F(·) :=K(2e ^j0t^m,·)−

m−1

X

n=1

bnK(2e ^j0tⁿ,·).

Thus, denoting by

wl,j,k : 1≤l≤2^N−1;j ∈Z,k ∈Z^N the sequence of the coefficients ofF(·), in the orthonormal basis of L²(R^N),

n2^jN/2F ψ_l(2^j· −k)

: 1≤l≤2^N−1;j ∈Z,k ∈Z^N o; it follows from Parseval identity that

A2=

2^N−1

X

l=1

X

j∈Z

X

k∈Z^N

|wl,j,k|²≥ X

k∈Z^N

|w1,0,k|². (3.18) From now on, we setw_k :=w_1,0,k andψ:=ψ₁.

Our main result and its proof

wk :=

Z

R^N

F(η)e^−ik·ηψ(η)b dη

= Z

R^N

K(2e ^j0t^m, η)e^−ik·ηψ(η)b dη−

m−1

X

n=1

bn

Z

R^N

K(2e ^j0tⁿ, η)e^−ik·ηψ(η)b dη

= Z

R^N

e^i2j0tm·η−1

e^−ik·ηgb(η)ψ(η)b

|η|^H+N/α dη

−

m−1

X

n=1

bn

Z

R^N

e^i2j0tn·η−1

e^−ik·ηgb(η)ψ(η)b

|η|^H+N/αdη

= Θ 2^j0t^m−k

−

m−1

X

n=0

bnΘ 2^j0tⁿ−k

, (3.19)

where,t⁰:= 0,b0:= 1−Pm−1

n=1 bn and Θ is the function ofS(R^N) defined as, Θ(x) :=

Z

R^N

e^ix·ηgb(η)ψ(η)b

|η|^H+N/α dη, ∀x ∈R^N. (3.20)

Recall that

ρ:= 2⁻¹εN^−1/2min{|t^m−tⁿ|: 1≤n<m}; (3.21) and thatj0≥1 is defined as the unique positive integer such that

2^−j0−1<c_∗ρ≤2^−j0, (3.22) where, till now,c_∗∈ 0,(16N)^−1/2

is an arbitrary constant. Letν=ν(t^m, ρ,c_∗) be the set of indicesk located ”neart^m”, defined as,

ν :=

k ∈Z^N :|t^m−2^−j0k| ≤2⁻¹ρ . (3.23) One clearly has,

A^1/2₂ ≥ X

k∈Z^N

|wk|^21/2

≥ X

k∈ν

|wk|^21/2

; (3.24)

thus, using the expression ofwk in terms of Θ, and the triangle inequality, one gets,

Our main result and its proof

A^1/2₂ ≥ X

k∈ν

Θ 2^j0t^m−k

21/2

−

m−1

X

n=0

|b_n| X

k∈ν

Θ 2^j0tⁿ−k

21/2

. (3.25) From now on, our goal is to show that the second term in the right-hand side of the inequality (3.25), is negligible with respect to the first one; to this end, we will use the fact that Θ is a well localized function, namely:

|Θ(x)| ≤c₄ 1 +|x|−1

, ∀ x∈R^N. (3.26)

It follows from (3.26),|tⁿ−t^m| ≥ρ,|t^m−2^−j0k| ≤2⁻¹ρ, and the triangle inequality, that|2^j0tⁿ−k| ≥4⁻¹c_∗⁻¹ and consequently that

X

k∈ν

Θ 2^j0tⁿ−k

2≤c₅c_∗, (3.27)

wherec₅>0 is some constant non depending onj₀, thetⁿ’s and c_∗.

On the other hand, using the fact that, for allx∈R^N andk ∈Z^N, Θ(x−k) :=

Z

R^N

e^i(x−k)·η bg(η)ψ(η)b

|η|^H+N/α dη (3.28)

= Z

[0,2π]^N

e^−ik·η



 X

q∈Z^N

e^{ix·(η+2πq)} bg(η+ 2πq)ψ(ηb + 2πq)

|η+ 2πq|^H+N/α



dη,

one can show that

c6:= infn X

k∈Z^N

Θ(x−k)

2:x ∈R^N o

>0. (3.29)

Then, the localization property of Θ and the inequality|2^j0t^m−k| ≥4⁻¹c_∗⁻¹for allk ∈/ν, imply that

X

k∈ν

Θ 2^j0t^m−k

2≥c6−c7c_∗, (3.30)

wherec >0 is some constant non depending onj , thetⁿ’s and c .

Our main result and its proof

Finally, using (3.25), (3.27), (3.30), and the inequality max0≤n≤m|bn| ≤m+ 1, one gets that

A^1/2₂ ≥ X

k∈ν

Θ 2^j0t^m−k

21/2

−

m−1

X

n=0

|bn| X

k∈ν

Θ 2^j0tⁿ−k

21/2

≥ √

c6−c7c_∗−m(m+ 1)√

c5c_∗, (3.31)

which in turn entails that

A^1/2₂ ≥2⁻¹√

c6>0, (3.32)

provided that one imposes to the constantc∗ to be small enough.

Organization of the talk

1 Introduction

2 Recalls on LND for stable fields

3 Our main result and its proof

4 Joint continuity of the local times

Joint continuity of the local times

Now, we consider theR^d-valued HFSFX~ ={X~(t),t∈R^N}defined by X~(t) = X₁(t), . . . ,X_d(t)

, ∀t ∈R^N, (4.1)

whereX1, . . . ,Xd are independent copies of the real-valued HFSFX. Thanks to the LND property ofX we have just derived, we will establish the joint continuity for the local times ofX~.

Let us first make a brief recall on local times and their joint continuity. For any fixed compact rectangleT ⊂R^N, the pathwise occupation measure ofX~ onT, denoted byµT, is defined as the following Borel measure onR^d:

µT(•) =λN

t ∈T :X~(t)∈ • . (4.2)

IfµT is almost surely absolutely continuous with respect to the Lebesgue measure λ_d onR^d, thenX~ is said to have local times onT, and its local timeL(·,T) is defined as the Radon-Nikod´ym derivative of µ_T with respect toλ_d, i.e.,

L(x,T) = dµ_T

dλ_d(x), ∀x ∈R^d. In the above,x is thespace variable, andT is the time variable.

Notice that whenX~ has local times onT, then it also has local times on any rectangleS ⊆T (consequence of Radon-Nikod´ym-Lebesgue Theorem).

Express the rectangleT asT =QN

l=1[a_l,a_l+h_l] wherea_l ∈Randh_l ∈R+, for alll= 1, . . . ,N. ThenX~ is said to have jointly continuous local times onT, whenever we can choose a version of the local times, still denoted byL ·,•

, such that, almost surely, the function

(x,t1,· · · ,tN)7−→L x,

N

Y

l=1

[al,al+tl] ,

is continuous onR^d×QN i=1[0,h_i].

We refer to Geman and Horowitz (1980) and Dozzi (2002) for further information on local times of random fields. At last, it is worth mentioning that it is shown in Adler (1981) that, when local times are jointly continuous, then, for each fixed x∈R^d,L(x,•) can be extended to be a finite Borel measure supported on the level set

X~_T⁻¹(x) =

t∈T : X~(t) =x ; (4.3)

thus, local times are useful in studying various fractal properties of level sets and

Joint continuity of the local times

Theorem 4.1 (Ayache and Xiao (2013))

X~ ={X~(t),t ∈R^N}, theR^d-valued HFSF with Hurst parameter H, has a jointly continuous local times on T when N >Hd .

The strategy of the proof of Theorem 4.1 is rather classical. It is based on Kolmogorov continuity Theorem, which can be used thanks to the following two estimates:

(i) For all integerm≥1, there exists a finite constantc1>0, which depends on m, such that for any cubeB ⊆T andx∈R^d,

E

L(x,B)^m

≤c1 diam(B)^m(N−Hd)

. (4.4)

(ii) For all even integerm≥2 andγ∈ 0,1∧¹₂(_Hd^N −1)

, there exists a finite constantc2>0, which depends onmandγ, such that for any cubeB ⊆T andx,y ∈R^d with|x−y| ≤1,

E h

L(x,B)−L(y,B)^mi

≤c2|x−y|^mγ diam(B)m(N−H(d+γ))

. (4.5)

For proving (i) and (ii), let us recall the following two identities from Geman and Horowitz (1980):

(i⁰) For allx ∈R^d and integerm≥1, E

h

L(x,B)^mi

= (2π)^−md Z

B^m

Z

R^md

exp

−i

m

X

j=1

u^j·x

Φ(u,t)d u d t

. (4.6)

(ii⁰) For allx,y ∈R^d and even integerm≥2, E

h

L(x,B)−L(y,B)mi

=(2π)^−md Z

B^m

Z

R^md m

Y

j=1

h

e^−iuj·x−e^−iuj·yi

×Φ(u,t)d u d t, (4.7)

whereu= (u¹, . . . ,u^m),t = (t¹, . . . ,t^m) and Φ(·,t) is the characteristic function of X~(t¹), . . . , ~X(t^m)

.

Joint continuity of the local times

From now on, our goal is to explain the main ideas for bounding the integrals in (4.6) and (4.7), by expressing, in a convenient way

Φ(u,t) :=Eexp

i

m

X

j=1

u^j·X~(t^j)

, ∀(u,t)∈R^md×B^m. (4.8) Observe that for anyc∈(0,+∞) and t⁰∈(0,+∞)^N,

Φ(u,t) =Eexp

i

m

X

j=1

c^Hu^j·

X c~ ⁻¹(t^j+t⁰)

−X c~ ⁻¹t⁰

, (4.9)

thanks to the stationarity of increments and self-similarity ofX~. Moreover, it can be seen that, given any compact rectangleT ⊂R^N, we can chooseε∈(0,1), c∈(0,+∞) andt⁰∈(0,∞)^N, such thats⁰:=c⁻¹t⁰∈I := [ε,1]^N and c⁻¹(t+t⁰)∈I for eacht ∈T. Thus, the change of variables

v^j =c^Hu^j, s^j =c⁻¹(t^j+t⁰), j= 1, . . . ,m, (4.10) allows to replace in the integrals (4.6) and (4.7), Φ(u,t),by

Φ(ve ,s) :=Eexp ^m

X

j=1

v^j·(X~(s^j)−X~(s⁰))

. (4.11)

Using an Abel transform one has

Φ(v,e s) = ˇΦ(w,s), (4.12)

where







w^j :=Pm

k=jv^k, j= 1, . . . ,m, Φ(wˇ ,s) :=Eexp

iPm

j=1w^j·(X~(s^j)−X~(s^j−1))

. (4.13)

Next, denoting byw_l^j,l= 1, . . . ,Nthe coordinates w^j, one gets Φ(wˇ ,s) =

N

Y

l=1

Eexp

i

m

X

j=1

w_l^j(X(s^j)−X(s^j−1))

, (4.14)

since theN coordinates ofX~ are independent copies of the HFSFX. Next, noticing thatEexp

iPm

j=1w_l^j(X(s^j)−X(s^j−1))

is the value at 1 of the characteristic function of the symmetricα-stable random variable

Pm

j=1w_l^j(X(s^j)−X(s^j−1)), one obtains

Joint continuity of the local times

Eexp

i

m

X

j=1

w_l^j·(X(s^j)−X(s^j−1))

= exp

−

m

X

j=1

w_l^j(X(s^j)−X(s^j−1))

α α

. (4.15) It is at this stage that the LND property ofX plays a crucial role. Indeed, thanks to it, we know that the incrementsX(s^j)−X(s^j−1),j= 1, . . .m, are

approximately independent; therefore there exists a constantc>0, non depending on thew_l^j’s and thes^j’s, such that,

m

X

j=1

w_l^j(X(s^j)−X(s^j−1))

α

α ≥ c

m

X

j=1

|w_l^j|^α

X(s^j)−X(s^j−1)

α α

= c

m

X

j=1

|w_l^j|^α|s^j−s^j−1|^αH. (4.16)