Convergence in variation of the laws of multiple stable integrals

Probability Theory

Convergence in variation of the laws of multiple stable integrals

Jean-Christophe Breton

Laboratoire de mathématiques et applications, Université de La Rochelle, avenue Michel Crépeau, 17042 La Rochelle cedex, France

Received 1 December 2002; accepted after revision 17 November 2003 Presented by Marc Yor

Abstract

Multiple stable integrals generalize Wiener–Itô integrals, their construction being based upon a generalized LePage representation. This approach allows one to study their behaviour. We are interested in this Note in the continuity for total variation norm of the laws of these integralsI_d(f )with respect tof. To cite this article: J.-C. Breton, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

2003 Académie des sciences. Published by Elsevier SAS. All rights reserved.

Résumé

Convergence en variation des lois des intégrales stables multiples. Les intégrales stables multiples généralisent celles de Wiener–Itô, leur construction est fondée sur une représentation de LePage généralisée. Cette approche permet d’étudier leur loi.

Nous nous intéressons dans cette Note à la continuité pour la variation totale des lois de ces intégralesI_d(f )par rapport àf. Pour citer cet article : J.-C. Breton, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

2003 Académie des sciences. Published by Elsevier SAS. All rights reserved.

Version française abrégée

Résultat principal

Nous considérons l’intégrale stable multipleI_d(f )par rapport à la mesure α-stable M sur [0,1] (que l’on suppose en plus symétrique siα1) définie pour f ∈L^α(log₊)^d−1([0,1]^d) dans [1] par et avec la propriété de représentationId(f )=^LSd(f )oùSd(f ):=C^d/αα

i>0[γ_i][Γ_i]^−1/αf (V_i)est une série de type LePage où les variables aléatoiresΓi,γi,Visont données après (3). Ce type d’intégrales généralise celles de Wiener–Itô et vérifie certaines de leurs propriétés telles que l’absolue continuité des lois (cf. [1]). Le résultat principal de cette Note vise à généraliser pour les intégrales stables celui de [3] obtenu dans le cas d’intégrales de Wiener–Itô. Plus précisément, en notant−→^var la convergence en variation, on montre que :

Théorème 0.1. SoientM comme précédemment (c’est-à-dire vérifiant (2)) et (f_n)_n>0 convergeant vers f ≡0 dansL^α(log₊)^d−1([0,1]^d), alors on aL(I_d(f_n))−→^var L(I_d(f )),n→ +∞.

Grâce à la régularité des lois prouvée dans [1], on a aussi :

E-mail address: [email protected] (J.-C. Breton).

1631-073X/$ – see front matter 2003 Académie des sciences. Published by Elsevier SAS. All rights reserved.

doi:10.1016/j.crma.2003.11.020

Corollaire 0.2. Avec les mêmes hypothèses que dans le Théorème 0.1, on a la convergence dansL¹(R)des densités deL(Id(fn))vers celle deL(Id(f )).

La preuve consiste à se ramener à des polynômes aléatoires en conditionnantS_d(f )puis à les étudier grâce aux propriétés des séries de LePage. On commence par quelques rappels sur la méthode de la superstructure utilisée.

Méthode de superstructure

Pour étudier une distribution fonctionnellePf⁻¹, Davydov se ramène à des distributions auxiliairesQεF_ε⁻¹ sur un espace plus large [2] : pour cela, on dispose en général de transformationsGc de l’espace X avec la propriétéP G⁻_c¹−→^var P,c→0, et on considère sur[0, ε] ⊗X la mesureQ_ε=(1/ε)λ|_[0,ε]×P et la fonctionnelle Fε(c, x)=f (Gcx). On a alorsQεF_ε⁻¹−→^var Pf⁻¹,ε→0, etQεF_ε⁻¹s’exprime comme mélange de mesures en dimension 1 qui sont les mesures images des restrictions def, comme en (6).

Esquisse de preuve

Soientf_n etf comme dans l’énoncé du le théorème etµ_n,µles lois de leurs intégrales. On trouve un multi- indice i^∗tel quef (V_i∗) =0 p.s. En supposant que l’espace est un espace produit(Ω,F, P )⊗(Ω,F, P)avec (Γ_i)_i>0, et Y:=(γ_i, V_i)_i>0 ne dépendant resp. que de (Ω,F, P ) et de (Ω,F, P), on se ramène d’abord à l’étude deµ_n,y=L(S_d(f_n)|Y=y)=^LF_n,y(Γ^−1/α)où avecA_n,i(y)=d!C_α^d/α[ i]f_n(u_i)poury =( , u), on a notéF_n,y(x)=

i∈T^dA_n,i(y)x_i1· · ·x_id. Notons alors p(y):=i_d^∗, et pour une suite t=(t_i)_i0, t_j =(t_i)_ij, t_j=(t_i)_ij les suites tronquées. Alors, en notant

ν:=L

Γ₁^−1/α, Γ₂^−1/α, . . . , Γ_p(y)^−1/αy=(γ_i, V_i)_i ett_p(y)+1=

Γ_i, ip(y)+1

qui admet une densité car∀n1, la loi de(Γ1, Γ2, . . . , Γn)conditionnellement à(Γn+1, Γn+2, . . .)en admet une, et en conditionnant maintenant par le futur deΓ,µetµns’expriment comme mélanges deν(Fn,y(·, tt_p(y)+1)⁻¹et ν(F_y(·, t_tp(y)+1)⁻¹respectivement par rapport aux lois de Y et(Γ_i)_ip(y)+1. D’après les propriétés des variations, il suffit dès lors de montrer

L ν

Fn,y(·, t_tp(y)+1)₋1 var

−→L ν

Fy(·, t_tp(y)+1)₋1

. (1)

Observons d’après la définition de F_n,y, que R_n,y,t(s) = F_n,y(s, t^−1/α_p(y)+1) est un polynôme en s = (s₁, . . . , s_p(y)). De même, nous associons un polynômeR_y,t àF_y.

Par choix de i^∗ et par conditionnement par le futur de Γ, le polynôme R_y,t admet comme terme de degré maximaldl’unique monômeX_i∗

1· · ·X_i∗

d. Fixonsv∈(R_∗)^p(y)de sorte quec→R_y,t(s+cv)est un polynôme non nul de degré maximald égal àA_i∗(y)v_i∗

1· · ·v_i∗

d =0. Considérons alors la famille de transformations deR^p(y)₊ données par G_c(s)=s+cv pour laquelle, par continuité des translations dans L¹(R^p(y)), on a νG⁻_c¹−→^varν, c→0. On applique maintenant la méthode de superstructure sur (R^p(y), ν) dont le principe a été décrit ci- dessus. En utilisant les notations correspondantes : par l’analogue de (5), on déduit la convergence uniforme en n quand ε →0 : νR⁻_n,y,t¹ −Q^ενF_n,ε⁻¹ →0, ainsi que νR⁻_y,t¹−Q^ενF_ε⁻¹ →0. Mais grâce à (8), on se ramène à l’étude deQ^ε_νF_ε⁻¹−Q^ε_νF_n,ε⁻¹ puis grâce à une écriture du type (6) et aux propriétés de la variation, àλ|_[0,ε](ϕ_y,t^s,v)⁻¹−λ|_[0,ε](ϕ_n,y,t^s,v )⁻¹ →0, quandn→ +∞oùϕ_n,y,t^s,v ,ϕ_y,t^s,vsont des polynômes à une indéterminée obtenus à partir deR_n,y,t etR_y,t ens+cv.

Étudions pour cela la convergence des coefficients du polynômeϕ_n,y,t^s,v vers ceux de ϕ_y,t^s,v ce qui permettra d’appliquer la Proposition 3.1 ci-dessous avec P = {λ|[0,ε]} pour obtenir (9). En fait, il suffit d’étudier la convergence des coefficients deRn,y,t vers ceux deRy,t pourP_(Γi)ip(y)+1-presque chaquet_p(y)+1etP_Y-presque chaquey. Compte tenu des définitions deRn,y,tet deFn,y, on étudie les coefficients aléatoires du type (10), ce qui est possible grâce à l’étude de la continuité en probabilité des séries de LePage dans [1]. Nous obtenons alors la

convergence dans le sens précédent des coefficients des polynômesRn,y,t vers ceux deRy,t et donc celle de ceux deϕ_n,y,t^s,v :c→R_n,y,t(s+cv).

Finalement par la Proposition 3.1, on a Q^ενF_ε⁻¹−Q^ενF_n,ε⁻¹ →0 quand n → +∞ puis d’après (8), νR_y,t⁻¹−νR_n,y,t⁻¹ →0, c’est-à-dire (1), ce qui permet finalement de conclure la preuve du théorème.

1. Introduction

We consider on a probability space(Ω,F,P), anα-stable random measure M on([0,1],B([0,1]), λ)with control measure the Lebesgue measureλand skewness functionβ:[0,1] → [−1,1]as Samorodnitsky-Taqqu’s terminology in [6]. We suppose moreover

for 0< α <1, Mis arbitrary, and for 1α <2, M is symmetric(β≡0). (2) Forf ∈L^α(log₊)^d−1([0,1]^d):= {f: [0,1]^d→Rmeasurable:

[0,1]^d|f|^α(1+log₊|f|)^d−1dλ^d<∞}, we have shown in [1] we can construct multiple stable integralI_d(f )=

[0,1]^dfdM^d using LePage type series with the representation propertyI_d(f )=^LS_d(f )where

Sd(f ):=C_α^d/α

i>0

[γi][Γi]^−1/αf (Vi) (3)

andC_α is a normalization constant given in [6, Eq. (1.2.9)],(Γ_i)_i∈N∗are Gamma random variables independent of the i.i.d. sequence(V_i, γ_i)withV_i uniform on[0,1]and withγ_i given by

P{γi= −1|Vi} =1−β(V_i)

2 , P{γi= +1|Vi} =1+β(V_i) 2

and where for multi-index i=(i₁, . . . , i_d),[γ_i] =γ_i1γ_i2· · ·γ_id,[Γ_i] =Γ_i1Γ_i2· · ·Γ_id. We refer to references in [1,5]

for other approaches of multiple stable integrals. Such integrals can be seen as counterparts of Wiener–Itô integrals in the stable context. Several properties of the law of the latest such as absolute continuity can also be derived for the former (see [1]). These integrals are also connected to stable random series (see [4]). In this Note, the goal is to generalize for multiple stable integrals the continuity for variation of the laws of stochastic integrals, obtained in the Wiener–Itô case in [3]. In the sequelµis the total variation of a finite signed measureµon a space(X,B_X) and−→^var stands for the convergence in variation. We state the main result of this Note:

Theorem 1.1. LetMbe as in (2) and(fn)n>0converges tof ≡0 inL^α(log₊)^d−1([0,1]^d), then we have L

Id(fn) var

−→L Id(f )

, n→ ∞. (4)

From [1], we knowL(I_d(f_n))λ,L(I_d(f ))λ, therefore due to properties of variation, Theorem 1.1 can also be stated as a local limit theorem:

Corollary 1.2. With the same hypothesis as in Theorem 1.1, the convergence of densities ofL(I_d(f_n))to those of L(I_d(f ))holds inL¹(R).

The general argument of the proof is to use suitable conditioning in order to apply the superstructure method and to get back to random polynomials that we study using the properties of representation (3). We begin with the description of superstructure and then sketch the proof.

2. Superstructure argument

This method is proposed by Davydov [2, Section 5] to study a functional distributionPf⁻¹where on a spaceX, P is a probability measure andf a functional. It consists in introducing auxiliary families of measuresQ_εand of functionalsFε on a larger space such thatQεF_ε⁻¹−→^var Pf⁻¹whenε→0 and such thatQεF_ε⁻¹can be expressed as a mixture of measures whose study is more tractable. In a simple case, it works as follows: let be given a family {G_c}c∈[0,a] of transformations of X withP G⁻_c¹−→^var P, c→0. Consider for ε∈ [0, a], on the product space ([0, ε], B[0,ε])⊗(X,U), the following auxiliary families of measures and functionals:Qε =(1/ε)λ|[0,ε]×P, Fε(c, x)=f (Gcx).We derive

Pf⁻¹−Q_εF_ε⁻¹= 1 ε ε

0

Pf⁻¹−P G⁻_c¹f⁻¹ dc

1

ε ε

0

P −P G⁻_c¹dc→0, ε→0. (5) To studyQ_εF_ε⁻¹, decompose now the product space into strata parallel to the factor space[0, ε], and bring back to the study ofϕ_x(c)=f (G_cx), c∈ [0, ε], that is the restrictions off over orbits of{G_c}c:

QεF_ε⁻¹=1 ε

X

λ|[0,ε]ϕ_x⁻¹P (dx). (6)

3. Sketch of the proof

Letfnandf be as in Theorem 1.1. Noteµn=L(Id(fn))andµ=L(Id(f )). Since it is enough to see (4) for a subsequence, there is no restriction in assuming(f_n)_n>0converges tof both inL^α(log₊)^d−1([0,1]^d)andλ^d-a.e.

in[0,1]^d. We suppose moreoverf is symmetric and zero over diagonal terms (see [1]). NoteT^d:= {i∈N^d, 0<

i₁< i₂<· · ·< i_d}. We easily show there is i∈T^d with V_i:=(V_i1, V_i2, . . . , V_id)∈A_f := {x∈ [0,1]^d|f (x) =0} withλ^d(A_f) >0. Choose i^∗ to be the unique of those indices satisfying the following minimal requirements by order of preference minid, minid−1, . . . ,mini2, mini1. We have easilyf (V_i∗) =0 andP{fn(V_i∗)→f (V_i∗)} =1.

3.1. Conditioning by(γ_i, V_i)_i>0

As(γ_i, V_i)_i>0and(Γ_i)_i>0are independent, suppose the basic probability space is a product space(Ω,F, P )⊗ (Ω,F, P), withP=P⊗Pand(Γ_i)_i>0,(γ_i, V_i)_i>0being defined on each of the factor space, that isΓ_i(ω, ω)= Γ_i(ω),γ_i(ω, ω)=γ_i(ω),V_i(ω, ω)=V_i(ω). With Y=(γ_i, V_i)_i>0depending only on(Ω,F, P),µ_n can be expressed as a mixture ofµn,y=L(Sd(fn)|Y=y)=P Fn,y(Γ^−1/α)⁻¹with respect toP_Y, where fory=( , u) with ∈ {−1,+1}^Nandu∈ [0,1]^N, we noteAn,i(y)=d!Cα^d/α[ i]fn(u_i), andFn,y(x)=

i∈T^dAn,i(y)xi₁· · ·xi_d. The same holds also forµwith an analogousF_y.

3.2. Conditioning by the future ofΓ

In the sequel, notet_j=(t_i)_ij andt_j=(t_i)_ij for the truncated sequences obtained fromt=(t_i)_i>0, note alsop(y)=i_d^∗and condition with respect to(Γ_i)_ip(y)+1. To this way, consider the following measure onR^p(y):

ν:=L

Γ₁^−1/α, Γ₂^−1/α, . . . , Γ_p(y)^−1/αy=(γ_i, V_i)_i andt_p(y)+1=

Γ_i, ip(y)+1 .

It has a density since∀n1, the law of(Γ₁, . . . , Γ_n)conditioned by(Γ_n+1, Γ_n+2, . . .)gets one. Measuresµand µncan now be expressed as mixtures of resp.ν(Fn,y(·, t_tp(y)+1))⁻¹andν(Fy(·, t_tp(y)+1))⁻¹with respect to the laws of Y and of(Γi)ip(y)+1. Using elementary properties of variation, it is enough from now on to derive for P_Y-almost allyandP_(Γi)ip(y)+1-almost allt_p(y)+1that

L ν

F_n,y(·, t_tp(y)+1)₋1 var

−→L ν

F_y(·, t_tp(y)+1)₋1

. (7)

To this way, we shall apply superstructure method on (R^p(y), ν) and functionals F_n,y(·, t_p(y)+1) and F_y(·, t_p(y)+1)that are in fact (from the definition ofF_n,y)d-multivariate polynomials. We note them henceforth Rn,y,t(s1, . . . , s_p(y))andRy,t(s1, . . . , s_p(y)).

3.3. Superstructure

Consider(R^p(y), ν). From the choice of i^∗and the conditioning byΓ, it appears thatRy,t is of degreed with unique monomial of degreed, X_i∗

1· · ·X_i∗

d. For a fixedv∈(R∗)^p(y),c∈R→Ry,t(s+cv) is a polynomial of degreedwith maximal coefficient given byAi^∗(y)v_i∗

1. . . v_i∗

d =0. Consider the following transformations ofR^p(y)₊ : Gc(s)=s+cv. Sinceνbelongs toL¹(R^p(y)₊ ), convergence in variation ofνG⁻_c¹toνis equivalent to convergence of their densities inL¹(R^p(y)₊ )which follows from continuity of translation operatorGc, so thatνG⁻_c¹−→^var νholds asc→0.

Apply then superstructure on ([0, ε],B([0, ε]))⊗(R^p(y)₊ ,B(R^p(y)₊ )) with the following auxiliary families:

Q^ε_ν =(1/ε)λ|_[0,ε]×ν, F_ε(c, s)=R_y,t(s +cv). Expressing Q^ε_νF_n,ε⁻¹ as in (5), we derive as ε→ 0 uniform convergence innto 0 ofνR_n,y,t⁻¹ −Q^ενF_n,ε⁻¹.SimilarlyνR_y,t⁻¹−Q^ενF_ε⁻¹ →0. But

νR⁻_y,t¹−νR⁻_n,y,t¹ νR⁻_y,t¹−Q^ενF_ε⁻¹+Q^ενF_ε⁻¹−Q^ενF_n,ε⁻¹+Q^ενF_n,ε⁻¹−νR⁻_n,y,t¹ . (8) It remains therefore to deal with the second summand of the right-hand side of (8) that we first express using ϕ_n,y,t^s,v (c)=Rn,y,t(s +cv) as in (6) in Section 2: Q^ενF_n,ε⁻¹ = ¹_ε

R^p(y)₊ λ|[0,ε](ϕ_n,y,t^s,v )⁻¹ν(ds). From elementary property of variation, we are brought back to study forν-almost alls:

λ|[0,ε] ϕ_y,t^s,v₋₁

−λ|[0,ε]

ϕ^s,v_n,y,t−1−→0, n→ ∞. (9)

To this way, we study the convergence of the coefficients of polynomialsϕ_n,y,t^s,v to their analogous forϕ_y,t^s,v. The convergence in the · 1-sense (to be defined just below) is therefore easily derived and we apply the following proposition withP= {λ|_[0,ε]}so that (9) finally holds.

Proposition 3.1 [2, Theorem 4.5]. For a bounded interval∆, letP be a family of measures defined onB(∆)such that their densities are equicontinuous. Letf∈C¹(∆)withf ≡0 a.e. Then

lim

δ→0sup

µ∈P

µf⁻¹−µg⁻¹ f−g1:=sup

∆

|f −g| +sup

∆

|f−g|δ =0.

Remark that ϕ^s,v_y,t is a polynomial with non zero dominant coefficient so that the requirement about non degeneracy of derivative is satisfied and the result applies. Sinceλ|_[0,ε](ϕ_n,y,t^s,v )⁻¹λ|_[0,ε]<∞, dominated convergence ensures therefore, whenn→ ∞:

Q^ενF_ε⁻¹−Q^ενF_n,ε⁻¹1 ε

R^p(y)₊

λ|[0,ε] ϕ_y,t^s,v₋1

−λ|[0,ε]

ϕ_n,y,t^s,v ₋1ν(ds)−→0

forP_(Γi)ip(y)+1-almost allt_p(y)+1andP_Y-almost ally. Finally, we derive from (8), that

nlim→∞νR_y,t⁻¹−νR_n,y,t⁻¹ =0

forP_(Γi)ip(y)+1-almost allt_p(y)+1andP_Y-almost ally, that is convergence (7), which is enough to conclude by conditioning. It remains therefore to deal with the convergence of coefficients of polynomialsϕ^s,v_n,y,t.

3.4. Study of the coefficients of polynomialsϕ_n,y,t^s,v

Coefficients of ϕ^s,v_y,t are linear combinations of coefficients of polynomial Ry,t, the coefficients of the combinations being polynomials in fixedsandv. The same holds true forϕ_n,y,t^s,v with identical linear combinations.

It is thus sufficient to show the convergence of the coefficients ofR_n,y,t to those ofR_y,t forP_(Γi)ip(y)+1-almost all t_p(y)+1andP_Y-almost ally. In fact, we study the coefficients of random polynomials

d!C_α^d/α

i|i1<···<iki_d^∗<ik+1<···<id

[γ_i]fn(V_i)Xi₁· · ·Xi_kΓ_i^−1/α

k+1 · · ·Γ_i^−1/α

d .

Let j∈T^dwithj₁<· · ·< j_ki_d^∗< j_k+1, the coefficient ofX_j1· · ·X_jk relative to j is d!C_α^d/α

i|i_d^∗<ik+1<···<id

i1=j1,...,ik=jk

[γ_i]f_n(V_i) Γ_i^−1/α

k+1 · · ·Γ_i^−1/α

d . (10)

We study the convergence of coefficients (10) to their counterparts forf. To this way, we use the study of continuity in probability ofS_d derived in [1] (Section 4.1.2 for 0< α <1 and Section 4.2.3 for 1α <2, β ≡0). We condition oni_d^∗=p: since(V_i, γ_i)_i>0are independent, their laws remain unchanged fori > p. The study of (10) is thus brought back to those of a(d−k)-multiple series of LePage type. Since

– fn→f inL^α(log₊)^d−1([0,1]^d)yields convergence inL^α(log₊)^d−k−1([0,1]^d−k), – (Vi, γi),i > p, are independent and independent of(Γi)i>0,

a careful reading of the proof of [1, Sections 4.1.2, 4.2.3] shows that it works even with the restrictionp < i_k+1<

· · ·< i_dover indices. We derive therefore from continuity in probability ofS_d−pthat

i|p<ik+1<···<id

i₁=j₁,...,i_k=j_k

[γi]fn(Vi)Γ_i^−1/α

k+1 · · ·Γ_i^−1/α

d

−→P

i|p<ik+1<···<id

i₁=j₁,...,i_k=j_k

[γi]f (Vi)Γ_i^−1/α

k+1 · · ·Γ_i^−1/α

d .

We deduce convergence in probability of (10) to its counterpart and extracting a subsequence, the almost sure convergence of (10). Finally, forP_Y-almost ally, forP_(Γi)ip(y)+1-almost allt_p(y)+1the following convergence holds:

i|p<i_k+1<···<i_d i₁=j₁,...,i_k=j_k

A_n,i(y)t_i^−1/α

k+1 · · ·t_i^−1/α

d −→

i|p<i_k+1<···<i_d i₁=j₁,...,i_k=j_k

A_i(y)t_i^−1/α

k+1 · · ·t_i^−1/α

d .

We obtain convergence in the same sense of the coefficients ofRn,y,t to those ofRy,t and the same holds true for the coefficients of polynomialc→Rn,y,t(s+cv). This allows us to achieve the proof of Theorem 1.1 as explained previously.

Acknowledgements

The author thanks an anonymous referee for suggestions lightening the proof.

References

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