• Aucun résultat trouvé

# Central and non-central limit theorems for weighted power variations of fractional brownian motion

N/A
N/A
Protected

Partager "Central and non-central limit theorems for weighted power variations of fractional brownian motion"

Copied!
25
0
0
En savoir plus ( Page)

Texte intégral

(1)

www.imstat.org/aihp 2010, Vol. 46, No. 4, 1055–1079

DOI:10.1214/09-AIHP342

© Association des Publications de l’Institut Henri Poincaré, 2010

## Central and non-central limit theorems for weighted power variations of fractional Brownian motion

a

b,1

### and Ciprian A. Tudor

c

aLaboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, Boîte courrier 188, 4 Place Jussieu, 75252 Paris Cedex 5, France. E-mail:ivan.nourdin@upmc.fr

bDepartment of Mathematics, University of Kansas, 405 Snow Hall, Lawrence, KS 66045-2142, USA. E-mail:nualart@math.ku.edu cSAMOS/MATISSE, Centre d’Économie de La Sorbonne, Université de Panthéon-Sorbonne Paris 1, 90 rue de Tolbiac, 75634 Paris Cedex 13,

France. E-mail:tudor@univ-paris1.fr

Received 4 December 2008; revised 15 August 2009; accepted 18 September 2009

Abstract. In this paper, we prove some central and non-central limit theorems for renormalized weighted power variations of orderq≥2 of the fractional Brownian motion with Hurst parameterH(0,1), whereqis an integer. The central limit holds for

1

2q< H≤1−2q1, the limit being a conditionally Gaussian distribution. IfH < 2q1 we show the convergence inL2to a limit which only depends on the fractional Brownian motion, and ifH >1−2q1 we show the convergence inL2to a stochastic integral with respect to the Hermite process of orderq.

Résumé. Dans ce papier, nous prouvons des théorèmes de la limite centrale et non-centrale pour les variations à poids d’ordreq du mouvement brownien fractionnaire d’indiceH(0,1), pourqun entier supérieur ou égal à 2. Il y a trois cas, suivant la position deH par rapport à2q1 et 1−2q1. Si2q1 < H≤1−2q1, nous montrons un théorème de la limite centrale vers une variable aléatoire de loi conditionnellement gaussienne. SiH <2q1, nous montrons la convergence dansL2vers une limite qui dépend seulement du mouvement brownien fractionnaire. SiH >1−2q1, nous montrons la convergence dansL2vers une intégrale stochastique par rapport au processus d’Hermite d’ordreq.

MSC:60F05; 60H05; 60G15; 60H07

Keywords:Fractional Brownian motion; Central limit theorem; Non-central limit theorem; Hermite process

1. Introduction

The study of single path behavior of stochastic processes is often based on the study of their power variations, and there exists a very extensive literature on the subject. Recall that, a realq >0 being given, theq-power variation of a stochastic processX, with respect to a subdivisionπn= {0=tn,0< tn,1<· · ·< tn,κ(n)=1}of[0,1], is defined to be the sum

κ(n)

k=1

|Xtn,kXtn,k1|q.

1The work of D. Nualart is supported by the NSF Grant DMS-06-04207.

(2)

For simplicity, consider from now on the case wheretn,k =k2n for n∈ {1,2,3, . . .} andk∈ {0, . . . ,2n}. In the present paper we wish to point out some interesting phenomena whenX=B is a fractional Brownian motion of Hurst indexH(0,1), and whenq≥2 is an integer. In fact, we will also drop the absolute value (whenq is odd) and we will introduce some weights. More precisely, we will consider

2n

k=1

f (B(k1)2n)(Bk2n)q, q∈ {2,3,4, . . .}, (1.1)

where the functionf:R→Ris assumed to be smooth enough and whereBk2ndenotes, here and in all the paper, the incrementBk2nB(k1)2n.

The analysis of the asymptotic behavior of quantities of type (1.1) is motivated, for instance, by the study of the exact rates of convergence of some approximation schemes of scalar stochastic differential equations driven byB (see [7,12] and ) besides, of course, the traditional applications of quadratic variations to parameter estimation problems.

Now, let us recall some known results concerningq-power variations (forq=2,3,4, . . .), which are today more or less classical. First, assume that the Hurst index isH=12, that isBis a standard Brownian motion. Letμqdenote the qth moment of a standard Gaussian random variable GN (0,1). By the scaling property of the Brownian motion and using the central limit theorem, it is immediate that, asn→ ∞:

2n/2

2n

k=1

2n/2Bk2n

q

μqLaw

−→N

0, μ2qμ2q

. (1.2)

When weights are introduced, an interesting phenomenon appears: instead of Gaussian random variables, we rather obtain mixing random variables as limit in (1.2). Indeed, whenqis even andf:R→Ris continuous and has poly- nomial growth, it is a very particular case of a more general result by Jacod  (see also Section 2 in Nourdin and Peccati  for related results) that we have, asn→ ∞:

2n/2

2n

k=1

f (B(k1)2n)

2n/2Bk2nq

μq

Law

−→

μ2qμ2q 1

0

f (Bs)dWs. (1.3)

Here,Wdenotes another standard Brownian motion, independent ofB. Whenqis odd, still forf:R→Rcontinuous with polynomial growth, we have, this time, asn→ ∞:

2n/2

2n

k=1

f (B(k1)2n)

2n/2Bk2n

q Law

−→

1

0

f (Bs)

μ2qμ2q+1dWs+μq+1dBs

, (1.4)

see for instance .

Secondly, assume thatH=12, that is the case where the fractional Brownian motionB has not independent in- crements anymore. Then (1.2) has been extended by Breuer and Major , Dobrushin and Major , Giraitis and Surgailis  or Taqqu . Precisely, five cases are considered, according to the evenness ofqand the value ofH:

• ifqis even and ifH(0,34), asn→ ∞,

2n/2

2n

k=1

2nHBk2n

q

μq Law

−→N 0,σH,q2

, (1.5)

• ifqis even and ifH=34, asn→ ∞,

√1 n2n/2

2n

k=1

23n/4Bk2n

q

μqLaw

−→N (0,σ3/4,q2 ), (1.6)

(3)

• ifqis even and ifH(34,1), asn→ ∞,

2n2nH

2n

k=1

2nHBk2nq

μqLaw

−→“Hermite r.v.,” (1.7)

• ifqis odd and ifH(0,12], asn→ ∞,

2n/2

2n

k=1

2nHBk2n

q Law

−→N 0,σH,q2

, (1.8)

• ifqis odd and ifH(12,1), asn→ ∞,

2nH

2n

k=1

2nHBk2n

q Law

−→N 0,σH,q2

. (1.9)

Here,σH,q >0 denotes some constant depending only onH andq. The term “Hermite r.v.” denotes a random variable whose distribution is the same as that ofZ(2)at time one, forZ(2)defined in Definition7below.

Now, let us proceed with the results concerning theweightedpower variations in the case whereH=12. Consider the following condition on a functionf:R→R, whereq≥2 is an integer:

(Hq) f belongs toC2qand, for anyp(0,)and0≤i≤2q:supt∈[0,1]E{|f(i)(Bt)|p}<∞.

Suppose thatf satisfies(Hq). Ifq is even andH(12,34), then by Theorem 2 in León and Ludeña  (see also Corcuera et al.  for related results on the asymptotic behavior of thep-variation of stochastic integrals with respect toB) we have, asn→ ∞:

2n/2

2n

k=1

f (B(k1)2−n)

2nHBk2−nq

μq Law

−→σH,q 1

0

f (Bs)dWs, (1.10)

where, once again,W denotes a standard Brownian motion independent ofB whileσH,q is the constant appearing in (1.5). Thus, (1.10) shows for (1.1) a similar behavior to that observed in the standard Brownian case, compare with (1.3). In contradistinction, the asymptotic behavior of (1.1) can be completely different of (1.3) or (1.10) for other values ofH. The first result in this direction has been observed by Gradinaru et al. . Namely, ifq≥3 is odd andH(0,12), we have, asn→ ∞:

2nHn

2n

k=1

f (B(k1)2n)

2nHBk2nq L2

−→ −μq+1 2

1

0

f(Bs)ds. (1.11)

Also, whenq=2 andH(0,14), Nourdin  proved that we have, asn→ ∞:

22H nn

2n

k=1

f (B(k1)2n)

2nHBk2n2

−1 L2

−→1 4

1

0

f(Bs)ds. (1.12)

In view of (1.3), (1.4), (1.10), (1.11) and (1.12), we observe that the asymptotic behaviors of the power variations of fractional Brownian motion (1.1) can be really different, depending on the values ofqandH. The aim of the present paper is to investigate what happens in the whole generality with respect toq andH. Our main tool is the Malliavin calculus that appeared, in several recent papers, to be very useful in the study of the power variations for stochastic processes. As we will see, the Hermite polynomials play a crucial role in this analysis. In the sequel, for an integer q≥2, we writeHqfor the Hermite polynomial with degreeq defined by

Hq(x)=(−1)q

q! ex2/2 dq dxq

ex2/2 ,

(4)

and we consider, whenf:R→Ris a deterministic function, the sequence ofweighted Hermite variations of orderq defined by

Vn(q)(f ):=

2n

k=1

f (B(k1)2−n)Hq

2nHBk2−n

. (1.13)

The following is the main result of this paper.

Theorem 1. Fix an integerq≥2,and suppose thatf satisfies(Hq).

1. Assume that0< H <2q1.Then,asn→ ∞,it holds 2nqHnVn(q)(f ) L

−→2 (−1)q 2qq!

1

0

f(q)(Bs)ds. (1.14)

2. Assume that 2q1 < H <1−2q1.Then,asn→ ∞,it holds B,2n/2Vn(q)(f ) Law

−→

B, σH,q

1 0

f (Bs)dWs

, (1.15)

whereW is a standard Brownian motion independent ofB and

σH,q= 1

2qq!

r∈Z

|r+1|2H+ |r−1|2H−2|r|2Hq

. (1.16)

3. Assume thatH=1−2q1.Then,asn→ ∞,it holds

B, 1

n2n/2Vn(q)(f ) Law

−→

B, σ11/(2q),q 1

0

f (Bs)dWs

, (1.17)

whereW is a standard Brownian motion independent ofB and

σ11/(2q),q=2 log 2 q!

1− 1

2q q

1−1 q

q

. (1.18)

4. Assume thatH >1−2q1.Then,asn→ ∞,it holds 2nq(1H )nVn(q)(f ) L

−→2

1

0

f (Bs)dZ(q)s , (1.19)

whereZ(q)denotes the Hermite process of orderqintroduced in Definition7below.

Remark 1. Whenq=1,we haveVn(1)(f )=2nH2n

k=1f (B(k1)2n)Bk2n.ForH =12, 2nHVn(1)(f )converges inL2to the Itô stochastic integral1

0f (Bs)dBs.ForH >12, 2nHVn(1)(f )converges inL2and almost surely to the Young integral1

0f (Bs)dBs.ForH <12, 23nHnVn(1)(f )converges inL2to121

0 f(Bs)ds.

Remark 2. After the first draft of the present paper have been submitted,Burdzy and Swansonand,independently, Nourdin and Réveillachave shown,in the critical caseH=14,that

B,2n/2Vn(2)(f ) Law

−→

B, σ1/4,2

1 0

f (Bs)dWs+1 8

1 0

f(Bs)ds

.

(5)

(The reader is also referred tofor the study of the weighted variations associated with iterated Brownian motion, which is a non-Gaussian self-similar process of order 14.)Later,it has finally been shown by Nourdin and Nualart

that,for any integerq≥2and in the critical caseH=2q1, B,2n/2Vn(q)(f ) Law

−→

B, σ1/(2q),q

1

0

f (Bs)dWs+(−1)q 2qq!

1

0

f(q)(Bs)ds

.

Consequently,the understanding of the asymptotic behavior of the weighted Hermite variations of the fractional Brownian motion is now complete.

WhenHis between14 and34, one can refine point 2 of Theorem1as follows:

Proposition 2. Letq≥2be an integer,f:R→Rbe a function such that(Hq)holds and assume thatH(14,34).

Then,asn→ ∞,

B,2n/2Vn(2)(f ), . . . ,2n/2Vn(q)(f )

−→Law

B, σH,2

1 0

f (Bs)dWs(2), . . . , σH,q 1

0

f (Bs)dWs(q)

, (1.20)

where(W(2), . . . , W(q))is a(q−1)-dimensional standard Brownian motion independent ofBand theσH,p’s, 2pq,are given by(1.16).

Theorem 1, together with Proposition 2, allows us to complete the missing cases in the understanding of the asymptotic behavior of weightedpowervariations of fractional Brownian motion:

Corollary 3. Letq≥2be an integer,andf:R→Rbe a function such that(Hq)holds.Then,asn→ ∞:

1. WhenH >12 andqis odd, 2nH

2n

k=1

f (B(k1)2n)

2nHBk2n

q L2

−→q1 1

0

f (Bs)dBs=q1 B1

0

f (x)dx. (1.21)

2. WhenH <14 andqis even, 22nHn

2n

k=1

f (B(k1)2n)

2nHBk2nq

μq L2

−→1 4

q 2

μq2 1

0

f(Bs)ds. (1.22)

(We recover(1.12)by choosingq=2).

3. WhenH=14 andqis even,

B,2n/2

2n

k=1

f (B(k1)2−n)

2n/4Bk2−nq

μq

−→Law

B,1

4 q

2

μq2 1

0

f(Bs)ds+σ1/4,q 1

0

f (Bs)dWs

, (1.23)

whereW is a standard Brownian motion independent ofBandσ1/4,q is the constant given by(1.25)just below.

4. When 14< H <34 andqis even,

B,2n/2

2n

k=1

f (B(k1)2n)

2nHBk2nq

μq

−→Law

B,σH,q

1

0

f (Bs)dWs

, (1.24)

(6)

forWa standard Brownian motion independent ofBand

σH,q= q

p=2

p! q

p 2

μ2qp2p

r∈Z

|r+1|2H+ |r−1|2H−2|r|2Hp

. (1.25)

5. WhenH=34andq is even,

B, 1

n2n/2

2n

k=1

f (B(k1)2−n)

2nHBk2−nq

μq

−→Law

B,σ3/4,q

1 0

f (Bs)dWs

, (1.26)

forWa standard Brownian motion independent ofBand

σ3/4,q= q

p=2

2 log 2p! q

p 2

μ2qp

1− 1 2q

q 1−1

q q

.

6. WhenH > 34andq is even,

2n2H n

2n

k=1

f (B(k1)2−n)

2nHBk2−nq

μq L2

−→2μq2

q 2

1 0

f (Bs)dZs(2), (1.27)

forZ(2)the Hermite process introduced in Definition7.

Finally, we can also give a new proof of the following result, stated and proved by Gradinaru et al.  and Cheridito and Nualart  in acontinuoussetting:

Theorem 4. Assume thatH >16,and thatf:R→Rverifies(H6).Then the limit in probability,asn→ ∞,of the symmetric Riemann sums

1 2

2n

k=1

f(Bk2−n)+f(B(k1)2−n)

Bk2−n (1.28)

exists and is given byf (B1)f (0).

Remark 3. WhenH16,quantity(1.28)does not converge in probability in general.As a counterexample,one can consider the case wheref (x)=x3,see Gradinaru et al. or Cheridito and Nualart.

2. Preliminaries and notation

We briefly recall some basic facts about stochastic calculus with respect to a fractional Brownian motion. One refers to [18,19] for further details. LetB=(Bt)t∈[0,1]be a fractional Brownian motion with Hurst parameterH(0,1).

That is,B is a zero mean Gaussian process, defined on a complete probability space(Ω,A, P ), with the covariance function

RH(t, s)=E(BtBs)=1 2

s2H+t2H− |ts|2H

, s, t∈ [0,1].

We suppose thatAis the sigma-field generated byB. LetE be the set of step functions on[0, T], andHbe the Hilbert space defined as the closure ofE with respect to the inner product

1[0,t],1[0,s]H=RH(t, s).

(7)

The mapping1[0,t]Bt can be extended to an isometry betweenHand the Gaussian spaceH1associated withB.

We will denote this isometry byϕB(ϕ).

LetS be the set of all smooth cylindrical random variables, i.e. of the form F =φ(Bt1, . . . , Btm),

wherem≥1,φ:Rm→R∈Cband 0≤t1<· · ·< tm≤1. The derivative ofF with respect toB is the element of L2(Ω,H)defined by

DsF = m

i=1

∂φ

∂xi(Bt1, . . . , Btm)1[0,ti](s), s∈ [0,1].

In particular DsBt =1[0,t](s). For any integerk≥1, we denote by Dk,2 the closure of the set of smooth random variables with respect to the norm

F2k,2=E F2

+ k j=1

EDjF2

H⊗j

.

The Malliavin derivativeDsatisfies the chain rule. Ifϕ:Rn→RisCb1and if(Fi)i=1,...,nis a sequence of elements ofD1,2, thenϕ(F1, . . . , Fn)∈D1,2and we have

Dϕ(F1, . . . , Fn)= n i=1

∂ϕ

∂xi(F1, . . . , Fn)DFi.

We also have the following formula, which can easily be proved by induction onq. Letϕ, ψCbq (q≥1), and fix 0≤u < v≤1 and 0≤s < t≤1. Thenϕ(BtBs)ψ (BvBu)∈Dq,2and

Dq

ϕ(BtBs)ψ (BvBu)

= q a=0

q a

ϕ(a)(BtBs(qa)(BvBu)1[s,ta]⊗1[u,v(q]a), (2.1)

where⊗means the symmetric tensor product.

The divergence operatorI is the adjoint of the derivative operatorD. If a random variableuL2(Ω,H)belongs to the domain of the divergence operator, that is, if it satisfies

EDF, uHcu

E(F2) for anyFS, thenI (u)is defined by the duality relationship

E F I (u)

=E

DF, uH

,

for everyF ∈D1,2.

For everyn≥1, letHn be thenth Wiener chaos ofB,that is, the closed linear subspace ofL2(Ω,A, P )gener- ated by the random variables{Hn(B(h)), h∈H,hH=1}, whereHnis thenth Hermite polynomial. The mapping In(hn)=n!Hn(B(h))provides a linear isometry between the symmetric tensor product Hn (equipped with the modified norm · Hn=1n! · Hn) and Hn. ForH =12,In coincides with the multiple Wiener–Itô integral of ordern. The following duality formula holds

E F In(h)

=E DnF, h

Hn

, (2.2)

for any elementh∈Hnand any random variableF ∈Dn,2.

(8)

Let{ek, k≥1}be a complete orthonormal system inH. Givenf ∈Hnandg∈Hm, for everyr=0, . . . , n∧m, the contraction off andgof orderris the element ofH(n+m2r)defined by

frg= k1,...,kr=1

f, ek1⊗ · · · ⊗ekrH⊗rg, ek1⊗ · · · ⊗ekrH⊗r.

Notice thatfrgis not necessarily symmetric: we denote its symmetrization byfrg∈H(n+m2r). We have the following product formula: iff ∈Hnandg∈Hmthen

In(f )Im(g)=

nm r=0

r!n r

m r

In+m2r(frg). (2.3)

We recall the following simple formula for anys < tandu < v:

E

(BtBs)(BvBu)

=1 2

|tv|2H+ |su|2H− |tu|2H− |sv|2H

. (2.4)

We will also need the following lemmas:

Lemma 5.

1. Lets < tbelong to[0,1].Then,ifH <1/2,one has E

Bu(BtBs)(ts)2H (2.5)

for allu∈ [0,1]. 2. For allH(0,1),

2n

k,l=1

E(B(k1)2nBl2n)=O 2n

. (2.6)

3. For anyr≥1,we have,ifH <1−2r1,

2n

k,l=1

E(Bk2nBl2n)r=O

2n2rH n

. (2.7)

4. For anyr≥1,we have,ifH=1−2r1,

2n

k,l=1

E(Bk2nBl2n)r=O

n22n2rn

. (2.8)

Proof. To prove inequality (2.5), we just write E

Bu(BtBs)

=1 2

t2Hs2H +1

2

|su|2H− |tu|2H ,

and observe that we have|b2Ha2H| ≤ |ba|2H for anya, b∈ [0,1], becauseH <12. To show (2.6) using (2.4), we write

2n

k,l=1

E(B(k1)2−nBl2−n)=22H n1

2n

k,l=1

|l−1|2Hl2H− |lk+1|2H+ |lk|2H

C2n,

(9)

the last bound coming from a telescoping sum argument. Finally, to show (2.7) and (2.8), we write

2n

k,l=1

E(Bk2−nBl2−n)r =22nrHr

2n

k,l=1

|kl+1|2H+ |kl−1|2H−2|kl|2Hr

≤2n2nrHr p=−∞

|p+1|2H+ |p−1|2H−2|p|2Hr,

and observe that, since the function||p+1|2H+ |p−1|2H−2|p|2H|behaves asCHp2H2for largep, the series in the right-hand side is convergent becauseH <1−2r1. In the critical caseH=1−2r1, this series is divergent, and

2n

p=−2n

|p+1|2H+ |p−1|2H−2|p|2Hr

behaves as a constant timen.

Lemma 6. Assume thatH > 12. 1. Lets < tbelong to[0,1].Then

E

Bu(BtBs)≤2H (t−s) (2.9)

for allu∈ [0,1].

2. Assume thatH >1−2l1 for somel≥1.Letu < vands < tbelong to[0,1].Then E(BuBv)(BtBs)H (2H−1)

2 2H l+1−2l

1/ l

(uv)(l1)/ l(ts). (2.10)

3. Assume thatH >1−2l1 for somel≥1.Then

2n

i,j=1

E(Bi2nBj2n)l=O 22n2ln

. (2.11)

Proof. We have E

Bu(BtBs)

=1 2

t2Hs2H +1

2

|su|2H− |tu|2H . But, when 0≤a < b≤1:

b2Ha2H=2H ba

0

(u+a)2H1du≤2H b2H1(ba)≤2H (b−a).

Thus,|b2Ha2H| ≤2H|ba|and the first point follows.

Concerning the second point, using Hölder inequality, we can write E(BuBv)(BtBs)=H (2H−1)

v

u

t

s

|yx|2H2dydx

H (2H−1)|uv|(l1)/ l 1

0

t

s

|yx|2H2dy l

dx 1/ l

H (2H−1)|uv|(l1)/ l|ts|(l1)/ l 1

0

t s

|yx|(2H2)ldydx 1/ l

.

(10)

Denote byH=1+(H−1)land observe thatH>12 (becauseH >1−2l1). Since 2H−2=(2H−2)l, we can write

H(2H−1) 1

0

t s

|yx|(2H2)ldydx=EB1H

BtHBsH≤2H|ts| by the first point of this lemma. This gives the desired bound.

We prove now the third point. We have

2n

i,j=1

E(Bi2−nBj2−n)l=22H nll

2n

i,j=1

|ij+1|2H+ |ij−1|2H−2|ij|2Hl

≤2n2H nl+1l

2n1 k=−2n+1

|k+1|2H+ |k−1|2H−2|k|2Hl

and the function|k+1|2H+ |k−1|2H−2|k|2H behaves as|k|2H2for largek. As a consequence, sinceH >1−2l1, the sum

2n1 k=−2n+1

|k+1|2H+ |k−1|2H−2|k|2Hl

behaves as 2(2H2)ln+nand the third point follows.

Now, let us introduce the Hermite process of orderq ≥2 appearing in (1.19). FixH >1/2 and t∈ [0,1]. The sequencen(t))n1, defined as

ϕn(t)=2nqn1 q!

[2nt] j=1

1[(jq1)2n,j2n],

is a Cauchy sequence in the spaceHq. Indeed, sinceH >1/2, we have 1[a,b],1[u,v]H=E

(BbBa)(BvBu)

=H (2H−1) b

a

v u

ss2H2dsds, so that, for anymn

ϕn(t), ϕm(t)

Hq

=Hq(2H−1)q

q!2 2nq+mqnm

[2mt] j=1

[2nt] k=1

j2m

(j1)2−m

k2n

(k1)2−n

s−s2H2dsds q

.

Hence

m,nlim→∞

ϕn(t), ϕm(t)

Hq

=Hq(2H−1)q q!2

t

0

t

0

s−s(2H2)qdsds=cq,Ht(2H2)q+2,

(11)

wherecq,H =q!2(H qHqq+(2H1)(2H q1)q2q+1). Let us denote byμ(q)t the limit inHqof the sequence of functionsϕn(t). For anyf∈Hq, we have

ϕn(t), f

H⊗q = 2nqn1 q!

[2nt] j=1

1[(jq1)2−n,j2−n], f

H⊗q

= 2nqn1

q!Hq(2H−1)q

[2nt] j=1

1 0

ds1

j2n (j1)2n

ds1s1s12H2· · ·

× 1

0

dsq

j2n

(j1)2−n

dsq|sqsq|2H2f (s1, . . . , sq)

n→∞−→ 1

q!Hq(2H−1)q t

0

ds

[0,1]qds1· · ·dsqs1s2H2· · · |sqs|2H2f (s1, . . . , sq)

= μ(q)t , f

H⊗q.

Definition 7. Fix q≥2 and H >1/2. The Hermite process Z(q)=(Z(q)t )t∈[0,1] of orderq is defined byZ(q)t = Iq(q)t )fort∈ [0,1].

LetZn(q)be the process defined byZn(q)(t)=Iqn(t))fort∈ [0,1]. By construction, it is clear thatZ(q)n (t) L

−→2

Z(q)(t)asn→ ∞, for all fixedt∈ [0,1]. On the other hand, it follows, from Taqqu  and Dobrushin and Major , thatZ(q)n converges in law to the “standard” and historicalqth Hermite process, defined through its moving average representation as a multiple integral with respect to a Wiener process with time horizonR. In particular, the process introduced in Definition7has the same finite dimensional distributions as the historical Hermite process.

Let us finally mention that it can be easily seen thatZ(q)isq(H−1)+1 self-similar, has stationary increments and admits moments of all orders. Moreover, it has Hölder continuous paths of order strictly less thanq(H−1)+1.

For further results, we refer to Tudor .

3. Proof of the main results

In this section we will provide the proofs of the main results. For notational convenience, from now on, we write ε(k1)2n(resp.δk2n) instead of1[0,(k1)2n](resp.1[(k1)2n,k2n]). The following proposition provides information on the asymptotic behavior ofE(Vn(q)(f )2), asntends to infinity, forH≤1−2q1.

Proposition 8. Fix an integerq≥2.Suppose thatf satisfies(Hq).Then,ifH2q1,then E

Vn(q)(f )2

=O

2n(2H q+2)

. (3.1)

If 2q1H <1−2q1,then E

Vn(q)(f )2

=O 2n

. (3.2)

Finally,ifH=1−2q1,then E

Vn(q)(f )2

=O n2n

. (3.3)

Références

Documents relatifs

The observation times are possibly random, and we try to find conditions on those times, allowing for limit theorems when the number of observations increases (the “high

Keywords: Mean central limit theorem; Wasserstein distance; Minimal distance; Martingale difference sequences; Strong mixing; Stationary sequences; Weak dependence; Rates

The existence of these three asymptotic regimes can be explained by the fact that the unitary Brownian motion is the “wrapping”, on the unitary group, of a Brownian motion on

The CLTs in the space of lattices are in turn deduced from an abstract Central Limit Theorem (Theorem 4.2) for weakly dependent random variables which is formulated and proven

It would be of interest to measure the magnetic moments of the other6 mono- capped octahedral molydenurn( ~ v ) complex which has longer molybdenum-ligand bond

At the end of the protocol, blood was removed and serum collected to quantify (B) total IgG1 and IgE as well as (C) Der f-specific and OVA-specific IgE by ELISA in controls (CTRL,

Cette clinique (SAC) fournit des soins aux adolescents et aux jeunes adultes atteints d'une infection par le VIH (soins médicaux, thérapie in- dividuelle/en groupe, éduca-

Comme nous l’avons mentionné dans l’introduction, nous avons choisi de travailler sur le thème de Noël ; c’est un effet un sujet qui convient

The present article is devoted to a fine study of the convergence of renor- malized weighted quadratic and cubic variations of a fractional Brownian mo- tion B with Hurst index H..

Central and noncentral limit theorems for weighted power variations of fractional Brownian motion.. Weighted power variations of iterated Brownian

During the 2- month follow-up period after vancomycin therapy, global MRSA carriage increased from 25% to 55% in the group receiving the standard dose and decreased from 43% to 36%

Nourdin (2008): Asymptoti behavior of some weighted quadrati and ubi variations of the fra tional Brownian motion. Nualart (2008): Central limit theorems for multiple Skor

Keywords and phrases: Central limit theorem; Non-central limit theorem; Convergence in distribution; Fractional Brownian motion; Free Brownian motion; Free probability;

Keywords: Total variation distance; Non-central limit theorem; Fractional Brownian motion; Hermite power variation; Multiple stochastic integrals; Hermite random

The purpose of this paper is to establish the multivariate normal convergence for the average of certain Volterra processes constructed from a fractional Brownian motion with

than {100} and crystal orientation dependency is roughly consistent with the Schmid law . As ½&lt;110&gt;{111} slip systems are not observed under single mode conditions, no CRSS

Effets dominos Changements climatiques Changements des surcharges environnementales sur les bâtiments et

The CLTs in the space of lattices are in turn deduced from an abstract Central Limit Theorem (Theorem 4.2) for weakly dependent random variables which is formulated and proven

We prove functional central and non-central limit theorems for generalized varia- tions of the anisotropic d-parameter fractional Brownian sheet (fBs) for any natural number d..

We give two main results, which are necessary tools for the study of stability in section 3: we prove uniqueness, up to an additive constant, of the optimal transport

Key words: Total variation distance; Non-central limit theorem; Fractional Brownian motion; Her- mite power variation; Multiple stochastic integrals; Hermite random variable..

We let S m,p (Γ\G) be the space of m times continuously differentiable functions on Γ\G such that the L p (Γ\G) norms of all derivatives of order less or equal to m are finite.. We

REPUBLIQUE ALGERIENNE DEMOCRATIQUE ET POPULAIRE ةعماج سيداب نب ديمحلا دبع مناغتسم ةيلك مولع ةايحلا و ةعيبطلا. Effet

Sélectionne ta langue

Le site sera traduit dans la langue de ton choix.

Langues suggérées pour toi :

Autres langues