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

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

Ivan Nourdin

, David Nualart

and Ciprian A. Tudor

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 à_2q¹ et 1−_2q¹. Si_2q¹ < 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 subdivisionπ_n= {0=t_n,0< t_n,1<· · ·< t_n,κ(n)=1}of[0,1], is defined to be the sum

κ(n)

k=1

|Xt_n,k−Xt_n,k−1|^q.

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

For simplicity, consider from now on the case wheret_n,k =k2⁻ⁿ 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_(k−1)2−n)(B_k2−n)^q, q∈ {2,3,4, . . .}, (1.1)

where the functionf:R→Ris assumed to be smooth enough and whereB_k2−ndenotes, here and in all the paper, the incrementB_k2−n−B_(k−1)2−n.

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 [13]) 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. Letμ_qdenote the qth moment of a standard Gaussian random variable G∼N (0,1). By the scaling property of the Brownian motion and using the central limit theorem, it is immediate that, asn→ ∞:

2^−n/2

2ⁿ

k=1

2^n/2B_k2−n

q

−μ_qLaw

−→N

0, μ2q−μ²_q

. (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 [10] (see also Section 2 in Nourdin and Peccati [16] for related results) that we have, asn→ ∞:

2^−n/2

2ⁿ

k=1

f (B_(k−1)2−n)

2^n/2B_k2−nq

−μq

Law

−→

μ2q−μ²_{q 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→ ∞:

2^−n/2

2ⁿ

k=1

f (B_(k−1)2−n)

2^n/2B_k2−n

q Law

−→

₁

0

f (B_s)

μ_2q−μ²_q+1dWs+μ_q+1dBs

, (1.4)

see for instance [16].

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 [1], Dobrushin and Major [5], Giraitis and Surgailis [6] or Taqqu [21]. Precisely, five cases are considered, according to the evenness ofqand the value ofH:

• ifqis even and ifH∈(0,³₄), asn→ ∞,

2^−n/2

2ⁿ

k=1

2^nHB_k2−n

q

−μ_q Law

−→N 0,σ_H,q²

, (1.5)

• ifqis even and ifH=³₄, asn→ ∞,

√1 n2^−n/2

2ⁿ

k=1

2^3n/4B_k2−n

q

−μ_qLaw

−→N (0,σ_3/4,q² ), (1.6)

• ifqis even and ifH∈(³₄,1), asn→ ∞,

2^n−2nH

2ⁿ

k=1

2^nHB_k2−nq

−μ_qLaw

−→“Hermite r.v.,” (1.7)

• ifqis odd and ifH∈(0,¹₂], asn→ ∞,

2^−n/2

2ⁿ

k=1

2^nHB_k2−n

q Law

−→N 0,σ_H,q²

, (1.8)

• ifqis odd and ifH∈(¹₂,1), asn→ ∞,

2^−nH

2ⁿ

k=1

2^nHB_k2−n

q Law

−→N 0,σ_H,q²

. (1.9)

Here,σ_H,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 toC^2qand, for anyp∈(0,∞)and0≤i≤2q:sup_t∈[0,1]E{|f⁽ⁱ⁾(B_t)|^p}<∞.

Suppose thatf satisfies(Hq). Ifq is even andH∈(¹₂,³₄), then by Theorem 2 in León and Ludeña [11] (see also Corcuera et al. [4] for related results on the asymptotic behavior of thep-variation of stochastic integrals with respect toB) we have, asn→ ∞:

2^−n/2

2ⁿ

k=1

f (B_(k−1)2−n)

2^nHB_k2−nq

−μ_q Law

−→σ_H,q 1

0

f (B_s)dWs, (1.10)

where, once again,W denotes a standard Brownian motion independent ofB whileσ_H,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. [9]. Namely, ifq≥3 is odd andH∈(0,¹₂), we have, asn→ ∞:

2^nH−n

2ⁿ

k=1

f (B_(k−1)2−n)

2^nHB_k2−nq L²

−→ −μ_q+1 2

₁

0

f(Bs)ds. (1.11)

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

2^{2H n−n}

2ⁿ

k=1

f (B_(k−1)2−n)

2^nHB_k2−n2

−1 L²

−→1 4

₁

0

f(B_s)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 writeH_qfor the Hermite polynomial with degreeq defined by

H_q(x)=(−1)^q

q! e^x2/2 d^q dx^q

e^−x2/2 ,

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

V_n^(q)(f ):=

2ⁿ

k=1

f (B_(k−1)2−n)Hq

2^nHB_k2−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 2^nqH−nVn^(q)(f ) ^L

−→2 (−1)^q 2^qq!

₁

0

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

2. Assume that _2q¹ < H <1−_2q¹.Then,asn→ ∞,it holds B,2^−n/2V_n^(q)(f ) Law

−→

B, σ_H,q

1 0

f (B_s)dWs

, (1.15)

whereW is a standard Brownian motion independent ofB and

σH,q= 1

2^qq!

r∈Z

|r+1|^2H+ |r−1|^2H−2|r|^2Hq

. (1.16)

3. Assume thatH=1−_2q¹.Then,asn→ ∞,it holds

B, 1

√n2^−n/2V_n^(q)(f ) Law

−→

B, σ_{1−1/(2q),q 1}

0

f (B_s)dWs

, (1.17)

whereW is a standard Brownian motion independent ofB and

σ1−1/(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 2^{nq(1−H )−n}Vn^(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 haveV_n⁽¹⁾(f )=2^−nH₂ⁿ

k=1f (B_(k−1)2−n)B_k2−n.ForH =¹₂, 2^nHV_n⁽¹⁾(f )converges inL²to the Itô stochastic integral1

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

0f (Bs)dBs.ForH <¹₂, 2^3nH−nV_n⁽¹⁾(f )converges inL²to−¹₂1

0 f(Bs)ds.

Remark 2. After the first draft of the present paper have been submitted,Burdzy and Swanson[2]and,independently, Nourdin and Réveillac[17]have shown,in the critical caseH=¹₄,that

B,2^−n/2V_n⁽²⁾(f ) Law

−→

B, σ1/4,2

1 0

f (B_s)dWs+1 8

1 0

f(B_s)ds

.

(The reader is also referred to[16]for 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[15]

that,for any integerq≥2and in the critical caseH=_2q¹, B,2^−n/2V_n^(q)(f ) Law

−→

B, σ1/(2q),q

₁

0

f (Bs)dWs+(−1)^q 2^qq!

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,2^−n/2V_n⁽²⁾(f ), . . . ,2^−n/2Vn^(q)(f )

−→Law

B, σ_H,2

1 0

f (B_s)dW_s⁽²⁾, . . . , σ_H,q 1

0

f (B_s)dWs^(q)

, (1.20)

where(W⁽²⁾, . . . , W^(q))is a(q−1)-dimensional standard Brownian motion independent ofBand theσ_H,p’s, 2≤ p≤q,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, 2^−nH

2ⁿ

k=1

f (B_(k−1)2−n)

2^nHB_k2−n

q L²

−→qμ_{q−1 1}

0

f (B_s)dBs=qμ_{q−1 B1}

0

f (x)dx. (1.21)

2. WhenH <¹₄ andqis even, 2^2nH−n

2ⁿ

k=1

f (B_(k−1)2−n)

2^nHB_k2−nq

−μ_q L²

−→1 4

q 2

μ_{q−2 1}

0

f(B_s)ds. (1.22)

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

3. WhenH=¹₄ andqis even,

B,2^−n/2

2ⁿ

k=1

f (B_(k−1)2−n)

2^n/4B_k2−nq

−μ_q

−→Law

B,1

4 q

2

μ_{q−2 1}

0

f(B_s)ds+σ_{1/4,q 1}

0

f (B_s)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,2^−n/2

2ⁿ

k=1

f (B_(k−1)2−n)

2^nHB_k2−nq

−μq

−→Law

B,σH,q

₁

0

f (Bs)dWs

, (1.24)

forWa standard Brownian motion independent ofBand

σ_H,q= ^q

p=2

p! q

p 2

μ²_q−p2^−p

r∈Z

|r+1|^2H+ |r−1|^2H−2|r|^2Hp

. (1.25)

5. WhenH=³₄andq is even,

B, 1

√n2^−n/2

2ⁿ

k=1

f (B_(k−1)2−n)

2^nHB_k2−nq

−μq

−→Law

B,σ3/4,q

1 0

f (Bs)dWs

, (1.26)

forWa standard Brownian motion independent ofBand

σ_3/4,q= ^q

p=2

2 log 2p! q

p 2

μ²_q−p

1− 1 2q

q 1−1

q q

.

6. WhenH > ³₄andq is even,

2^{n−2H n}

2ⁿ

k=1

f (B_(k−1)2−n)

2^nHB_k2−nq

−μ_q L²

−→2μq−2

q 2

1 0

f (B_s)dZ_s⁽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. [8] and Cheridito and Nualart [3] 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(B_k2−n)+f(B_(k−1)2−n)

B_k2−n (1.28)

exists and is given byf (B₁)−f (0).

Remark 3. WhenH≤¹₆,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. [8]or Cheridito and Nualart[3].

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=(B_t)_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

s^2H+t^2H− |t−s|^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).

The mapping1_[0,t]→B_t can be extended to an isometry betweenHand the Gaussian spaceH1associated withB.

We will denote this isometry byϕ→B(ϕ).

LetS be the set of all smooth cylindrical random variables, i.e. of the form F =φ(B_t1, . . . , B_tm),

wherem≥1,φ:R^m→R∈C_b^∞and 0≤t₁<· · ·< t_m≤1. The derivative ofF with respect toB is the element of L²(Ω,H)defined by

D_sF = m

i=1

∂φ

∂x_i(B_t1, . . . , B_tm)1_[0,ti](s), s∈ [0,1].

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

F²_k,2=E F²

+ k j=1

ED^jF²

H^⊗j

.

The Malliavin derivativeDsatisfies the chain rule. Ifϕ:Rⁿ→RisC_b¹and if(F_i)_i=1,...,nis a sequence of elements ofD^1,2, thenϕ(F₁, . . . , F_n)∈D^1,2and we have

Dϕ(F₁, . . . , F_n)= n i=1

∂ϕ

∂x_i(F₁, . . . , F_n)DF_i.

We also have the following formula, which can easily be proved by induction onq. Letϕ, ψ∈C_b^q (q≥1), and fix 0≤u < v≤1 and 0≤s < t≤1. Thenϕ(B_t−B_s)ψ (B_v−B_u)∈D^q,2and

D^q

ϕ(B_t−B_s)ψ (B_v−B_u)

= q a=0

q a

ϕ^(a)(B_t−B_s)ψ^(q−a)(B_v−B_u)1^⊗_[s,t^a_]⊗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 variableu∈L²(Ω,H)belongs to the domain of the divergence operator, that is, if it satisfies

EDF, uH≤c_u

E(F²) for anyF∈S, thenI (u)is defined by the duality relationship

E F I (u)

=E

DF, uH

,

for everyF ∈D^1,2.

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{H_n(B(h)), h∈H,hH=1}, whereH_nis thenth Hermite polynomial. The mapping In(h^⊗n)=n!Hn(B(h))provides a linear isometry between the symmetric tensor product Hⁿ (equipped with the modified norm · Hⁿ=^√1_n! · H^⊗n) and Hn. ForH =¹₂,In coincides with the multiple Wiener–Itô integral of ordern. The following duality formula holds

E F I_n(h)

=E DⁿF, h

H^⊗n

, (2.2)

for any elementh∈Hⁿand any random variableF ∈D^n,2.

Let{e_k, k≥1}be a complete orthonormal system inH. Givenf ∈Hⁿandg∈H^m, for everyr=0, . . . , n∧m, the contraction off andgof orderris the element ofH^{⊗(n+m−2r)}defined by

f⊗rg= ∞ k1,...,kr=1

f, ek₁⊗ · · · ⊗ek_rH^⊗r ⊗ g, ek₁⊗ · · · ⊗ek_rH^⊗r.

Notice thatf⊗rgis not necessarily symmetric: we denote its symmetrization byf⊗rg∈H^(n+m−2r). We have the following product formula: iff ∈Hⁿandg∈H^mthen

I_n(f )I_m(g)=

n∧m r=0

r!n r

m r

I_n+m−2r(f⊗rg). (2.3)

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

E

(Bt−Bs)(Bv−Bu)

=1 2

|t−v|^2H+ |s−u|^2H− |t−u|^2H− |s−v|^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

B_u(B_t−B_s)≤(t−s)^2H (2.5)

for allu∈ [0,1]. 2. For allH∈(0,1),

2ⁿ

k,l=1

E(B_(k−1)2−nB_l2−n)=O 2ⁿ

. (2.6)

3. For anyr≥1,we have,ifH <1−_2r¹,

2ⁿ

k,l=1

E(B_k2−nB_l2−n)^r=O

2^{n−2rH n}

. (2.7)

4. For anyr≥1,we have,ifH=1−_2r¹,

2ⁿ

k,l=1

E(B_k2−nB_l2−n)^r=O

n2^2n−2rn

. (2.8)

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

B_u(B_t−B_s)

=1 2

t^2H−s^2H +1

2

|s−u|^2H− |t−u|^2H ,

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

2ⁿ

k,l=1

E(B_(k−1)2−nB_l2−n)=2^{−2H n−1}

2ⁿ

k,l=1

|l−1|^2H−l^2H− |l−k+1|^2H+ |l−k|^2H

≤C2ⁿ,

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

2ⁿ

k,l=1

E(B_k2−nB_l2−n)^r =2^−2nrH−r

2ⁿ

k,l=1

|k−l+1|^2H+ |k−l−1|^2H−2|k−l|^2Hr

≤2^n−2nrH−r ∞ 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 asC_Hp^2H−2for 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

B_u(B_t−B_s)≤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(B_u−B_v)(B_t−B_s)≤H (2H−1)

2 2H l+1−2l

1/ l

(u−v)^{(l−1)/ l}(t−s). (2.10)

3. Assume thatH >1−_2l¹ for somel≥1.Then

2ⁿ

i,j=1

E(B_i2−nB_j2−n)^l=O 2^2n−2ln

. (2.11)

Proof. We have E

B_u(B_t−B_s)

=1 2

t^2H−s^2H +1

2

|s−u|^2H− |t−u|^2H . But, when 0≤a < b≤1:

b^2H−a^2H=2H _b−a

0

(u+a)^2H−1du≤2H b^2H−1(b−a)≤2H (b−a).

Thus,|b^2H−a^2H| ≤2H|b−a|and the first point follows.

Concerning the second point, using Hölder inequality, we can write E(B_u−B_v)(B_t−B_s)=H (2H−1)

_v

u

_t

s

|y−x|^2H−2dydx

≤H (2H−1)|u−v|^{(l−1)/ l} ₁

0

_t

s

|y−x|^2H−2dy l

dx 1/ l

≤H (2H−1)|u−v|^{(l−1)/ l}|t−s|^{(l−1)/ l} 1

0

t s

|y−x|^(2H−2)ldydx 1/ l

.

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

|y−x|^(2H−2)ldydx=EB₁^H

B_t^H−B_s^H≤2H|t−s| 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(B_i2−nB_j2−n)^l=2^{−2H nl−l}

2ⁿ

i,j=1

|i−j+1|^2H+ |i−j−1|^2H−2|i−j|^2Hl

≤2^{n−2H nl+1−l}

2ⁿ−1 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|^2H−2for largek. As a consequence, sinceH >1−_2l¹, the sum

2ⁿ−1 k=−2ⁿ+1

|k+1|^2H+ |k−1|^2H−2|k|^2Hl

behaves as 2^(2H−2)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 sequence(ϕn(t))n≥1, defined as

ϕn(t)=2^nq−n1 q!

[2ⁿt] j=1

1^⊗_[(j^q₋₁₎₂₋n,j2⁻ⁿ],

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

(Bb−Ba)(Bv−Bu)

=H (2H−1) b

a

v u

s−s^2H−2dsds, so that, for anym≥n

ϕ_n(t), ϕ_m(t)

H⊗q

=H^q(2H−1)^q

q!² 2^{nq+mq−n−m}

[2^mt] j=1

[2ⁿt] k=1

_j2−m

(j−1)2^−m

_k2−n

(k−1)2⁻ⁿ

s−s^2H−2dsds q

.

Hence

m,nlim→∞

ϕ_n(t), ϕ_m(t)

H⊗q

=H^q(2H−1)^q q!²

_t

0

_t

0

s−s^(2H−2)qdsds=c_q,Ht^(2H−2)q+2,

wherecq,H =_q!2(H q−^Hq^q+^(2H1)(2H q^−1)q−2q+1). Let us denote byμ^(q)_t the limit inH^⊗qof the sequence of functionsϕn(t). For anyf∈H^⊗q, we have

ϕ_n(t), f

H^⊗q = 2^nq−n1 q!

[2ⁿt] j=1

1^⊗_[(j^q_{−1)2−n,j2−n]}, f

H^⊗q

= 2^nq−n1

q!H^q(2H−1)^q

[2ⁿt] j=1

1 0

ds1

j2⁻ⁿ (j−1)2⁻ⁿ

ds₁s₁−s₁^2H−2· · ·

× ₁

0

dsq

_j2−n

(j−1)2⁻ⁿ

ds_q|s_q−s_q|^2H−2f (s₁, . . . , s_q)

n→∞−→ 1

q!H^q(2H−1)^q t

0

ds

[0,1]^qds1· · ·dsqs1−s^2H−2· · · |sq−s|^2H−2f (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(μ^(q)_t )fort∈ [0,1].

LetZ_n^(q)be the process defined byZ_n^(q)(t)=Iq(ϕn(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 [21] and Dobrushin and Major [5], 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 [22].

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_(k−1)2−n(resp.δ_k2−n) instead of1_[0,(k−1)2−n](resp.1_[(k−1)2−n,k2⁻ⁿ]). The following proposition provides information on the asymptotic behavior ofE(V_n^(q)(f )²), asntends to infinity, forH≤1−_2q¹.

Proposition 8. Fix an integerq≥2.Suppose thatf satisfies(Hq).Then,ifH≤_2q¹,then E

V_n^(q)(f )²

=O

2^{n(−2H q+2)}

. (3.1)

If _2q¹ ≤H <1−_2q¹,then E

V_n^(q)(f )²

=O 2ⁿ

. (3.2)

Finally,ifH=1−_2q¹,then E

V_n^(q)(f )²

=O n2ⁿ

. (3.3)