3 Proof of the main results

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

Ivan Nourdin^∗, David Nualart^†‡ and Ciprian A. Tudor^§

Abstract: In this paper, we prove some central and non-central limit theorems for renormalized weighted power variations of order q ≥ 2 of the fractional Brownian motion with Hurst parameter H ∈ (0,1), where q is an integer. The central limit holds for ₂¹_q < H ≤ 1− ₂¹_q, the limit being a conditionally Gaussian distribution. IfH < ₂¹_q we show the convergence in L² to a limit which only depends on the fractional Brownian motion, and ifH > 1− _2q¹ we show the convergence inL² to a stochastic integral with respect to the Hermite process of orderq.

Key words: fractional Brownian motion, central limit theorem, non-central limit theorem, Hermite process.

2000 Mathematics Subject Classification: 60F05, 60H05, 60G15, 60H07.

This version: August 2009.

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 real q > 0 being given, the q-power variation of a stochastic process X, 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)X

k=1

|X_tn,k −X_tn,k−1|^q.

For simplicity, consider from now on the case where t_n,k = k2⁻ⁿ for n ∈ {1,2,3, . . .} and k∈ {0, . . . ,2ⁿ}. In the present paper we wish to point out some interesting phenomena when X = B is a fractional Brownian motion of Hurst index H ∈ (0,1), and when q ≥ 2 is an integer. In fact, we will also drop the absolute value (when q is odd) and we will introduce

∗Laboratoire de Probabilit´es et Mod`eles Al´eatoires, Universit´e Pierre et Marie Curie, Boˆıte courrier 188, 4 Place Jussieu, 75252 Paris Cedex 5, France,ivan.nourdin@upmc.fr

†Department of Mathematics, University of Kansas, 405 Snow Hall, Lawrence, Kansas 66045-2142, USA, nualart@math.ku.edu

‡The work of D. Nualart is supported by the NSF Grant DMS-0604207

§SAMOS/MATISSE, Centre d’´Economie de La Sorbonne, Universit´e de Panth´eon-Sorbonne Paris 1, 90 rue de Tolbiac, 75634 Paris Cedex 13, France,tudor@univ-paris1.fr

some weights. More precisely, we will consider

2ⁿ

X

k=1

f(B_(k−1)2⁻ⁿ)(∆B_k2⁻ⁿ)^q, q ∈ {2,3,4, . . .}, (1.1) where the function f :R → R is assumed to be smooth enough and where ∆B_k2⁻ⁿ denotes, here and in all the paper, the increment B_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 by B (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 isB is a standard Brownian motion. Letµ_qdenote theqth moment of a standard Gaussian random variableG∼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

X

k=1

h(2^n/2∆B_k2⁻ⁿ)^q−µ_qi _Law

−→ N (0, µ_2q−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, whenq is even andf :R→Ris continuous and has polynomial 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

X

k=1

f(B_(k−1)2⁻ⁿ)h

(2^n/2∆B_k2⁻ⁿ)^q−µ_qi _Law

−→ q

µ_2q−µ²_q Z ₁

0

f(B_s)dW_s. (1.3) Here,W denotes another standard Brownian motion, independent ofB. Whenq is odd, still forf :R→R continuous with polynomial growth, we have, this time, asn→ ∞:

2^−n/2

2ⁿ

X

k=1

f(B_(k−1)2⁻ⁿ)(2^n/2∆B_k2⁻ⁿ)^q −→^Law Z ₁

0

f(B_s) q

µ_2q−µ²_q+1dW_s+µ_q+1dB_s

, (1.4) see for instance [16].

Secondly, assume thatH6= ¹₂, that is the case where the fractional Brownian motionB has not independent increments 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 ofq and the value ofH:

• ifq is even and ifH ∈(0,³₄), asn→ ∞, 2^−n/2

2ⁿ

X

k=1

(2^nH∆B_k2⁻ⁿ)^q−µ_q Law

−→ N(0,eσ_H,q² ). (1.5)

• ifq is even and ifH = ³₄, asn→ ∞,

√1 n2^−n/2

2ⁿ

X

k=1

(2³⁴ⁿ∆B_k2⁻ⁿ)^q−µ_q Law

−→ N(0,σe²3

4,q). (1.6)

• ifq is even and ifH ∈(³₄,1), asn→ ∞, 2^n−2nH

2ⁿ

X

k=1

(2^nH∆B_k2⁻ⁿ)^q−µq Law

−→ “Hermite r.v.”. (1.7)

• ifq is odd and ifH ∈(0,¹₂], as n→ ∞, 2^−n/2

2ⁿ

X

k=1

(2^nH∆B_k2⁻ⁿ)^q −→^Law N (0,σe²_H,q). (1.8)

• ifq is odd and ifH ∈(¹₂,1), asn→ ∞, 2^−nH

2ⁿ

X

k=1

(2^nH∆B_k2⁻ⁿ)^q −→^Law N (0,eσ²_H,q). (1.9)

Here,eσ_H,q >0 denote some constant depending only onH and q. The term “Hermite r.v.” denotes a random variable whose distribution is the same as that ofZ⁽²⁾ at time one, for Z⁽²⁾ defined in Definition 7 below.

Now, let us proceed with the results concerning the weighted power variations in the case whereH 6= ¹₂. Consider the following condition on a functionf :R→R, where q ≥2 is an integer:

(Hq) f belongs toC^2q and, for any p∈(0,∞) and 0≤i≤2q: sup_t∈[0,1]E

|f⁽ⁱ⁾(Bt)|^p <∞. Suppose that f satisfies (Hq). If q is even and H ∈ (¹₂,³₄), then by Theorem 2 in Le´on and Lude˜na [11] (see also Corcuera et al [4] for related results on the asymptotic behavior of the p-variation of stochastic integrals with respect to B) we have, as n→ ∞:

2^−n/2

2ⁿ

X

k=1

f(B_(k−1)2−n)

(2^nH∆B_k2−n)^q−µ_q Law

−→ eσ_H,q Z 1

0

f(B_s)dW_s, (1.10) where, once again,W denotes a standard Brownian motion independent ofB whileσe_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 Gradinaruet al [9]. Namely, ifq≥3 is odd andH∈(0,¹₂), we have, asn→ ∞:

2^nH−n

2ⁿ

X

k=1

f(B_(k−1)2⁻ⁿ)(2^nH∆B_k2⁻ⁿ)^q −→ −^L2 µ_q+1 2

Z ₁

0

f^′(B_s)ds. (1.11) Also, whenq= 2 and H∈(0,¹₄), Nourdin [14] proved that we have, as n→ ∞:

2^2Hn−n

2ⁿ

X

k=1

f(B_(k−1)2⁻ⁿ)

(2^nH∆B_k2⁻ⁿ)²−1 L²

−→ 1 4

Z 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 be- haviors of the power variations of fractional Brownian motion (1.1) can be really different, depending on the values of q and H. The aim of the present paper is to investigate what happens in the whole generality with respect to q and H. 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 write H_q for the Hermite polynomial with degreeq defined by

Hq(x) = (−1)^q q! e^x

2 2 d^q

dx^q

e^−x

2 2

,

and we consider, whenf :R→Ris a deterministic function, the sequence ofweighted Hermite variation of order q defined by

V_n^(q)(f) :=

2ⁿ

X

k=1

f B_(k−1)2⁻ⁿ

Hq 2^nH∆B_k2⁻ⁿ

. (1.13)

The following is the main result of this paper.

Theorem 1 Fix an integer q≥2, and suppose that f satisfies (H_q).

1. Assume that 0< H < _2q¹ . Then, as n→ ∞, it holds 2^nqH−nV_n^(q)(f) −→^L2 (−1)^q 2^qq!

Z 1

0

f^(q)(Bs)ds. (1.14) 2. Assume that _2q¹ < H <1− _2q¹. Then, as n→ ∞, it holds

B,2^−n/2V_n^(q)(f) Law

−→ B, σ_H,q Z ₁

0

f(B_s)dW_s

, (1.15)

where W is a standard Brownian motion independent of B and σ_H,q =

s 1 2^qq!

X

r∈Z

|r+ 1|^2H +|r−1|^2H −2|r|^2Hq

. (1.16)

3. Assume that H= 1−_2q¹. Then, as n→ ∞, it holds B, 1

√n2^−n/2V_n^(q)(f) Law

−→ B, σ_1−1/(2q),q Z 1

0

f(Bs)dWs

, (1.17)

where W is a standard Brownian motion independent of B and σ_1−1/(2q),q = 2 log 2

q! 1− 1 2q

q

1− 1 q

q

. (1.18)

4. Assume that H >1−_2q¹. Then, as n→ ∞, it holds 2^{nq(1−H)−n}V_n^(q)(f) −→^L2

Z ₁

0

f(B_s)dZ_s^(q), (1.19) where Z^(q) denotes the Hermite process of order q introduced in Definition 7 below.

Remark 1. When q = 1, we have V_n⁽¹⁾(f) = 2^−nHP2ⁿ

k=1f B_(k−1)2⁻ⁿ

∆B_k2⁻ⁿ. For H =

1

2, 2^nHVn⁽¹⁾(f) converges in L² to the Itˆo stochastic integral R₁

0 f(B_s)dB_s. For H > ¹₂, 2^nHVn⁽¹⁾(f) converges inL² and almost surely to the Young integralR₁

0 f(B_s)dB_s. For H < ¹₂, 2^3nH−nVn⁽¹⁾(f) converges inL² to −¹2

R1

0 f^′(Bs)ds.

Remark 2. In the critical case H = _2q¹ (q ≥ 2), we conjecture the following asymptotic behavior: asn→ ∞,

B,2^−n/2V_n^(q)(f) Law

−→ B, σ_1/(2q),q Z 1

0

f(B_s)dW_s+(−1)^q 2^qq!

Z 1

0

f^(q)(B_s)ds

, (1.20) for W a standard Brownian motion independent of B and σ_1/(2q),q the constant defined by (1.16). Actually, (1.20) forq = 2 andH = ¹₄ has been proved in [2, 15, 17] after that the first draft of the current paper have been submitted. 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.

WhenH is between ¹₄ and ³₄, one can refine point 2 of Theorem 1 as follows:

Proposition 2 Let q ≥ 2 be an integer, f :R → R be a function such that (H_q) holds and assume thatH ∈(¹₄,³₄). Then

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

(1.21)

−→Law

B, σ_H,2 Z 1

0

f(B_s)dW_s⁽²⁾, . . . , σ_H,q Z 1

0

f(B_s)dW_s^(q)

, where (W⁽²⁾, . . . , W^(q)) is a (q−1)-dimensional standard Brownian motion independent of B and theσ_H,p’s, 2≤p≤q, are given by (1.16).

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

Corollary 3 Let q ≥ 2 be an integer, and f : R → R be a function such that (H_q) holds.

Then, asn→ ∞:

1. When H > ¹₂ and q is odd,

2^−nH

2ⁿ

X

k=1

f(B_(k−1)2⁻ⁿ)(2^nH∆B_k2⁻ⁿ)^{q L}−→² qµ_q−1 Z ₁

0

f(B_s)dB_s=qµ_q−1 Z _B1

0

f(x)dx.

(1.22) 2. When H < ¹₄ and q is even,

2^2nH−n

2ⁿ

X

k=1

f(B_(k−1)2⁻ⁿ)

(2^nH∆B_k2⁻ⁿ)^q−µ_q L²

−→ 1 4

q 2

µ_q−2

Z ₁

0

f^′′(B_s)ds. (1.23) (We recover (1.12) by choosing q = 2).

3. When H= ¹₄ and q is even, B,2^−n/2

2ⁿ

X

k=1

f(B_(k−1)2−n)

(2^n/4∆B_k2−n)^q−µ_q!

−→Law

B,1

4 q

2

µ_q−2 Z 1

0

f^′′(B_s)ds +eσ_1/4,q

Z ₁

0

f(B_s)dW_s

, (1.24)

where W is a standard Brownian motion independent of B and eσ_1/4,q is the constant given by (1.26) just below.

4. When ¹₄ < H < ³₄ and q is even,

B,2^−n/2

2ⁿ

X

k=1

f(B_(k−1)2⁻ⁿ)

(2^nH∆B_k2⁻ⁿ)^q−µq!

−→Law

B,eσH,q

Z 1

0

f(Bs)dWs

, (1.25) for W a standard Brownian motion independent of B and

σe_H,q= vu ut

Xq

p=2

p!

q p

2

µ²_q−p2^−pX

r∈Z

|r+ 1|^2H +|r−1|^2H−2|r|^2Hp

. (1.26)

5. When H= ³₄ and q is even, B, 1

√n2^−n/2

2ⁿ

X

k=1

f(B_(k−1)2⁻ⁿ)

(2^nH∆B_k2⁻ⁿ)^q−µ_q!

−→Law

B,σe3

4,q

Z 1

0

f(B_s)dW_s

, (1.27) for W a standard Brownian motion independent of B and

σe3 4,q=

vu ut

Xq

p=2

2 log 2p!

q p

2

µ²_q−p 1− 1 2q

q

1−1 q

q

.

6. When H > ³₄ and q is even,

2^n−2Hn

2ⁿ

X

k=1

f(B_(k−1)2⁻ⁿ)

(2^nH∆B_k2⁻ⁿ)^q−µ_q L²

−→2µ_q−2 q

2 Z 1

0

f(B_s)dZ_s⁽²⁾, (1.28) for Z⁽²⁾ the Hermite process introduced in Definition 7.

Finally, we can also give a new proof of the following result, stated and proved by Gradinaruet al. [8] and Cheridito and Nualart [3] in a continuous setting:

Theorem 4 Assume that H > ¹₆, and that f : R → R verifies (H6). Then the limit in probability, asn→ ∞, of the symmetric Riemann sums

1 2

2ⁿ

X

k=1

f^′(B_k2⁻ⁿ) +f^′(B_(k−1)2⁻ⁿ)

∆B_k2⁻ⁿ (1.29)

exists and is given by f(B₁)−f(0).

Remark 3 When H ≤ ¹6, quantity (1.29) does not converge in probability in general. As a counterexample, one can consider the case where f(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 Brown- ian motion. One refers to [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

R_H(t, s) =E(B_tB_s) = 1

2 s^2H+t^2H − |t−s|^2H

, s, t∈[0,1].

We suppose that A is the sigma-field generated by B. Let E be the set of step functions on [0, T], andHbe the Hilbert space defined as the closure ofE with respect to the inner product

h1_[0,t],1_[0,s]i^H=R_H(t, s).

The mapping1_[0,t]7→ B_t can be extended to an isometry betweenH and the Gaussian space H1 associated withB. We will denote this isometry by ϕ7→B(ϕ).

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

wherem≥1,φ:R^m→R∈C∞

b and 0≤t₁ < . . . < t_m≤1. The derivative ofF with respect to B is the element of L²(Ω,H) defined by

D_sF = Xm

i=1

∂φ

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

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

kFk²k,2 = E(F²) + Xk

j=1

E

kD^jFk²H^⊗j

.

The Malliavin derivativeD satisfies the chain rule. Ifϕ:Rⁿ→R isC¹

b and if (F_i)_i=1,...,n is a sequence of elements ofD^1,2, then ϕ(F₁, . . . , F_n)∈D^1,2 and we have

D ϕ(F₁, . . . , F_n) = Xn

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^q

b

(q≥1), and fix 0≤u < v≤1 and 0≤s < t≤1. Then ϕ(Bt−Bs)ψ(Bv−Bu)∈D^q,2 and D^q ϕ(B_t−B_s)ψ(B_v−B_u)

= Xq

a=0

q a

ϕ^(a)(B_t−B_s)ψ^(q−a)(B_v−B_u)1^⊗_[s,t]^a⊗e1^⊗_[u,v]^(q−a), (2.30) where⊗e means the symmetric tensor product.

The divergence operator I is the adjoint of the derivative operator D. If a random variableu∈L²(Ω,H) belongs to the domain of the divergence operator, that is, if it satisfies

|EhDF, ui^H| ≤c_up

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

E F I(u)

=E hDF, ui^H ,

for everyF ∈D^1,2.

For every n ≥ 1, let Hⁿ be the nth Wiener chaos of B, that is, the closed linear subspace of L²(Ω,A, P) generated by the random variables {H_n(B(h)), h ∈ H,khk^H = 1}, where H_n is the nth Hermite polynomial. The mapping I_n(h^⊗n) = n!H_n(B(h)) provides a linear isometry between the symmetric tensor productH^⊙n(equipped with the modified norm k · kH^⊙n = √¹

n!k · kH^⊗n) andHn. For H = ¹₂,I_ncoincides with the multiple Wiener-Itˆo integral of ordern. The following duality formula holds

E(F I_n(h)) =E hDⁿF, hiH^⊗n

, (2.31)

for any elementh∈H^⊙n and any random variableF ∈D^n,2.

Let{e_k, k≥1}be a complete orthonormal system inH. Given f ∈H^⊙nand g∈H^⊙m, for everyr= 0, . . . , n∧m, the contraction of f and gof order r is the element of H^{⊗(n+m−2r)} defined by

f ⊗rg= X∞ k1,...,kr=1

hf, e_k1 ⊗. . .⊗e_kriH^⊗r ⊗ hg, e_k1 ⊗. . .⊗e_kriH^⊗r.

Notice that f ⊗r g is not necessarily symmetric: we denote its symmetrization by f⊗erg ∈ H^{⊙(n+m−2r)}. We have the following product formula: iff ∈H^⊙n and g∈H^⊙m then

I_n(f)I_m(g) =

nX∧m

r=0

r!

n r

m r

I_n+m−2r(f⊗erg). (2.32) We recall the following simple formula for anys < tand u < v:

E((Bt−Bs)(Bv−Bu)) = 1

2 |t−v|^2H+|s−u|^2H − |t−u|^2H − |s−v|^2H

. (2.33) We will also need the following lemmas:

Lemma 5 1. Let s < t belong to [0,1]. Then, ifH <1/2, one has

E B_u(B_t−B_s)≤(t−s)^2H (2.34) for all u∈[0,1].

2. For allH ∈(0,1),

2ⁿ

X

k,l=1

E B_(k−1)2−n∆B_l2⁻ⁿ=O(2ⁿ). (2.35)

3. For anyr ≥1, we have, if H <1− _2r¹,

2ⁿ

X

k,l=1

|E(∆B_k2⁻ⁿ∆B_l2⁻ⁿ)|^r =O(2^n−2rHn). (2.36)

4. For anyr ≥1, we have, if H= 1− _2r¹,

2ⁿ

X

k,l=1

|E(∆B_k2⁻ⁿ∆B_l2⁻ⁿ)|^r=O(n2^2n−2rn). (2.37) Proof: To prove inequality (2.34), 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 (2.35) using (2.33), we write

2ⁿ

X

k,l=1

E B_(k−1)2⁻ⁿ∆B_l2⁻ⁿ = 2^−2Hn−1

2ⁿ

X

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 (2.36) and (2.37), we write

2ⁿ

X

k,l=1

|E(∆B_k2⁻ⁿ∆B_l2⁻ⁿ)|^r = 2^−2nrH−r

2ⁿ

X

k,l=1

|k−l+ 1|^2H +|k−l−1|^2H −2|k−l|^{2H r}

≤ 2^n−2nrH−r X∞ p=−∞

|p+ 1|^2H +|p−1|^2H −2|p|^{2H r},

and observe that, since the function

|p+ 1|^2H +|p−1|^2H −2|p|^2H

behaves as C_Hp^2H−2 for largep, the series in the right-hand side is convergent becauseH <1−_2r¹. In the critical case H= 1−_2r¹, this series is divergent, and

2ⁿ

X

p=−2ⁿ

|p+ 1|^2H +|p−1|^2H −2|p|^2Hr

behaves as a constant time n.

Lemma 6 Assume thatH > ¹₂. 1. Let s < t belong to [0,1]. Then

E B_u(B_t−B_s)≤2H(t−s) (2.38) for all u∈[0,1].

2. Assume that H >1−_2l¹ for some l≥1. Let u < v and s < t belong to [0,1]. Then

|E(B_u−B_v)(B_t−B_s)| ≤H(2H−1)

2 2Hl+ 1−2l

¹

l

(u−v)^l−1l (t−s). (2.39) 3. Assume that H >1−2l¹ for some l≥1. Then

2ⁿ

X

i,j=1

E ∆B_i2⁻ⁿ∆B_j2^−nl=O(2^2n−2ln). (2.40)

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 Z _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 H¨older inequality, we can write

|E(B_u−B_v)(B_t−B_s)| = H(2H−1) Z v

u

Z t

s |y−x|^2H−2dydx

≤ H(2H−1)|u−v|^l−1l Z 1

0

Z t

s |y−x|^2H−2dy l

dx

!¹

l

≤ H(2H−1)|u−v|^l−1l |t−s|^l−1l Z 1

0

Z t

s |y−x|^(2H−2)ldydx ¹_l

. Denote byH^′ = 1 + (H−1)land observe thatH^′ > ¹₂ (because H >1−_2l¹). Since 2H^′−2 = (2H−2)l, we can write

H^′(2H^′−1) Z 1

0

Z t

s |y−x|^(2H−2)ldydx=E

B₁^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

X

i,j=1

E ∆B_i2⁻ⁿ∆B_j2^−nl = 2^−2Hnl−l

2ⁿ

X

i,j=1

|i−j+ 1|^2H +|i−j−1|^2H −2|i−j|^{2H l}

≤ 2^{n−2Hnl+1−l}

2Xⁿ−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

2Xⁿ−1

k=−2ⁿ+1

|k+ 1|^2H +|k−1|^2H −2|k|^{2H l}

behaves as 2^(2H−2)ln+n and the third point follows.

Now, let us introduce the Hermite process of order q ≥ 2 appearing in (1.19). Fix H >1/2 and t∈[0,1]. The sequence ϕ_n(t)

n≥1, defined as ϕ_n(t) = 2^nq−n1

q!

[2Xⁿt]

j=1

1^⊗_[(j^q

−1)2⁻ⁿ,j2⁻ⁿ], is a Cauchy sequence in the spaceH^⊗q. Indeed, sinceH >1/2, we have

h1_[a,b],1_[u,v]i^H=E (B_b−Ba)(Bv−Bu)

=H(2H−1) Z b

a

Z v

u |s−s^′|^2H−2dsds^′, so that, for anym≥n

hϕ_n(t), ϕ_m(t)iH^⊗q = H^q(2H−1)^q

q!² 2^{nq+mq−n−m}

[2X^mt]

j=1 [2Xⁿt]

k=1

Z j2^−m (j−1)2^−m

Z _k2⁻ⁿ

(k−1)2⁻ⁿ|s−s^′|^2H−2dsds^′

!q

. Hence

m,nlim→∞hϕn(t), ϕm(t)iH^⊗q = H^q(2H−1)^q q!²

Z t

0

Z t

0 |s−s^′|^(2H−2)qdsds^′ =cq,Ht^(2H−2)q+2, wherec_q,H = _q!2(Hq−^Hq+1)(2Hq^q(2H−1)q−2q+1). Let us denote by µ^(q)_t the limit in H^⊗q of the sequence of functionsϕn(t). For any f ∈H^⊗q, we have

hϕ_n(t), fiH^⊗q = 2^nq−n1 q!

[2ⁿt]

X

j=1

h1^⊗_[(j^q₋₁₎₂−n,j2⁻ⁿ], fiH^⊗q

= 2^nq−n1

q!H^q(2H−1)^q

[2ⁿt]

X

j=1

Z ₁

0

ds₁ Z _j2⁻ⁿ

(j−1)2⁻ⁿ

ds^′₁|s₁−s^′₁|^2H−2. . .

× Z 1

0

ds_q Z 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 Z _t

0

ds^′ Z

[0,1]^q

ds₁. . . ds_q|s₁−s^′|^2H−2. . .|s_q−s^′|^2H−2f(s₁, . . . , s_q)

=hµ^(q)_t , fiH^⊗q.

Definition 7 Fix q ≥2 and H >1/2. The Hermite process Z^(q) = (Z_t^(q))_t∈[0,1] of order q is defined byZ_t^(q) =Iq(µ^(q)_t ) for t∈[0,1].

LetZn^(q) be the process defined byZn^(q)(t) =I_q(ϕ_n(t)) for t∈[0,1]. By construction, it is clear thatZ_n^(q)(t)−→^L2 Z^(q)(t) asn→ ∞, for all fixedt∈[0,1]. On the other hand, it follows, from Taqqu [21] and Dobrushin and Major [5], that Z_n^(q) converges in law to the “standard”

and historical qth Hermite process, defined through its moving average representation as a multiple integral with respect to a Wiener process with time horizon R. In particular, the process introduced in Definition 7 has 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 H¨older 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 writeε_(k−1)2⁻ⁿ (resp. δ_k2⁻ⁿ) instead of 1_[0,(k−1)2⁻ⁿ_] (resp. 1_[(k−1)2⁻ⁿ_,k2⁻ⁿ_]). The following proposition provides information on the asymptotic behavior of E

Vn^(q)(f)² , as n tends to infinity, forH ≤1−_2q¹.

Proposition 8 Fix an integer q≥2. Suppose that f satisfies (H_q). Then, if H≤ _2q¹ , then E

V_n^(q)(f)²

=O(2^n(−2Hq+2)). (3.41)

If _2q¹ ≤H <1−_2q¹, then

E

V_n^(q)(f)²

=O(2ⁿ). (3.42)

Finally, if H= 1−_2q¹ , then

E

V_n^(q)(f)²

=O(n2ⁿ). (3.43)

Proof. Using the relation between Hermite polynomials and multiple stochastic integrals, we haveH_q 2^nH∆B_k2⁻ⁿ

= _q!¹2^qnHI_q δ^⊗_k2^q−n

. In this way we obtain E

V_n^(q)(f)²

=

2ⁿ

X

k,l=1

E

f(B_(k−1)2⁻ⁿ)f(B_(l−1)2⁻ⁿ)H_q 2^nH∆B_k2⁻ⁿ

H_q 2^nH∆B_l2⁻ⁿ

= 1

q!² 2^2Hqn

2ⁿ

X

k,l=1

En

f(B_(k−1)2⁻ⁿ)f(B_(l−1)2⁻ⁿ)I_q δ^⊗_k2^q−n

I_q

δ_l2^⊗−n^q

o .

Now we apply the product formula (2.32) for multiple stochastic integrals and the duality relationship (2.31) between the multiple stochastic integral I_N and the iterated derivative operatorD^N, obtaining

E

V_n^(q)(f)²

= 2^2Hqn q!²

2ⁿ

X

k,l=1

Xq

r=0

r!

q r

2

×En

f(B_(k−1)2⁻ⁿ)f(B_(l−1)2⁻ⁿ)I_2q−2r

δ^⊗_k2^q−n^−r⊗eδ_l2^⊗−n^q−r

o

hδ_k2⁻ⁿ, δ_l2⁻ⁿi^rH

= 2^2Hqn

2ⁿ

X

k,l=1

Xq

r=0

1 r!(q−r)!²

×EnD

D^2q−2r f(B_(k−1)2⁻ⁿ)f(B_(l−1)2⁻ⁿ)

, δ_k2^⊗q−n^−r⊗eδ^⊗_l2^q−n^−r

E

H^⊗(2q−2r)

o

hδ_k2⁻ⁿ, δ_l2⁻ⁿi^rH, where ⊗e denotes the symmetrization of the tensor product. By (2.30), the derivative of the product D^2q−2r f(B_(k−1)2⁻ⁿ)f(B_(l−1)2⁻ⁿ)

is equal to a sum of derivatives:

D^2q−2r f(B_(k−1)2⁻ⁿ)f(B_(l−1)2⁻ⁿ)

= X

a+b=2q−2r

f^(a)(B_(k−1)2⁻ⁿ)f^(b)(B_(l−1)2⁻ⁿ)

×(2q−2r)!

a!b!

ε^⊗_(k^a

−1)2⁻ⁿ⊗eε^⊗_(l^b

−1)2⁻ⁿ

.

We make the decomposition E

V_n^(q)(f)²

=A_n+B_n+C_n+D_n, (3.44)

where

An= 2^2Hqn q!²

2ⁿ

X

k,l=1

En

f^(q)(B_(k−1)2⁻ⁿ)f^(q)(B_(l−1)2⁻ⁿ)o

hε_(k−1)2⁻ⁿ, δ_k2⁻ⁿi^q hε_(l−1)2⁻ⁿ, δ_l2⁻ⁿi^q,

B_n = 2^2Hqn X

c+d+e+f=2q d+e≥1

2ⁿ

X

k,l=1

En

f^(q)(B_(k−1)2⁻ⁿ)f^(q)(B_(l−1)2⁻ⁿ)o

α(c, d, e, f)

×hε_(k−1)2−n, δ_k2−ni^cHhε_(k−1)2−n, δ_l2−ni^dHhε_(l−1)2−n, δ_k2−ni^eHhε_(l−1)2−n, δ_l2−ni^fH, C_n = 2^2Hqn X

a+b=2q (a,b)6=(q,q)

2ⁿ

X

k,l=1

En

f^(a)(B_(k−1)2⁻ⁿ)f^(b)(B_(l−1)2⁻ⁿ)o (2q)!

q!²a!b!

×hε^⊗_(k^a₋₁₎₂−n⊗eε^⊗_(l−^b₁₎₂−n, δ^⊗_k2^q−n⊗eδ_l2^⊗q−niH^⊗(2q),

and

D_n = 2^2Hqn Xq

r=1

X

a+b=2q−2r 2ⁿ

X

k,l=1

En

f^(a)(B_(k−1)2⁻ⁿ)f^(b)(B_(l−1)2⁻ⁿ)o (2q−2r)!

r!(q−r)!²a!b!

×hε^⊗_(k^a₋₁₎₂−n⊗eε^⊗_(l−^b₁₎₂−n, δ_k2^⊗q−n^−r⊗eδ^⊗_l2^q−n^−riH^⊗(2q−2r)hδ_k2⁻ⁿ, δ_l2⁻ⁿi^rH,

for some combinatorial constantsα(c, d, e, f). That is, A_n and B_n contain all the terms with r= 0 and (a, b) = (q, q);C_ncontains the terms withr = 0 and (a, b)6= (q, q); andD_ncontains the remaining terms.

For any integerr≥1, we set α_n = sup

k,l=1,...,2ⁿ

hε_(k−1)2⁻ⁿ, δ_l2⁻ⁿi^H

, (3.45)

β_r,n =

2ⁿ

X

k,l=1

hδ_k2−n, δ_l2−niH

^r, (3.46)

γ_n =

2ⁿ

X

k,l=1

hε_(k−1)2⁻ⁿ, δ_l2⁻ⁿi^H

. (3.47)

Then, under assumption (Hq), we have the following estimates:

|An| ≤ C2^2Hqn+2n(αn)^2q,

|B_n|+|C_n| ≤ C2^2Hqn(α_n)^2q−1γ_n,

|Dn| ≤ C2^2Hqn Xq

r=1

(αn)^2q−2rβr,n,

whereCis a constant depending only onqand the functionf. Notice that the second inequality follows from the fact that when (a, b) 6= (q, q), or (a, b) = (q, q) and c+d+e+f = 2q with d≥1 or e≥1, there will be at least a factor of the formhε_(k−1)2⁻ⁿ, δ_l2⁻ⁿi^Hin the expression ofB_nor C_n.

In the caseH < ¹₂, we have by (2.34) thatα_n≤2^−2nH, by (2.36) thatβ_r,n≤C2^n−2rHn, and by (2.35) thatγn≤C2ⁿ. As a consequence, we obtain

|A_n| ≤ C2^n(−2Hq+2), (3.48)

|B_n|+|C_n| ≤ C2^{n(−2Hq+2H+1)}, (3.49)

|D_n| ≤ C Xq

r=1

2^{n(−2(q−r)H+1)}, (3.50)

which implies the estimates (3.41) and (3.42).