Ivan Nourdin
email: ivan.nourdin@upmc.fr

David Nualart
email: nualart@math.ku.edu

Ciprian A Tudor
email: tudor@univ-paris1.fr

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

Keywords: fractional Brownian motion, central limit theorem, non-central limit theorem, Hermite process. 2000 Mathematics Subject Classification: 60F05, 60H05, 60G15, 60H07

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 1 2q < H ≤ 1 -1 2q , the limit being a conditionally Gaussian distribution. If H < 1 2q we show the convergence in L 2 to a limit which only depends on the fractional Brownian motion, and if H > 1 -1 2q we show the convergence in L 2 to a stochastic integral with respect to the Hermite process of order q.

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 subdivision π 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 |X t n,k -X t n,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, . . . , 2 n }. 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 1 some weights. More precisely, we will consider

2 n 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 [START_REF] Gradinaru | Milstein's type scheme for fractional SDEs[END_REF], [START_REF] Neuenkirch | Exact rate of convergence of some approximation schemes associated to SDEs driven by a fractional Brownian motion[END_REF] and [START_REF] Nourdin | A simple theory for the study of SDEs driven by a fractional Brownian motion, in dimension one[END_REF]) besides, of course, the traditional applications of quadratic variations to parameter estimation problems. Now, let us recall some known results concerning q-power variations (for q = 2, 3, 4, . . .), which are today more or less classical. First, assume that the Hurst index is H = 1 2 , that is B is a standard Brownian motion. Let µ q denote 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, as n → ∞:

2 -n/2 2 n k=1 (2 n/2 ∆B k2 -n ) q -µ q Law -→ N (0, µ 2q -µ 2 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, when q is even and f : R → R is continuous and has polynomial growth, it is a very particular case of a more general result by Jacod [START_REF] Jacod | Limit of random measures associated with the increments of a Brownian semimartingale[END_REF] (see also Section 2 in Nourdin and Peccati [START_REF] Nourdin | Weighted power variations of iterated Brownian motion[END_REF] for related results) that we have, as n → ∞:

2 -n/2 2 n k=1 f (B (k-1)2 -n ) (2 n/2 ∆B k2 -n ) q -µ q Law -→ µ 2q -µ 2 q 1 0 f (B s )dW s . (1.3)
Here, W denotes another standard Brownian motion, independent of B. When q is odd, still for f : R → R continuous with polynomial growth, we have, this time, as n → ∞:

2 -n/2 2 n k=1 f (B (k-1)2 -n )(2 n/2 ∆B k2 -n ) q Law -→ 1 0 f (B s ) µ 2q -µ 2 q+1 dW s + µ q+1 dB s , (1.4)
see for instance [START_REF] Nourdin | Weighted power variations of iterated Brownian motion[END_REF].

Secondly, assume that H = 1 2 , that is the case where the fractional Brownian motion B has not independent increments anymore. Then (1.2) has been extended by Breuer and Major [START_REF] Breuer | Central limit theorems for nonlinear functionals of Gaussian fields[END_REF], Dobrushin and Major [START_REF] Dobrushin | Non-central limit theorems for nonlinear functionals of Gaussian fields[END_REF], Giraitis and Surgailis [START_REF] Giraitis | CLT and other limit theorems for functionals of Gaussian processes[END_REF] or Taqqu [START_REF] Taqqu | Convergence of integrated processes of arbitrary Hermite rank[END_REF]. Precisely, five cases are considered, according to the evenness of q and the value of H:

• if q is even and if H ∈ (0, 3 4 ), as n → ∞,

2 -n/2 2 n k=1
(2 nH ∆B k2 -n ) qµ q Law -→ N (0, σ 2 H,q ). (1.5)

• if q is even and if H = 3 4 , as n → ∞,

1 √ n 2 -n/2 2 n k=1 ( 2
3 4 n ∆B k2 -n ) q -µ q Law -→ N (0, σ 2 3 4 ,q ).
(1.6)

• if q is even and if H ∈ ( 3 4 , 1), as n → ∞,

2 n-2nH 2 n k=1 (2 nH ∆B k2 -n ) q -µ q Law -→ "Hermite r.v.".
(1.7)

• if q is odd and if H ∈ (0, 1 2 ], as n → ∞, 2 -n/2 2 n k=1 (2 nH ∆B k2 -n ) q Law -→ N (0, σ 2 H,q ).
(1.8)

• if q is odd and if H ∈ ( 1 2 , 1), as n → ∞, 2 -nH 2 n k=1 (2 nH ∆B k2 -n ) q Law -→ N (0, σ 2 H,q ). (1.9)
Here, σ H,q > 0 denote some constant depending only on H and q. The term "Hermite r.v." denotes a random variable whose distribution is the same as that of Z (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 where H = 1 2 . Consider the following condition on a function f : R → R, where q ≥ 2 is an integer: (H q ) f belongs to C 2q and, for any p ∈ (0, ∞) and

0 ≤ i ≤ 2q: sup t∈[0,1] E |f (i) (B t )| p < ∞.
Suppose that f satisfies (H q ). If q is even and H ∈ ( 1 2 , 3 4 ), then by Theorem 2 in León and Ludeña [START_REF] León | Limits for weighted p-variations and likewise functionals of fractional diffusions with drift[END_REF] (see also Corcuera et al [START_REF] Corcuera | Power variation of some integral fractional processes[END_REF] 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 2 n k=1 f (B (k-1)2 -n ) (2 nH ∆B k2 -n ) q -µ q Law -→ σ H,q 1 0 f (B s )dW s , (1.10)
where, once again, W denotes a standard Brownian motion independent of B while σ 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 of H. The first result in this direction has been observed by Gradinaru et al [START_REF] Gradinaru | Generalized covariations, local time and Stratonovich Itô's formula for fractional Brownian motion with Hurst index H ≥ 1 4[END_REF]. Namely, if q ≥ 3 is odd and H ∈ (0, 1 2 ), we have, as n → ∞:

2 nH-n 2 n k=1 f (B (k-1)2 -n )(2 nH ∆B k2 -n ) q L 2 -→ - µ q+1 2 1 0 f ′ (B s )ds. (1.11)
Also, when q = 2 and H ∈ (0, 1 4 ), Nourdin [START_REF] Nourdin | Asymptotic behavior of some weighted quadratic and cubic variations of the fractional Brownian motion[END_REF] proved that we have, as n → ∞:

2 2Hn-n 2 n k=1 f (B (k-1)2 -n ) (2 nH ∆B k2 -n ) 2 -1 L 2 -→ 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 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 degree q defined by

H q (x) = (-1) q q! e x 2 2 d q dx q e -x 2 2 ,
and we consider, when f : R → R is a deterministic function, the sequence of weighted Hermite variation of order q defined by

V (q) n (f ) := 2 n k=1 f B (k-1)2 -n H q 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 < 1 2q . Then, as n → ∞, it holds

2 nqH-n V (q) n (f ) L 2 -→ (-1) q 2 q q! 1 0 f (q) (B s )ds. (1.14) 2. Assume that 1 2q < H < 1 -1 2q . Then, as n → ∞, it holds B, 2 -n/2 V (q) n (f ) Law -→ B, σ H,q 1 0 f (B s )dW s , (1.15)
where W is a standard Brownian motion independent of B and

σ H,q = 1 2 q q! r∈Z |r + 1| 2H + |r -1| 2H -2|r| 2H q .
(1.16)

Assume that

H = 1 -1 2q . Then, as n → ∞, it holds B, 1 √ n 2 -n/2 V (q) n (f ) Law -→ B, σ 1-1/(2q),q 1 0 f (B s )dW s , (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 -1 2q . Then, as n → ∞, it holds

2 nq(1-H)-n V (q) n (f ) L 2 -→ 1 0 f (B s )dZ (q) s , (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

(1)

n (f ) = 2 -nH 2 n k=1 f B (k-1)2 -n ∆B k2 -n . For H = 1 2 , 2 nH V (1)
n (f ) converges in L 2 to the Itô stochastic integral

1 0 f (B s )dB s . For H > 1 2 , 2 nH V (1)
n (f ) converges in L 2 and almost surely to the Young integral

1 0 f (B s )dB s . For H < 1 2 , 2 3nH-n V (1) n (f ) converges in L 2 to -1 2 1 0 f ′ (B s )ds.
Remark 2. In the critical case H = 1 2q (q ≥ 2), we conjecture the following asymptotic behavior: as n → ∞,

B, 2 -n/2 V (q) n (f ) Law -→ B, σ 1/(2q),q 1 0 f (B s )dW s + (-1) q 2 q q! 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) for q = 2 and H = 1 4 has been proved in [START_REF] Burdzy | A change of variable formula with Itô correction term[END_REF][START_REF] Nourdin | Central limit theorems for multiple Skorohod integrals[END_REF][START_REF] Nourdin | Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: the critical case H=1/4[END_REF] after that the first draft of the current paper have been submitted. The reader is also referred to [START_REF] Nourdin | Weighted power variations of iterated Brownian motion[END_REF] for the study of the weighted variations associated with iterated Brownian motion, which is a non-Gaussian self-similar process of order 1 4 .

When H is between 1 4 and 3 4 , 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 that H ∈ ( 1 4 , 3 4 ). Then

B, 2 -n/2 V (2) n (f ), . . . , 2 -n/2 V (q) n (f ) (1.21) Law -→ B, σ H,2 1 0 f (B s )dW (2) s , . . . , σ H,q 1 0 f (B s )dW (q) s ,
where (W (2) , . . . , W (q) ) is a (q -1)-dimensional standard Brownian motion independent of B and the σ 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 weighted power variations 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, as n → ∞:

1. When H > 1 2 and q is odd,

2 -nH 2 n k=1 f (B (k-1)2 -n )(2 nH ∆B k2 -n ) q L 2 -→ qµ q-1 1 0 f (B s )dB s = qµ q-1 B 1 0 f (x)dx.
(1.22)

2. When H < 1 4 and q is even,

2 2nH-n 2 n k=1 f (B (k-1)2 -n ) (2 nH ∆B k2 -n ) q -µ q L 2 -→ 1 4 q 2 µ q-2 1 0 f ′′ (B s )ds. (1.23)
(We recover (1.12) by choosing q = 2).

3. When H = 1 4 and q is even,

B, 2 -n/2 2 n 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 1 0 f ′′ (B s )ds + σ 1/4,q 1 0 f (B s )dW s , (1.24)
where W is a standard Brownian motion independent of B and σ 1/4,q is the constant given by (1.26) just below.

4. When 1 4 < H < 3 4 and q is even,

B, 2 -n/2 2 n k=1 f (B (k-1)2 -n ) (2 nH ∆B k2 -n ) q -µ q Law -→ B, σ H,q 1 0 f (B s )dW s ,
(1.25) for W a standard Brownian motion independent of B and

σ H,q = q p=2 p! q p 2 µ 2 q-p 2 -p r∈Z |r + 1| 2H + |r -1| 2H -2|r| 2H p .
(1.26)

5. When H = 3 4 and q is even,

B, 1 √ n 2 -n/2 2 n k=1 f (B (k-1)2 -n ) (2 nH ∆B k2 -n ) q -µ q Law -→ B, σ 3 4 ,q 1 0 f (B s )dW s ,
(1.27) for W a standard Brownian motion independent of B and

σ 3 4 ,q = q p=2 2 log 2 p! q p 2 µ 2 q-p 1 - 1 2q q 1 - 1 q q .
6. When H > 3 4 and q is even,

2 n-2Hn 2 n k=1 f (B (k-1)2 -n ) (2 nH ∆B k2 -n ) q -µ q L 2 -→ 2µ q-2 q 2 1 0 f (B s )dZ (2) s , (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 Gradinaru et al. [START_REF] Gradinaru | m-order integrals and Itô's formula for non-semimartingale processes; the case of a fractional Brownian motion with any Hurst index[END_REF] and Cheridito and Nualart [START_REF] Cheridito | Stochastic integral of divergence type with respect to fractional Brownian motion with Hurst parameter H in (0, 1/2)[END_REF] in a continuous setting: Theorem 4 Assume that H > 1 6 , and that f : R → R verifies (H 6 ). Then the limit in probability, as n → ∞, of the symmetric Riemann sums 1 2

2 n k=1 f ′ (B k2 -n ) + f ′ (B (k-1)2 -n ) ∆B k2 -n (1.29)
exists and is given by f (B 1 )f (0).

Remark 3 When H ≤ 1 6 , quantity (1.29) does not converge in probability in general. As a counterexample, one can consider the case where f (x) = x 3 , see Gradinaru et al. [START_REF] Gradinaru | m-order integrals and Itô's formula for non-semimartingale processes; the case of a fractional Brownian motion with any Hurst index[END_REF] or Cheridito and Nualart [START_REF] Cheridito | Stochastic integral of divergence type with respect to fractional Brownian motion with Hurst parameter H in (0, 1/2)[END_REF].

Preliminaries and notation

We briefly recall some basic facts about stochastic calculus with respect to a fractional Brownian motion. One refers to [START_REF] Nualart | Stochastic calculus with respect to the fractional Brownian motion and applications[END_REF] for further details. Let B = (B t ) t∈[0,1] be a fractional Brownian motion with Hurst parameter H ∈ (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 t B 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 ], and H be the Hilbert space defined as the closure of E with respect to the inner product

1 [0,t] , 1 [0,s] H = R H (t, s).
The mapping 1 [0,t] → B t can be extended to an isometry between H and the Gaussian space H 1 associated with B. We will denote this isometry by ϕ → B(ϕ).

Let S be the set of all smooth cylindrical random variables, i.e. of the form

F = φ(B t 1 , . . . , B tm ) where m ≥ 1, φ : R m → R ∈ C ∞ b and 0 ≤ t 1 < . . . < t m ≤ 1. The derivative of F with respect to B is the element of L 2 (Ω, H) defined by D s F = m i=1 ∂φ ∂x i (B t 1 , . . . , B tm )1 [0,t i ] (s), s ∈ [0, 1].
In particular D s B t = 1 [0,t] (s). For any integer k ≥ 1, we denote by D k,2 the closure of the set of smooth random variables with respect to the norm

F 2 k,2 = E(F 2 ) + k j=1 E D j F 2 H ⊗j .
The Malliavin derivative D satisfies the chain rule.

If ϕ : R n → R is C 1 b and if (F i ) i=1,...,n is a sequence of elements of D 1,2 , then ϕ(F 1 , . . . , F n ) ∈ D 1,2 and we have D ϕ(F 1 , . . . , F n ) = n i=1 ∂ϕ ∂x i (F 1 , . . . , F n )DF i .
We also have the following formula, which can easily be proved by induction on q. Let ϕ, ψ ∈ C q b (q ≥ 1), and fix 0

≤ u < v ≤ 1 and 0 ≤ s < t ≤ 1. Then ϕ(B t -B s )ψ(B v -B u ) ∈ D q,2 and 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 ⊗a [s,t] ⊗1 ⊗(q-a) [u,v] , (2.30)
where ⊗ means the symmetric tensor product.

The divergence operator I is the adjoint of the derivative operator D. If a random variable u ∈ L 2 (Ω, H) belongs to the domain of the divergence operator, that is, if it satisfies

|E DF, u H | ≤ c u E(F 2 ) for any F ∈ S , then I(u) is defined by the duality relationship E F I(u) = E DF, u H , for every F ∈ D 1,2 .
For every n ≥ 1, let H n be the nth Wiener chaos of B, that is, the closed linear subspace of L 2 (Ω, A, P ) generated by the random variables {H n (B (h)) , h ∈ H, h 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 product H ⊙n (equipped with the modified norm

• H ⊙n = 1 √ n!
• H ⊗n ) and H n . For H = 1 2 , I n coincides with the multiple Wiener-Itô integral of order n. The following duality formula holds

E (F I n (h)) = E D n F, h H ⊗n , (2.31)
for any element h ∈ H ⊙n and any random variable

F ∈ D n,2 .
Let {e k , k ≥ 1} be a complete orthonormal system in H. Given f ∈ H ⊙n and g ∈ H ⊙m , for every r = 0, . . . , n ∧ m, the contraction of f and g of order r is the element of H ⊗(n+m-2r) defined by

f ⊗ r g = ∞ k 1 ,...,kr=1 f, e k 1 ⊗ . . . ⊗ e kr H ⊗r ⊗ g, e k 1 ⊗ . . . ⊗ e kr H ⊗r .
Notice that f ⊗ r g is not necessarily symmetric: we denote its symmetrization by f ⊗ r g ∈ H ⊙(n+m-2r) . We have the following product formula: if f ∈ H ⊙n and g ∈ H ⊙m then

I n (f )I m (g) = n∧m r=0 r! n r m r I n+m-2r (f ⊗ r g). (2.32)
We recall the following simple formula for any s < t and u < v:

E ((B t -B s )(B v -B u )) = 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, if H < 1/2, one has E B u (B t -B s ) ≤ (t -s) 2H (2.34) for all u ∈ [0, 1].
2. For all H ∈ (0, 1),

2 n k,l=1 E B (k-1)2 -n ∆B l2 -n = O(2 n ). (2.35) 3. For any r ≥ 1, we have, if H < 1 -1 2r , 2 n k,l=1 |E (∆B k2 -n ∆B l2 -n )| r = O(2 n-2rHn ). (2.36) 4. For any r ≥ 1, we have, if H = 1 -1 2r , 2 n 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 any a, b ∈ [0, 1], because H < 1 2 .
To show (2.35) using (2.33), we write

2 n k,l=1 E B (k-1)2 -n ∆B l2 -n = 2 -2Hn-1 2 n k,l=1 |l -1| 2H -l 2H -|l -k + 1| 2H + |l -k| 2H ≤ C2 n ,
the last bound coming from a telescoping sum argument. Finally, to show (2.36) and (2.37), we write

2 n k,l=1 |E (∆B k2 -n ∆B l2 -n )| r = 2 -2nrH-r 2 n k,l=1 |k -l + 1| 2H + |k -l -1| 2H -2|k -l| 2H r ≤ 2 n-2nrH-r ∞ 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 H p 2H-2 for large p, the series in the right-hand side is convergent because H < 1 -1 2r . In the critical case H = 1 -1 2r , this series is divergent, and

2 n p=-2 n |p + 1| 2H + |p -1| 2H -2|p| 2H r
behaves as a constant time n.

Lemma 6 Assume that H > 1 2 . 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].

Assume that

H > 1 -1 2l 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-1 l (t -s).
(2.39)

3. Assume that H > 1 -1 2l for some l ≥ 1. Then

2 n i,j=1 E ∆B i2 -n ∆B j2 -n l = 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 b-a 0 (u + a) 2H-1 du ≤ 2H b 2H-1 (b -a) ≤ 2H(b -a).
Thus, |b 2Ha 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-2 dydx ≤ H(2H -
Denote by H ′ = 1 + (H -1)l and observe that

H ′ > 1 2 (because H > 1 -1 2l ). Since 2H ′ -2 = (2H -2)l, we can write H ′ (2H ′ -1) 1 0 t s |y -x| (2H-2)l dydx = E B H ′ 1 (B H ′ t -B H ′ s ) ≤ 2H ′ |t -s|
by the first point of this lemma. This gives the desired bound.

We prove now the third point. We have

2 n i,j=1 E ∆B i2 -n ∆B j2 -n l = 2 -2Hnl-l 2 n i,j=1 |i -j + 1| 2H + |i -j -1| 2H -2|i -j| 2H l ≤ 2 n-2Hnl+1-l 2 n -1 k=-2 n +1 |k + 1| 2H + |k -1| 2H -2|k| 2H l
and the function |k +1| 2H +|k -1| 2H -2|k| 2H behaves as |k| 2H-2 for large k. As a consequence, since H > 1 -1 2l , the sum

2 n -1 k=-2 n +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-n 1 q! [2 n t] j=1 1 ⊗q [(j-1)2 -n ,j2 -n ] ,
is a Cauchy sequence in the space H ⊗q . Indeed, since H > 1/2, we have

1 [a,b] , 1 [u,v] H = E (B b -B a )(B v -B u ) = H(2H -1) b a v u |s -s ′ | 2H-2 dsds ′ , so that, for any m ≥ n ϕ n (t), ϕ m (t) H ⊗q = H q (2H -1) q q! 2 2 nq+mq-n-m [2 m t] j=1 [2 n t] k=1 j2 -m (j-1)2 -m k2 -n (k-1)2 -n |s -s ′ | 2H-2 dsds ′ q . Hence lim m,n→∞ ϕ n (t), ϕ m (t) H ⊗q = H q (2H -1) q q! 2 t 0 t 0 |s -s ′ | (2H-2)q dsds ′ = c q,H t (2H-2)q+2 ,
where c q,H = H q (2H-1) q q! 2 (Hq-q+1)(2Hq-2q+1) . Let us denote by µ (q)

t the limit in H ⊗q of the sequence of functions ϕ n (t). For any f ∈ H ⊗q , we have

ϕ n (t), f H ⊗q = 2 nq-n 1 q! [2 n t] j=1 1 ⊗q [(j-1)2 -n ,j2 -n ] , f H ⊗q = 2 nq-n 1 q! H q (2H -1) q [2 n t] j=1 1 0 ds 1 j2 -n (j-1)2 -n ds ′ 1 |s 1 -s ′ 1 | 2H-2 . . . × 1 0 ds q j2 -n (j-1)2 -n ds ′ q |s q -s ′ q | 2H-2 f (s 1 , . . . , s q ) -→ n→∞ 1 q! H q (2H -1) q t 0 ds ′ [0,1] q ds 1 . . . ds q |s 1 -s ′ | 2H-2 . . . |s q -s ′ | 2H-2 f (s 1 , . . . , s q ) = 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 order q is defined by Z (q) t = I q (µ (q) t ) for t ∈ [0, 1]. Let Z (q)
n be the process defined by

Z (q) n (t) = I q (ϕ n (t)) for t ∈ [0, 1]. By construction, it is clear that Z (q) n (t) L 2
-→ Z (q) (t) as n → ∞, for all fixed t ∈ [0, 1]. On the other hand, it follows, from Taqqu [START_REF] Taqqu | Convergence of integrated processes of arbitrary Hermite rank[END_REF] and Dobrushin and Major [START_REF] Dobrushin | Non-central limit theorems for nonlinear functionals of Gaussian fields[END_REF], that Z (q) n 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 that Z (q) is q(H -1)+1 self-similar, has stationary increments and admits moments of all orders. Moreover, it has Hölder continuous paths of order strictly less than q(H -1) + 1. For further results, we refer to Tudor [START_REF] Tudor | Analysis of the Rosenblatt process[END_REF].

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 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 V (q) n (f ) 2 , as n tends to infinity, for H ≤ 1 -1 2q .

Proposition 8 Fix an integer q ≥ 2. Suppose that f satisfies (H q ). Then, if H ≤ 1 2q , then

E V (q) n (f ) 2 = O(2 n(-2Hq+2) ).
(3.41)

If 1 2q ≤ H < 1 -1 2q , then E V (q) n (f ) 2 = O(2 n ). (3.42) Finally, if H = 1 -1 2q , then E V (q) n (f ) 2 = O(n2 n ). ( 3

.43)

Proof. Using the relation between Hermite polynomials and multiple stochastic integrals, we have H q 2 nH ∆B k2 -n = 1 q! 2 qnH I q δ ⊗q k2 -n . In this way we obtain

E V (q) n (f ) 2 = 2 n 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 2 n k,l=1 E f (B (k-1)2 -n ) f (B (l-1)2 -n ) I q δ ⊗q k2 -n I q δ ⊗q l2 -n
. 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 operator D N , obtaining

E V (q) n (f ) 2 = 2 2Hqn q! 2 2 n k,l=1 q r=0 r! q r 2 ×E f (B (k-1)2 -n ) f (B (l-1)2 -n ) I 2q-2r δ ⊗q-r k2 -n ⊗δ ⊗q-r l2 -n δ k2 -n , δ l2 -n r H = 2 2Hqn 2 n k,l=1 q r=0 1 r!(q -r)! 2 ×E D 2q-2r f (B (k-1)2 -n ) f (B (l-1)2 -n ) , δ ⊗q-r k2 -n ⊗δ ⊗q-r l2 -n H ⊗(2q-2r) δ k2 -n , δ l2 -n r H ,
where ⊗ 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 ) = a+b=2q-2r f (a) (B (k-1)2 -n ) f (b) (B (l-1)2 -n ) × (2q -2r)! a!b! ε ⊗a (k-1)2 -n ⊗ε ⊗b (l-1)2 -n .
We make the decomposition

E V (q) n (f ) 2 = A n + B n + C n + D n , (3.44)
where

A n = 2 2Hqn q! 2 2 n k,l=1 E f (q) (B (k-1)2 -n ) f (q) (B (l-1)2 -n ) ε (k-1)2 -n , δ k2 -n q ε (l-1)2 -n , δ l2 -n q , B n = 2 2Hqn c+d+e+f =2q d+e≥1 2 n k,l=1 E f (q) (B (k-1)2 -n ) f (q) (B (l-1)2 -n ) α(c, d, e, f ) × ε (k-1)2 -n , δ k2 -n c H ε (k-1)2 -n , δ l2 -n d H ε (l-1)2 -n , δ k2 -n e H ε (l-1)2 -n , δ l2 -n f H , C n = 2 2Hqn a+b=2q (a,b) =(q,q) 2 n k,l=1 E f (a) (B (k-1)2 -n ) f (b) (B (l-1)2 -n ) (2q)! q! 2 a!b! × ε ⊗a (k-1)2 -n ⊗ε ⊗b (l-1)2 -n , δ ⊗q k2 -n ⊗δ ⊗q l2 -n H ⊗(2q) ,
and

D n = 2 2Hqn q r=1 a+b=2q-2r 2 n k,l=1 E f (a) (B (k-1)2 -n ) f (b) (B (l-1)2 -n ) (2q -2r)! r!(q -r)! 2 a!b! × ε ⊗a (k-1)2 -n ⊗ε ⊗b (l-1)2 -n , δ ⊗q-r k2 -n ⊗δ ⊗q-r l2 -n H ⊗(2q-2r) δ k2 -n , δ l2 -n r H ,
for some combinatorial constants α(c, d, e, f ). That is, A n and B n contain all the terms with r = 0 and (a, b) = (q, q); C n contains the terms with r = 0 and (a, b) = (q, q); and D n contains the remaining terms.

For any integer r ≥ 1, we set

α n = sup k,l=1,...,2 n ε (k-1)2 -n , δ l2 -n H , (3.45) β r,n = 2 n k,l=1 δ k2 -n , δ l2 -n H r , (3.46)
γ n = 2 n k,l=1 ε (k-1)2 -n , δ l2 -n H . (3.47)
Then, under assumption (H q ), we have the following estimates:

|A n | ≤ C2 2Hqn+2n (α n ) 2q , |B n | + |C n | ≤ C2 2Hqn (α n ) 2q-1 γ n , |D n | ≤ C2 2Hqn q r=1 (α n ) 2q-2r β r,n ,
where C is a constant depending only on q and the function f . Notice that the second inequality follows from the fact that when (a, b) = (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 form ε (k-1)2 -n , δ l2 -n H in the expression of B n or C n .

In the case H < 1 2 , we have by (2.34) that α n ≤ 2 -2nH , by (2.36) that β r,n ≤ C2 n-2rHn , and by (2.35) that γ n ≤ C2 n . 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 q r=1
2 n(-2(q-r)H+1) , (3.50) which implies the estimates (3.41) and (3.42).

In the case 1 2 ≤ H < 1 -1 2q , we have by (2.38) that α n ≤ C2 -n , by (2.36) that β r,n ≤ C2 n-2rHn , and by (2.35) that γ n ≤ C2 n . As a consequence, we obtain

|A n | + |B n | + |C n | ≤ C2 n(2q(H-1)+2) , |D n | ≤ C q r=1
2 n((2q-2r)(H-1)+1) , which also implies (3.42).

Finally, if H = 1 -1 2q , we have by (2.38) that α n ≤ C2 -n , by (2.37) that β r,n ≤ Cn2 2n-2rn , and by (2.35) that γ n ≤ C2 n . As a consequence, we obtain

|A n | + |B n | + |C n | ≤ C2 n , |D n | ≤ C q r=1 n2 n r q ,
which implies (3.43).

Proof of Theorem 1 in the case

0 < H < 1 2q
In this subsection we are going to prove the first point of Theorem 1. The proof will be done in three steps. Set

V (q) 1,n (f ) = 2 n(qH-1) V (q)
n (f ). We first study the asymptotic behavior of E V (q) 1,n (f ) 2 , using Proposition 8.

Step 1. The decomposition (3.44) leads to

E V (q) 1,n (f ) 2 = 2 2n(qH-1) (A n + B n + C n + D n ) .
From the estimate (3.49) we obtain 2 2n(qH-1) 1) , which converges to zero as n goes to infinity since H < 1 2q < 1 2 . On the other hand (3.50) yields

(|B n | + |C n |) ≤ C2 n(2H-
2 2n(qH-1) |D n | ≤ C q r=1
2 n(2rH-1) , which tends to zero as n goes to infinity since 2rH -1 ≤ 2qH -1 < 0 for all r = 1, . . . , q.

In order to handle the term A n , we make use of the following estimate, which follows from (2.34) and (2.33):

ε (k-1)2 -n , δ k2 -n q H -- 2 -2Hn 2 q = ε (k-1)2 -n , δ k2 -n H + 2 -2Hn 2 q-1 s=0 ε (k-1)2 -n , δ k2 -n s H - 2 -2Hn 2 q-1-s ≤ C k 2H -(k -1) 2H 2 -2Hqn . (3.51) Thus, 2 4Hqn-2n q! 2 2 n k,l=1 E f (q) (B (k-1)2 -n ) f (q) (B (l-1)2 -n ) ε (k-1)2 -n , δ k2 -n q H ε (l-1)2 -n , δ l2 -n q H - 2 -2n-2q q! 2 2 n k,l=1 E f (q) (B (k-1)2 -n ) f (q) (B (l-1)2 -n ) ≤ C2 2Hn-n ,
which implies, as n → ∞:

E V (q) 1,n (f ) 2 = 2 -2n-2q q! 2 2 n k,l=1 E f (q) (B (k-1)2 -n ) f (q) (B (l-1)2 -n ) + o(1). (3.52)
Step 2: We need the asymptotic behavior of the double product

J n := E V (q) 1,n (f ) × 2 -n 2 n l=1 f (q) (B (l-1)2 -n ) .
Using the same arguments as in Step 1 we obtain

J n = 2 Hqn-2n 2 n k,l=1 E f (B (k-1)2 -n ) f (q) (B (l-1)2 -n ) H q 2 nH ∆B k2 -n = 1 q! 2 2Hqn-2n 2 n k,l=1 E f (B (k-1)2 -n ) f (q) (B (l-1)2 -n ) I q δ ⊗q k2 -n = 1 q! 2 2Hqn-2n 2 n k,l=1 E D q f (B (k-1)2 -n ) f (q) (B (l-1)2 -n ) , δ ⊗q k2 -n H ⊗q = 2 2Hqn-2n 2 n k,l=1 q a=0 1 a!(q -a)! E f (a) (B (k-1)2 -n ) f (2q-a) (B (l-1)2 -n ) × ε (k-1)2 -n , δ k2 -n a H ε (l-1)2 -n , δ k2 -n q-a H .
It turns out that only the term with a = q will contribute to the limit as n tends to infinity. For this reason we make the decomposition

J n = 2 2Hqn-2n 2 n k,l=1 1 q! E f (q) (B (k-1)2 -n ) f (q) (B (l-1)2 -n ) ε (k-1)2 -n , δ k2 -n q H + S n ,
where

S n = 2 2Hqn-2n 2 n k,l=1 ε (l-1)2 -n , δ k2 -n H q-1 a=0 1 a!(q -a)! E f (a) (B (k-1)2 -n ) f (2q-a) (B (l-1)2 -n ) × ε (k-1)2 -n , δ k2 -n a H ε (l-1)2 -n , δ k2 -n q-a-1 H .
By (2.34) and (2.35), we have

|S n | ≤ C2 2Hn-n ,
which tends to zero as n goes to infinity. Moreover, by (3.51), we have

2 2Hqn-2n q! 2 n k,l=1 E f (q) (B (k-1)2 -n ) f (q) (B (l-1)2 -n ) ε (k-1)2 -n , δ k2 -n q H -(-1) q 2 -2n-q q! 2 n k,l=1 E f (q) (B (k-1)2 -n ) f (q) (B (l-1)2 -n ) ≤ C 2 2Hn-n ,
which also tends to zero as n goes to infinity. Thus, finally, as n → ∞:

J n = (-1) q 2 -2n-q q! 2 n k,l=1 E f (q) (B (k-1)2 -n ) f (q) (B (l-1)2 -n ) + o(1). (3.53)
Step 3: By combining (3.52) and (3.53), we obtain that

E V (q) 1,n (f ) - (-1) q 2 q q! 2 -n 2 n k=1 f (q) (B (k-1)2 -n ) 2 = o(1),
as n → ∞. Thus, the proof of the first point of Theorem 1 is done using a Riemann sum argument.

3.2 Proof of Theorem 1 in the case H > 1 -1 2q : the weighted non-central limit theorem

We prove here that the sequence V 3,n (f ), given by

V (q) 3,n (f ) = 2 n(1-H)q-n V (q) n (f ) = 2 qn-n 1 q! 2 n k=1 f B (k-1)2 -n I q δ ⊗q k2 -n , converges in L 2 as n → ∞ to the pathwise integral 1 0 f (B s )dZ (q)
s with respect to the Hermite process of order q introduced in Definition 7.

Observe first that, by construction of Z (q) (precisely, see the discussion before Definition 7 in Section 2), the desired result is in order when the function f is identically one. More precisely:

Lemma 9 For each fixed t ∈ [0, 1], the sequence 2 qn-n 1 q! [2 n t] k=1 I q δ ⊗q k2 -n converges in L 2 to the Hermite random variable Z (q) t .
Now, consider the case of a general function f . We fix two integers m ≥ n, and decompose the sequence V (q) 3,m (f ) as follows:

V (q) 3,m (f ) = A (m,n) + B (m,n) ,
where

A (m,n) = 1 q! 2 m(q-1) 2 n j=1 f B (j-1)2 -n j2 m-n i=(j-1)2 m-n +1 I q δ ⊗q i2 -m , and
B (m,n) = 1 q! 2 m(q-1) 2 n j=1 j2 m-n i=(j-1)2 m-n +1 ∆ m,n i,j f (B) I q δ ⊗q i2 -m , with the notation ∆ m,n i,j f (B) = f (B (i-1)2 -m ) -f (B (j-1)2 -n ).
We shall study A (m,n) and B (m,n) separately.

Study of A (m,n) . When n is fixed, Lemma 9 yields that the random vector n) , where

  1 q! 2 m(q-1) j2 m-n i=(j-1)2 m-n +1 I q δ ⊗q i2 -m ; j = 1, . . . , 2 n   converges in L 2 , as m → ∞, to the vector Z (q) j2 -n -Z (q) (j-1)2 -n ; j = 1, . . . , 2 n . Then, as m → ∞, A (m,n) L 2 → A (∞,
A (∞,n) := 2 n j=1 f (B (j-1)2 -n ) Z (q) j2 -n -Z (q) (j-1)2 -n .
Finally, we claim that when n tends to infinity,

A (∞,n) converges in L 2 to 1 0 f (B s ) dZ (q)
s . Indeed, observe that the stochastic integral

1 0 f (B s ) dZ (q) s
is a pathwise Young integral. So, to get the convergence in L 2 it suffices to show that the sequence A (∞,n) is bounded in L p for some p ≥ 2. The integral

1 0 f (B s ) dZ (q)
s has moments of all orders, because for all p ≥ 2

E   sup 0≤s<t≤1   Z (q) t -Z (q) s |t -s| γ   p   < ∞ and E sup 0≤s<t≤1 |B t -B s | |t -s| β p < ∞,
if γ < q(H -1) + 1 and β < H. On the other hand, Young's inequality implies

A (∞,n) - 1 0 f (B s ) dZ (q) s ≤ c ρ,ν Var ρ f (B) Var ν Z (q) ,
where Var ρ denotes the variation of order ρ, and with ρ, ν > 1 such that

1 ρ + 1 ν > 1. Choosing ρ > 1
H and ν > 1 q(H-1)+1 , the result follows. This proves that, by letting m and then n go to infinity, A (m,n) converges in L 2 to

1 0 f (B s ) dZ (q)
s .

Study of the term

B (m,n) : We prove that lim n→∞ sup m E B (m,n) 2 = 0. (3.54)
We have, using the product formula (2.32) for multiple stochastic integrals,

E B (m,n) 2 = 2 2m(q-1) 2 n j=1 j2 m-n i=(j-1)2 m-n +1 2 n j ′ =1 j ′ 2 m-n i ′ =(j ′ -1)2 m-n +1 q l=0 l! q! 2 q l 2 ×b (m,n) l δ i2 -m , δ i ′ 2 -m l H , (3.55)
where b is equal to

(m,n) l = E ∆ m,n i,j f (B)∆ m,n i ′ ,j ′ f (B)I 2(q-l) δ ⊗(q-l) i2 -m ⊗δ ⊗(q-l) i ′ 2 -m . ( 3
E D 2(q-l) ∆ m,n i,j f (B)∆ m,n i ′ ,j ′ f (B) , δ ⊗(q-l) i2 -m ⊗δ ⊗(q-l) i ′ 2 -m H ⊗2(q-l) = 2q-2l a=0 2q -2l a E f (a) (B (i-1)2 -m )ε ⊗a (i-1)2 -m -f (a) (B (j-1)2 -n )ε ⊗a (j-1)2 -n ⊗ f (2q-2l-a) (B (i ′ -1)2 -m )ε ⊗b (i ′ -1)2 -m -f (2q-2l-a) (B (j ′ -1)2 -n )ε ⊗b (j ′ -1)2 -m , δ ⊗(q-l) i2 -m ⊗δ ⊗(q-l) i ′ 2 -m H ⊗2(q-l) .
The term in (3.55) corresponding to l = q can be estimated by

1 q! 2 2m(q-1) sup |x-y|≤2 -n E |f (B x ) -f (B y )| 2 β q,m ,
where β q,m has been introduced in (3.46). So it converges to zero as n tends to infinity, uniformly in m, because, by (2.40) and using that H > 1 -1 2q , we have sup m 2 2m(q-1) β q,m < ∞.

20

In order to handle the terms with 0 ≤ l ≤ q -1, we make the decomposition b

(m,n) l ≤ 2q-2l a=0 2q -2l a 4 h=1 B h , (3.57)
where

B 1 = E ∆ m,n i,j f (B)∆ m,n i ′ ,j ′ f (B) ε ⊗a (i-1)2 -m ⊗ε ⊗(2q-2l-a) (i ′ -1)2 -m , δ ⊗(q-l) i2 -m ⊗δ ⊗(q-l) i ′ 2 -m H ⊗2(q-l) , B 2 = E f (a) (B (j-1)2 -n )∆ m,n i ′ ,j ′ f (B) × ε ⊗a (i-1)2 -m -ε ⊗a (j-1)2 -n ⊗ε ⊗(2q-2l-a) (i ′ -1)2 -m , δ ⊗(q-l) i2 -m ⊗δ ⊗(q-l) i ′ 2 -m H ⊗2(q-l) , B 3 = E ∆ m,n i,j f (B)f (2q-2l-a) (B (j ′ -1)2 -n ) × ε ⊗a (i-1)2 -m ⊗ ε ⊗(2q-2l-a) (i ′ -1)2 -m -ε ⊗(2q-2l-a) (j ′ -1)2 -n , δ ⊗(q-l) i2 -m ⊗δ ⊗(q-l) i ′ 2 -m H ⊗2(q-l) , B 4 = E f (a) (B (j-1)2 -n )f (2q-2l-a) (B (j ′ -1)2 -n ) × ε ⊗a (i-1)2 -m -ε ⊗a (j-1)2 -n ⊗ ε ⊗(2q-2l-a) (i ′ -1)2 -m -ε ⊗(2q-2l-a) (j ′ -1)2 -n , δ ⊗(q-l) i2 -m ⊗δ ⊗(q-l) i ′ 2 -m H ⊗2(q-l) . (3.58)
By using (2.38) and the conditions imposed on the function f , one can bound the terms B 1 , B 2 and B 3 as follows:

|B 1 | ≤ c(q, f, H) sup |x-y|≤ 1 2 n ,0≤a≤2q E f (a) (B x ) -f (a) (B y ) 2 2 -2m(q-l) , |B 2 | + |B 3 | ≤ c(q, f, H) sup |x-y|≤ 1 2 n ,0≤a≤2q E f (2q-2l-a) (B x ) -f (2q-2l-a) (B y ) 2 -2m(q-l) ,
and, by using (2.39), we obtain that

|B 4 | ≤ c(q, f, H)2 -n q-1 q -2m(q-l) .
By setting

R n = 1 q! sup |x-y|≤2 -n E |f (B x ) -f (B y )| 2 sup m 2 2m(q-1) β q,m ,
we can finally write, by the estimate (2.40),

E B (m,n) 2 ≤ R n + c(H, f, q)2 2m(q-1) sup |x-y|≤ 1 2 n ,0≤a≤2q f (2q-2l-a) (B x ) -f (2q-2l-a) (B y ) + (2 -n ) q-1 q × 2 n j=1 j2 m-n i=(j-1)2 m-n +1 2 n j ′ =1 j ′ 2 m-n i ′ =(j ′ -1)2 m-n +1 q-1 l=0 2 -2m(q-l) δ i2 -m , δ i ′ 2 -m l H ≤ R n + c(H, f, q)2 2m(q-1) sup |x-y|≤ 1 2 n ,0≤a≤2q f (2q-2l-a) (B x ) -f (2q-2l-a) (B y ) + (2 -n ) q-1 q × q-1 l=0 2 -2m(q-l) 2 m i,j=0 δ i2 -m , δ i ′ 2 -m l H ≤ R n + c(H, f, q) sup |x-y|≤ 1 2 n ,0≤a≤2q f (2q-2l-a) (B x ) -f (2q-2l-a) (B y ) + (2 -n ) q-1 q
and this converges to zero due to the continuity of B and since q > 1.

3.3 Proof of Theorem 1 in the case 1 2q < H ≤ 1 -1 2q : the weighted central limit theorem Suppose first that 1 2q < H < 1 -1 2q . We study the convergence in law of the sequence

V (q) 2,n (f ) = 2 -n 2 V (q)
n (f ). We fix two integers m ≥ n, and decompose this sequence as follows: n) , where

V (q) 2,m (f ) = A (m,n) + B (m,
A (m,n) = 2 -m 2 2 n j=1 f B (j-1)2 -n j2 m-n i=(j-1)2 m-n +1 H q 2 mH ∆B i2 -m , and
B (m,n) = 1 q! 2 m(Hq-1 2 ) 2 n j=1 j2 m-n i=(j-1)2 m-n +1 ∆ m,n i,j f (B)I q δ ⊗q i2 -m ,
and where as before we make use of the notation ∆

m,n i,j f (B) = f (B (i-1)2 -m ) -f (B (j-1)2 -n ).
Let us first consider the term A (m,n) . From Theorem 1 in Breuer and Major [START_REF] Breuer | Central limit theorems for nonlinear functionals of Gaussian fields[END_REF], and taking into account that H < 1 -1 2q , it follows that the random vector

  B, 2 -m 2 j2 m-n i=(j-1)2 m-n +1 H q (2 mH ∆B i2 -m ); j = 1, . . . , 2 n   converges in law, as m → ∞, to B, σ H,q ∆W j2 -n ; j = 1, . . . , 2 n
where σ H,q is the constant defined by (1.16) and W is a standard Brownian motion independent of B (the independence is a consequence of the central limit theorem for multiple stochastic integrals proved in Peccati and Tudor [START_REF] Peccati | Gaussian limits for vector-valued multiple stochastic integrals[END_REF]). Since

2 n j=1 f B (j-1)2 -n ∆W j2 -n converges in L 2 as n → ∞ to the Itô integral 1 0 f (B s )dW s we conclude that, by letting m → ∞ and then n → ∞, we have B, A (m,n) Law -→ B, σ H,q 1 0 f (B s )dW s .
Then it suffices to show that lim

n→∞ sup m→∞ E B (m,n) 2 = 0. (3.59)
We have, as in (3.55),

E B (m,n) 2 = 2 m(2Hq-1) 2 n j=1 j2 m-n i=(j-1)2 m-n +1 2 n j ′ =1 j ′ 2 m-n i ′ =(j ′ -1)2 m-n +1 q l=0 l! q! 2 q l 2 ×b (m,n) l δ i2 -m , δ i ′ 2 -m l H , (3.60)
where b (m,n) l has been defined in (3.56). The term in (3.60) corresponding to l = q can be estimated by

1 q! 2 m(2Hq-1) sup |x-y|≤2 -n E |f (B x ) -f (B y )| 2 β q,m ,
which converges to zero as n tends to infinity, uniformly in m, because by (2.36) and using that H < 1 -1 2q , we have sup m 2 m(2Hq-1) β q,m < ∞.

In order to handle the terms with 0 ≤ l ≤ q -1, we will distinguish two different cases, depending on the value of H.

Case H < 1/2. Suppose 0 ≤ l ≤ q -1. By (2.35), we can majorize b (m,n) l as follows:

|b (m,n) l | ≤ C2 -4Hm(q-l) .
As a consequence, applying again (2.36), the corresponding term in (3.60) is bounded by C2 m(2Hq-1) 2 -4Hm(q-l) β l,m ≤ C2 2mH(l-q) , which converges to zero as m tends to infinity because l < q.

Case H > 1/2. Suppose 0 ≤ l ≤ q -1. By (2.38), we get the estimate

|b (m,n) l | ≤ C2 -2m(q-l) .
As a consequence, applying again (2.36), the corresponding term in (3.60) is bounded by C2 m(2Hq-1) 2 -2m(q-l) β l,m .

If H < 1 -1 2l , applying (2.36), this is bounded by C2 m(2H(q-l)-2(q-l)) , which converges to zero as m tends to infinity because H < 1 and l < q. In the case H = 1 -1 2l , applying (2.37), we get the estimate Cm2 m(2H(q-l)-2(q-l)) , which converges to zero as m tends to infinity because H < 1 and l < q. In the case H > 1 -1 2l , we apply (2.38) and we get the estimate C2 m(2H2+1-2q) , which converges to zero as m tends to infinity because H < 1 -1 2q .

The proof in the case H = 1 -1 2q is similar. The convergence of the term A (m,n) is obtained by applying Theorem 1' in [START_REF] Breuer | Central limit theorems for nonlinear functionals of Gaussian fields[END_REF], and the convergence to zero in L 2 of the term B (m,n) follows the same lines as before.

Proof of Proposition 2

We proceed as in Section 3.3. For p = 2, . . . , q, we set

V (p) 2,n (f ) = 2 -n 2 V (p)
n (f ). We fix two integers m ≥ n, and decompose this sequence as follows: where (W (2) , . . . , W (q) ) is a (q -1)-dimensional standard Brownian motion independent of B and the σ H,p 's are given by (1.16). Indeed, the convergence in law of each component follows from Theorem 1 in Breuer and Major [START_REF] Breuer | Central limit theorems for nonlinear functionals of Gaussian fields[END_REF], taking into account that H < 3 4 ≤ 1 -1 2q . The joint convergence and the fact that the processes W (p) for p = 2, . . . , q are independent (and also independent of B) is a direct application of the central limit theorem for multiple stochastic integrals proved in Peccati and Tudor [START_REF] Peccati | Gaussian limits for vector-valued multiple stochastic integrals[END_REF].

Since, for any p = 2, . . . , q, the quantity for all p = 2, . . . , q. This finishes the proof of Proposition 2.

Proof of Corollary 3

For any integer q ≥ 2, we have 2 nH ∆B k2 -n qµ q = q p=1 q p µ q-p 2 Hnp I p (δ ⊗p k2 -n ) = q p=1 p! q p µ q-p H p 2 nH ∆B k2 -n .

Indeed, the pth kernel in the chaos representation of 2 nH ∆B k2 -n q is 1 p! E(D p 2 nH ∆B k2 -n q ) = q p 2 nHp µ q-p δ ⊗p k2 -n .

Suppose first that q is odd and H > 1 2 . In this case, we have

2 -nH 2 n k=1 f (B (k-1)2 -n )(2 nH ∆B k2 -n ) q = q p=1 p! q p µ q-p 2 -nH V (p) n (f ).
The term with p = 1 converges in L 2 to qµ q-1 1 0 f (B s )dB s . For p ≥ 2, the limit in L 2 is zero. Indeed, if H ≤ 1 -1 2p , then E V (p) n (f ) 2 is bounded by a constant times n2 n by Proposition zero, because H > 1 6 . In the same way, we have 2 -2Hn V (1) n (f (3) ) = -

1 2 2 -2Hn 2 n k=1 f (4) (B (k-1)2 -n ) (∆B k2 -n ) 2 - 1 6 2 -2Hn
2 n k=1 f (5) (B (k-1)2 -n ) (∆B k2 -n ) 3 + o(1).

We have obtained f (B 1 ) = f (0) + 1 2

2 n k=1 f ′ (B k2 -n ) + f ′ (B (k-1)2 -n ) ∆B k2 -n + 1 4 × 2 -4Hn 2 n k=1 f (4) (B (k-1)2 -n )H 2 2 nH ∆B k2 -n - 1 24 × 2 -2Hn 2 n k=1 f (5) (B (k-1)2 -n ) (∆B k2 -n ) 3 + o(1).
As before 2 -4Hn V

n (f (4) ) converges to zero in L 2 . Finally, by (1.11),

-2Hn

2 n k=1 f (5) (B (k-1)2 -n ) (∆B k2 -n ) 3 also converges to zero. This completes the proof.

VH p 2

 2 i=(j-1)2 m-n +1 H p 2 mH ∆B i2 -m , andB (m,n) i=(j-1)2 m-n +1 ∆ m,n i,j f (B)I p δ ⊗p i2 -m ,and where as before we make use of the notation ∆ m,n i,jf (B) = f (B (i-1)2 -m )f (B (j-1)2 -n ). Let us first consider the term A (m,n) p. We claim that the random vector mH ∆B i2 -m ; j = 1, . . . , 2 n , as m → ∞, to B, {σ H,p ∆W (p)j2 -n ; j = 1, . . . , 2 n } 2≤p≤q

1 0 1 0f 1 0f.

 111 converges in L 2 as n → ∞ to the Itô integral f (B s )dW (p)s , we conclude that, by letting m → ∞ and then n → ∞, we haveB, A (m,n) 2 , . . . , A (m,n) q Law -→ B, σ H,2 (B s )dW(2) s , . . . , σ H,q (B s )dW(q) s On the other hand, and becauseH ∈ ( 1 4 , 3 4 ) (implying that H ∈ ( 1 2p , 1 -12p)), we have shown in Section

Acknowledgments. We are grateful to Jean-Christophe Breton and Nabil Kazi-Tani for helpful remarks. We also wish to thank the anonymous referee for his/her very careful reading.

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

If

n (f ) 2 is bounded by a constant times 2 -n2(1-H)p+2n by (1. [START_REF] Nualart | Stochastic calculus with respect to the fractional Brownian motion and applications[END_REF]), with -2(1 -H)p + 2 -2H = (1 -H)(2 -2p) < 0.

Suppose now that q is even. Then

p! q p µ q-p V (p) n (f ) .

If H < 1 4 , by (1.14), one has that 2 2nH-n ×2 q 2 µ q-2 V

(2)

) by (3.42) with 4H -1 < 0. Therefore (1.23) holds. In the case 1 4 < H < 3 4 , Proposition 2 implies that the vector

where (W (2) , . . . , W (q) ) is a (q -1)-dimensional standard Brownian motion independent of B and the σ H,p 's, 2 ≤ p ≤ q, are given by (1.16). This implies the convergence (1.25). The proof of (1.27) is analogous (with an adequate version of Proposition 2). The convergence (1.24) is obtained by similar arguments using the limit result (1.20) in the critical case H = 1 4 , p = 2. Finally, consider the case H > 3 4 . For p = 2, 2 n-2Hn V

(2)

s by (1.19).

Proof of Theorem 4

We can assume H < 1 2 , the case where H ≥ 1 2 being straightforward. By a Taylor's formula, we have

with R n converging towards 0 in probability as n → ∞, because H > 1/6. We can expand the monomials x m , m = 2, 3, 4, 5, in terms of the Hermite polynomials:

x 4 = 24 H 4 (x) + 12 H 2 (x) + 3

In this way we obtain

n (f (3) ), (3.62)

By (3.42) and using that H > 1 6 , we have E V

n (f (5) ) 2 ≤ C2 n . As a consequence, the first summand in (3.62) and the second one in (3.64) converge to zero in L 2 as n tends to infinity. Also, by (3.42), E V (4) n (f (4) ) 2 ≤ C2 n and E V [START_REF] Dobrushin | Non-central limit theorems for nonlinear functionals of Gaussian fields[END_REF] n (f (5) ) 2 ≤ C2 n . Hence, the first summand in (3.63) and the first summand in (3.64) converge to zero in L 2 as n tends to infinity. If

n (f (4) ) converges to zero in L 2 as n tends to infinity. Moreover, using the following identity, valid for regular functions h : R → R:

for some θ k2 -n lying between (k -1)2 -n and k2 -n , we deduce that 2 -4Hn V

n (f (5) ) tends to