Asymptotic behavior of weighted quadratic variation of bi-fractional Brownian motion

We prove, by means of Malliavin calculus, the convergence in L2 of some properly renormalized weighted quadratic variations of bi-fractional Brownian motion (biFBM) with parameters H and K, when H < 1/4 and K ∈ (0, 1]. Comportement asymptotique de la variation quadratique à poids du mouvement brownien bifractionnaire Résumé Nous utilisons le calcul de Malliavin pour montrer la convergence dans L2 de la variation quadratique à poids du mouvement brownien bifractionnaire (biFBM) d’indices H et K lorsque H < 1/4 et K ∈ (0, 1].


Introduction
There has been recently a lot of interests in the literature to the study of weighted power variations. More precisely, for a given integer p > 1, a smooth enough function h : R → R and a process X, the analysis of the asymptotic behavior, as n tends to infinity, of quantities such as n−1 l=0 h(X l/n )(∆X l/n ) p (1.1) This study origins in the work [5] by Nourdin, in the case where X is a fractional Brownian motion (f.B.m, in short). Then, the results of [5] have been improved in [6] by Nourdin, Nualart and Tudor. Let us also stress that the study in [6,5] has been used in [2,4] to deduce exact rate of convergence of some approximation schemes of scalar stochastic differential equations driven by a f.B.m. Moreover, for another motivation of this study, we can also mention that the analysis of the asymptotic behavior of (1.1), in the particular case p = 2 and X the fractional Brownian motion of Hurst parameter 1/4, allowed the authors of [7] to derive a new type of change of variable formula for X, with a correction term that is an ordinary Itô integral with respect to a Wiener process that is independent of X.
As we said, a complete description of the nature of the convergence of weighted p-power variation of the form (1.1) in the case where X is the fractional Brownian motion with Hurst parameter H ∈ (0, 1) has been given in [6,5,7]. More precisely, after adequate renormalization, central and non-central limit theorems have been derived there, depending on the value of p and H. In particular, it is shown in [5] that, for weighted quadratic variations (p = 2), the following convergence holds for h regular enough and H strictly between 0 and 1/4: As pointed out by Nourdin in [5], (1.2) is somewhat surprising when it is compared to the situation where h ≡ 1. Indeed, since the seminal work of Breuer and Major [1], we know that, for any 0 < H < 3/4: where 2) still holds in the case of a more general process, namely the bi-fractional Brownian motion (see below for a precise definition). As in [5], our main tool for the proof is based on the integration by parts formula of Malliavin calculus.
The note is organized as follows. In Section 2 we recall the definition of the bi-f.B.m and present some preliminary results about its Malliavin calculus. In Section 3 we state and prove our result concerning the convergence similar to (1.2), but in the case where X is a bi-f.B.m.

Preliminaries and notation
Here we recall the definition of the bi-fractional Brownian motion and present the elements of Malliavin calculus that will be needed in the sequel. Definition 2.1. Let H ∈ (0, 1) and K ∈ (0, 1]. A bi-fractional Brownian motion (B H,K t ) t≥0 of indices H and K is a centered Gaussian process, starting from zero, with covariance function given by In particular, by choosing K = 1 and H ∈ (0, 1) in (2.1), observe that we recover the covariance function of the fractional Brownian motion with Hurst parameter H.
The bi-fractional Brownian motion was introduced by Houdré and Villa in [3], and then further studied by Russo and Tudor in [9], and by Tudor and Xiao in [11]. It enjoys the self-similarity property, that is, for any constant c > 0, the processes {c −HK B H,K ct , t ≥ 0} and {B H,K t , t ≥ 0} have the same distribution. Moreover, if K = 1, B H,K does not have stationary increments (see e.g. [10]). It is precisely the main difference with respect to f.B.m.
Let us introduce some basic facts on the Malliavin calculus with respect to B H,K on the time interval [0, 1]. For a more complete exposition, we refer to [8]. Let H be the Hilbert space defined as the closure of the linear space E generated by the indicator functions (1 [0,t] , t ∈ [0, 1]) with respect to the following inner product can be extended to an isometry between H and the Gaussian space generated by B H,K . We denote this isometry by Let S be the set of all smooth cylindrical random variables of the form where n ≥ 0, f ∈ C ∞ has a compact support and ϕ i ∈ H. The Malliavin derivative of F with respect to B H,K is the element belonging to L 2 (Ω, H) defined by This operator can be extended to the closure D 1,2 of S with respect to the norm The Malliavin derivative satisfies the following chain rule. For every random vector F = (F 1 , . . . , F n ) with components in D 1,2 and for every continuously differentiable function ϕ : R n → R with bounded partial derivatives, we obtain ϕ(F 1 , ..., F 1 ) ∈ D 1,2 and we have, for any s ∈ [0, 1]: The divergence operator I is the adjoint of D in the following sense. A random process u ∈ L 2 (Ω, H) belongs to the domain of I if and only if where C u is a constant depending only on u. In that case, I(u) verifies the integration by part formula:

Asymptotic behavior of weighted quadratic variations of bifractional Brownian motion.
We will make use of the following assumption on the weight function h. Assumption (H m ): h : R → R belongs to C m and, for any p > 0 and any i = 1, . . . , m, The main result of this section is the following: Remark 3.2. When K = 1 (that is when B H,K is a fractional Brownian motion) we recover Theorem 1.1 in [5]. Our proof in the general case follows the same lines.
Proof of the theorem. Throughout the proof, we will denote for simplicity δ k/n = 1 [k/n,(k+1)/n] and ε k/n = 1 [0,k/n] and we let C stand for a positive generic constant independent of k, l, n that can be different from line to line.
We will need several lemmas. The first one is immediate to check, so its proof is left to the reader.

Lemma 3.3.
(1) If 2HK < 1, then the sequence ϕ defined by as l goes to infinity. In particular, ϕ is bounded.

Lemma 3.4.
(1) Assume that 2HK < 1. Then, as n → ∞, Then, as n → ∞, and therefore Concerning the second point, we use the elementary inequality ||x| K − |y| K | ≤ |x − y| K , valid for any x,y ∈ R because K ≤ 1, to see that Consequently, since (l + 1) 2H − l 2H K behaves as l 2HK−K for large l, we get Then, the series ∞ l=1 1/l K−2HK+γ is convergent and For the third point, we have with D k,l defined by (3.5). Then, we obtain as previously Thus, using (3.6) of Lemma 3.4 and the fact that H < 1/4, equality (3.7) follows since n = o(n 2−4HK ), which completes the proof.
Proof of Lemma 3.5. For k, l = 0, 1, . . . , n − 1, we use the integration by parts formula to write But, with φ defined as in (3.3). Thus, with ϕ defined by (3.2). Therefore, using Lemma 3.3, we get Since 2HK < 1, we can choose β > 0 such that 2HK < β < 1 and we set γ = 1 − β. Then l≥1 ϕ(l)/l β < ∞ and consequently, This implies that, under condition (H 2 ) Furthermore, using the fact that 2HK ≤ 2H ≤ 1, we see that is bounded independently of k and l. Now, since by telescoping sum, we deduce that Thus, under condition (H 2 ), we obtain . On the other hand, by Lemma 3.3 and once again using condition (H 2 ) Finally, by combining all the previous estimates with (3.10), the proof of Lemma 3.5 is done.
Proof of Lemma 3.6. Using the integration by part formula we have It follows from (3.9), that so that (recall that H < 1/4 < 1/2) For the last term n k,l=0 |(d) k,l,n |, we have This finishes the proof of Claim 3.1, and thus the proof of Lemma 3.6.
Combining these two lemmas, the proof of the theorem can be completed along the same lines as in [5]. Indeed, by Lemma  As a consequence, we obtain the convergence