Hsu-Robbins and Spitzer’s theorems for the variations of fractional Brownian motion

Using recent results on the behavior of multiple Wiener-Itˆo integrals based on Stein’s method, we prove Hsu-Robbins and Spitzer’s theorems for sequences of correlated random variables related to the increments of the fractional Brownian motion.


Introduction
A famous result by Hsu and Robbins [7] says that if X 1 , X 2 , . . . is a sequence of independent identically distributed random variables with zero mean and finite variance and S n := X 1 + . . .+ X n , then n≥1 P (|S n | > εn) < ∞ for every ε > 0. Later, Erdös ([3], [4]) showed that the converse implication also holds, namely if the above series is finite for every ε > 0 and X 1 , X 2 , . . .are independent and identically distributed, then EX 1 = 0 and EX 2  1 < ∞.Since then, many authors extended this result in several directions.
Spitzer's showed in [13] that for every ε > 0 if and only if EX 1 = 0 and E|X 1 | < ∞.Also, Spitzer's theorem has been the object of various generalizations and variants.One of the problems related to the Hsu-Robbins' and Spitzer's theorems is to find the precise asymptotic as ε → 0 of the quantities n≥1 P (|S n | > εn) and n≥1 1 n P (|S n | > εn).Heyde [5] showed that lim whenever EX 1 = 0 and EX 2 1 < ∞.In the case when X is attracted to a stable distribution of exponent α > 1, Spataru [12] proved that lim ε→0 1 − log ε n≥1 The purpose of this paper is to prove Hsu-Robbins and Spitzer's theorems for sequences of correlated random variables, related to the increments of fractional Brownian motion, in the spirit of [5] or [12].Recall that the fractional Brownian motion ( ).It can be also defined as the unique self-similar Gaussian process with stationary increments.Concretely, in this paper we will study the behavior of the tail probabilities of the sequence where B is a fractional Brownian motion with Hurst parameter H ∈ (0, 1) (in the sequel we will omit the superscript H for B) and H q is the Hermite polynomial of degree q ≥ 1 given by H q (x) = (−1) q e x 2 2 d q dx q (e − x 2 2 ).The sequence V n behaves as follows (see e.g.[9], Theorem 1; the result is also recalled in Section 3 of our paper): if 0 < H < 1 − 1 2q , a central limit theorem holds for the renormalized sequence Z (1) (Ω) to a Hermite random variable of order q (see Section 2 for the definition of the Hermite random variable and Section 3 for a rigorous statement concerning the convergence of V n ).Here c 1,q,H , c 2,q,H are explicit positive constants depending on q and H.
We note that the techniques generally used in the literature to prove the Hsu-Robbins and Spitzer's results are strongly related to the independence of the random variables X 1 , X 2 , . . . .In our case the variables are correlated.Indeed, for any k, l ≥ 1 we have which is not equal to zero unless H = 1 2 (which is the case of the standard Brownian motion).We use new techniques based on the estimates for the multiple Wiener-Itô integrals obtained in [2], [10] via Stein's method and Malliavin calculus.Concretely, we study in this paper the behavior as ε → 0 of the quantities and if 0 < H < 1 − 1 2q and of and The basic idea in our proofs is that, if we replace Z n and Z n by their limits (standard normal random variable or Hermite random variable) in the above expressions, the behavior as ε → 0 can be obtained by standard calculations.Then we need to estimate the difference between the tail probabilities of n and the tail probabilities of their limits.To this end, we will use the estimates obtained in [2], [10] via Malliavin calculus and we are able to prove that this difference converges to zero in all cases.We obtain that, as ε → 0, the quantities ( 4) and ( 6) are of order of log ε while the functions ( 5) and ( 7) are of order of ε 2 and ε 1−q(1−H) respectively.
The paper is organized as follows.Section 2 contains some preliminaries on the stochastic analysis on Wiener chaos.In Section 3 we prove the Spitzer's theorem for the variations of the fractional Brownian motion while Section 4 is devoted to the Hsu-Robbins theorem for this sequence.
Throughout the paper we will denote by c a generic strictly positive constant which may vary from line to line (and even on the same line).

Preliminaries
Let (W t ) t∈[0,1] be a classical Wiener process on a standard Wiener space (Ω, F, P).If f ∈ L 2 ([0, 1] n ) with n ≥ 1 integer, we introduce the multiple Wiener-Itô integral of f with respect to W .The basic reference is [11].
Let f ∈ S m be an elementary function with m variables that can be written as where the coefficients satisfy c i 1 ,...im = 0 if two indices i k and i l are equal and the sets A i ∈ B([0, 1]) are disjoint.For such a step function f we define where we put ).It can be seen that the mapping I n constructed above from S m to L 2 (Ω) is an isometry on S m , i.e. and Since the set S n is dense in L 2 ([0, 1] n ) for every n ≥ 1 the mapping I n can be extended to an isometry from L 2 ([0, 1] n ) to L 2 (Ω) and the above properties hold true for this extension.
We will need the following bound for the tail probabilities of multiple Wiener-Itô integrals (see [8], Theorem 4.1) The Hermite random variable of order q ≥ 1 that appears as limit in Theorem 1, point ii. is defined as (see [9]) where the kernel L ∈ L 2 ([0, 1] q ) is given by The constant d(q, H) is a positive normalizing constant that guarantees that EZ 2 = 1 and K H is the standard kernel of the fractional Brownian motion (see [11], Section 5).We will not need the explicit expression of this kernel.Note that the case q = 1 corresponds to the fractional Brownian motion and the case q = 2 corresponds to the Rosenblatt process.

Spitzer's theorem
Let us start by recalling the following result on the convergence of the sequence V n (3) (see [9], Theorem 1).
Theorem 1 Let q ≥ 2 an integer and let (B t ) t≥0 a fractional Brownian motion with Hurst parameter H ∈ (0, 1).Then, with some explicit positive constants c 1,q,H , c 2,q,H depending only on q and H we have ii where Z is a Hermite random variable given by (10).
In the case H = 1 − 1 2q the limit is still Gaussian but the normalization is different.However we will not treat this case in the present work.
We set with the constants c 1,q,H , c 2,q,H from Theorem 1.
Let us denote, for every ε > 0, and Remark 1 It is natural to consider the tail probability of order n 2−2q(1−H) in (15) because the L 2 norm of the sequence V n is in this case of order n 1−q(1−H) .
The first lemma gives the asymptotics of the functions f i (ǫ) as ε → 0 when n are replaced by their limits.
i. Let Z (1) be a standard normal random variable.Then as ii.Let Z (2) be a Hermite random variable or order q given by (10).Then, for any integer .
Proof: The case when Z (1) follows the standard normal law is hidden in [12].We will give the ideas of the proof.We can write (see [12]) is a bounded function and concerning the last term it is also trivial to observe that and Φ ′ Z (1) are bounded.Therefore the asymptotics of the function f 1 (ε) as ε → 0 will be given by Let us consider now the case of the Hermite random variable.We will have as above By making the change of variables cεx 1−q(1−H) = y we will obtain where we used the fact that Φ Z (2) (y) ≤ y −2 E|Z (2) | 2 and so lim y→∞ log yΦ Z (2) (y) = 0.
It remains to show that ) converges to zero as ε tends to 0 (note that actually it follows from a result by [1] that a Hermite random variable has a density, but we don't need it explicitly, we only use the fact that Φ Z (2) is differentiable almost everywhere).This is equal to which clearly goes to zero since P 1 is bounded and The next result estimates the limit of the difference between the functions f i (ε) given by ( 14), (15) and the sequence in Lemma 1.
n be given by ( 13) and let Z (1) be standard normal random variable.Then it holds ii.Let Z (2) be a Hermite random variable of order q ≥ 2 and H > 1 − 1 2q .Then Proof: Let us start with the point i.Assume H < 1 − 1 2q .We can write n≥1 It follows from [10], Theorem 4.1 that and this implies that (17) and the last sums are finite (for the last one we use H < 1 − 1 2q ).The conclusion follows.
Concerning the point ii.(the case H > 1 − 1 2q ), by using a result in Proposition 3.1 of [2] we have and as a consequence and the above series is convergent because H > 1 − 1 2q .
We state now the Spitzer's theorem for the variations of the fractional Brownian motion.
Proof: It is a consequence of Lemma 1 and Proposition 1.

Remark 2 Concerning the case
Because of the appearance of the term log n our approach is not directly applicable to this case.

Hsu-Robbins theorem for the variations of fractional Brownian motion
In this section we prove a version of the Hsu-Robbins theorem for the variations of the fractional Brownian motion.Concretely, we denote here by, for every ε > 0 and by ( 9) We state the main result of this section which is a consequence of Lemma 2 and Proposition 2.
ii.If 1 − 1 2q < H < 1 we have (c −1 2,q,H ε) Remark 5 In the case H = 1 2 we retrieve the result (1) of [5].The case q = 1 is trivial, because in this case, since V n = B n and EV 2 n = n 2H , we obtain the following (by applying Lemma 1 and 2 with q = 1) Remark 6 Let (ε i ) i∈Z be a sequence of i.i.d.centered random variable with finite variance and let (a i ) i≥1 a square summable real sequence.Define X n = i≥1 a i ε n−i .Then the sequence S N = N n=1 [K(X n ) − EK(X n )] satisfies a central limit theorem or a non-central limit theorem according to the properties of the measurable function K (see [6] or [14]).We think that our tools can be applied to investigate the tail probabilities of the sequence S N in the spirit of [5] or [12] at least the in particular cases (for example, when ε i represents the increment W i+1 − W i of a Wiener process because in this case ε i can be written as a multiple integral of order one and X n can be decomposed into a sum of multiple integrals.We thank the referee for mentioning the references [6] and [14].