Almost sure limit theorems on Wiener chaos: the non-central case

In \cite{BNT}, a framework to prove almost sure central limit theorems for sequences $(G_n)$ belonging to the Wiener space was developed, with a particular emphasis of the case where $G_n$ takes the form of a multiple Wiener-It\^o integral with respect to a given isonormal Gaussian process. In the present paper, we complement the study initiated in \cite{BNT}, by considering the more general situation where the sequence $(G_n)$ may not need to converge to a Gaussian distribution. As an application, we prove that partial sums of Hermite polynomials of increments of fractional Brownian motion satisfy an almost sure limit theorem in the long-range dependence case, a problem left open in \cite{BNT}.


Introduction and main results
Let us start with the following definition, which plays a pivotal role in the paper. Definition 1.1 Let (G n ) be a sequence of real-valued random variables defined on a common probability space (Ω, F , P).
1. We say that (G n ) satisfies an almost sure limit theorem (in short: ASLT) with limit G ∞ if, almost surely for any continuous and bounded function ϕ : R → R, But (1.3) cannot be true, as it would violate the classical arc sine law when x = 0 (see, e.g., [26,Chapter IV]). Another remark is that the validity (or not) of (1.1) is a property of the distribution of the whole sequence (G n ). Let us elaborate on this point. If G n → G ∞ in law, it is well-known by Skorokhod's representation theorem that there exists a family (G ⋆ n ) n∈N∪{∞} of random variables defined on a common probability space such that G ⋆ n law = G n for all n and G ⋆ n → G ⋆ ∞ almost surely. Moreover, Cesàro summation theorem implies that, almost surely, (log n) −1 k≤n ϕ(G ⋆ k )/k −→ ϕ(G ⋆ ∞ ) for any continuous bounded function ϕ : R → R; that is, (G ⋆ n ) does not satisfy an ASLT except if G ⋆ ∞ is a Dirac mass, since apart in this case ϕ(G ⋆ ∞ ) = E[ϕ(G ⋆ ∞ )] for at least one function ϕ. Now we have made these preliminary comments on general ASLTs, let us concentrate on the specific kind of random variables (G n ) we are interested in in this paper: namely, random variables taking the form of multiple Wiener-Itô integrals. Since the discovery by Nualart and Peccati [21] of their fourth moment theorem (according to which a normalized sequence of multiple Wiener-Itô integrals converges in law to N (0, 1) if and only if its fourth moment converges to 3), those stochastic integrals have become a probabilistic object of considerable interest. See for instance the constantly updated webpage [15] for a demonstration of the intense activity surrounding them.
More specifically, a sequence (G n ) of multiple Wiener-Itô integrals converging in law to some G ∞ being given, our goal in this paper is to provide a meaningful set of conditions under which an ASLT holds true. The central case (that is, when G ∞ ∼ N (0, 1)) has already been the object of [1]. Therein, a framework has been developed in order to show an ASCLT by taking advantage of the numerous estimates coming from the Malliavin-Stein approach [18]. One of the main results of [1], that we state here for the sake of comparison, is the following. The notation I q (g) refers to the qth Wiener-Itô integral with kernel g associated to a given isonormal Gaussian process X over a separable Hilbert space H, whereas the notation ⊗ r stands for the contraction operator, see Section 2.3 for more details. Theorem 1.2 Fix q 2, and let (G n ) be a sequence of the form G n = I q (g n ), n 1. Suppose (for simplicity) that E[G 2 n ] = 1 for all n. Suppose in addition that the following two conditions are met.
An application of high interest of Theorem 1.2 studied in [1] concerns the celebrated Breuer-Major theorem. Let {X k } be a stationary centered Gaussian family, characterized by its correlation function ρ : Z → R given by ρ(k − l) := E[X k X l ], and assume that ρ(0) = 1. Fix q 2 and let H q denote the qth Hermite polynomial. Finally, consider the sequence of the Hermite variations of X, defined as It is well known since the eighties and the seminal works by Breuer and Major [6], Giraitis and Surgailis [9] and Taqqu [27], that the study of the fluctuations of V n crucially depends on the summability of |ρ| q . More precisely, if we set then G n law → N (0, 1) as soon as k∈Z |ρ(k)| q < ∞. When k∈Z |ρ(k)| q = ∞, the sequence (G n ) may converge to a Gaussian (in some critical situations) or non-Gaussian limit (more likely), see below.
A classical example of a stationary Gaussian sequence {X k } k∈Z falling within this framework is given by the fractional Gaussian noise where B = (B t ) t∈R is a fractional Brownian motion of Hurst index H ∈ (0, 1), that is, B is a centered Gaussian process with covariance function More precisely, it is nowadays well-known that The proof of (i) follows directly from the Breuer-Major theorem when H < 1− 1 2q , whereas a little more effort are required in the critical case H = 1 − 1 2q (see, e.g., [5]). For a definition of Hermite distribution as well as a short proof of the weak convergence (ii) the reader can consult, e.g., [1, Proposition 6.1] and [18,Proposition 7.4.2].
In this paper, we aim to answer the following question: can we associate an almost sure limit theorem to the previous two convergences in law (i) and (ii)? This problem is actually not new, and has been first considered almost one decade ago. In [1], a positive answer has been indeed given for (i), with the help of Theorem 1.2: namely, if H 1 − 1 2q and if G n is defined by (1.4), then (1.1) holds true with G ∞ ∼ N (0, 1). What about the case H ∈ (1 − 1 2q , 1)? A solution to this problem was left open in [1], because the techniques developed therein do not cover the situation where the limit of (G n ) is not Gaussian.
In the present paper, our main motivation is then to study the case H ∈ (1 − 1 2q , 1) left open in [1]. In order to do so, we first develop a general abstract result (Theorem 1.3), of independent interest and valid in a framework similar to that of Theorem 1.2; then, we apply it to our specific situation (Corollary 1.4). Theorem 1.3 Fix q 2, and let (G n ) be any sequence of the form G n = I q (g n ), n 1. Suppose G n law → G ∞ , and denote by φ n (resp. φ ∞ ) the characteristic function of G n (resp. G ∞ ). Suppose in addition that the following two conditions are met.
Let G n be given by (1.4) with H belonging to (1 − 1 2q , 1). In [1, Section 6], the authors have constructed an explicit sequence ( G n ) such that G n law = G n for any n and G n does not satisfy an ASLT. But as we already said, from this we can learn nothing about the validity or not of an ASLT associated to the original sequence (G n ). Here, using our abstract Theorem 1.3, we are actually able to prove that (G n ) does satisfy an ASLT. Corollary 1.4 Let G n be given by (1.4), with X k as in (1.5) and H belonging to (1 − 1 2q , 1). Then (G n ) satisfies an ASLT, with G ∞ drawn from the Hermite distribution.
To conclude this introduction, we would like to stress that our Theorem 1.3 is a true extension of Theorem 1.2, in the sense that the latter can be obtained as a particular case of the former: see Section 5 for the details.
The rest of the paper is organized as follows. In Section 2 several preliminary results that are needed for the proofs are collected. Section 3 is devoted to the proof of Theorem 1.3, whereas the proof of Corollary 1.4 can be found in Section 4. Finally, Section 5 details how to deduce Theorem 1.2 from [1] from our Theorem 1.3.

Ibragimov-Lifshits criterion for ASLT
As we will see, we are not going to prove directly that (G n ) in Theorem 1.3 satisfies (1.2). Instead, we are going to check the validity of a powerful criterion due to Ibragimov and Lifshits [12], that we recall here in its general form for the convenience of the reader. Theorem 2.1 Let (G n ) be a sequence of random variables defined on a common probability space and converging in distribution towards a target random variable G ∞ , and set then (G n ) satisfies the ASLT (1.1).

An easy reduction lemma
In this section, we state and prove an easy reduction lemma, that we are going to use in the proof of Corollary 1.4.
Lemma 2.2 Suppose that (G n ) is such that G n law → G ∞ , and assume that G ∞ has a density. Let (a n ) be a real-valued sequence converging to a ∞ = 0. Then (G n ) satisfies an ASLT if and only if (a n G n ) does.
Proof. Without loss of generality, we may and will assume that a n > 0 for all n and that a ∞ = 1. By symmetry, only the implication "if (G n ) satisfies an ASLT then (a n G n ) does" has to be proved.
So, let us assume that (G n ) satisfies an ASLT. Since G ∞ has a density, we are going to use criterion (1.2). Fix ε > 0, and let k 0 be such that 1 − ε a k 1 + ε for all k k 0 . For any x ∈ R, we can write, for all k k 0 , Letting n → ∞, we deduce from (1.2) that, almost surely, Letting now ε → 0 yields n k=1 1 , which is desired conclusion.

Elements of Malliavin calculus
Our proofs of Theorem 1.3 and Corollary 1.4 are based on the use of the Malliavin calculus. This is why in this section we recall the few elements of Gaussian analysis and Malliavin calculus that will be needed. For more details or missing proofs, we invite the reader to consult one of the three books [18,22,23]. Let H be a real separable Hilbert space. For any q 1, we write H ⊗q and H ⊙q to indicate, respectively, the qth tensor power and the qth symmetric tensor power of H.
We denote by X = {X(h) : h ∈ H} an isonormal Gaussian process over H, that is, X is a centered Gaussian family defined a common probability space (Ω, F , P) satisfying E [X(g)X(h)] = g, h H . We also assume that F is generated by X, and use the shorthand notation L 2 (Ω) = L 2 (Ω, F , P).
For every q 1, the qth Wiener chaos of X is defined as the closed linear subspace of L 2 (Ω) generated by the family {H q (X(h)) : h ∈ H, h H = 1}, where H q is the qth Hermite polynomial : (2.8) For any q 1, the mapping I q (h ⊗q ) = H q (W (h)) (for h ∈ H, h H = 1) can be extended to a linear isometry between the symmetric tensor product H ⊙q (equipped with the modified norm √ q! · H ⊗q ) and the qth Wiener chaos. Let {e k , k 1} be a complete orthonormal system in H. Given f ∈ H ⊙p and g ∈ H ⊙q , for every r = 0, . . . , p∧q, the contraction of f and g of order r is the element of H ⊗(p+q−2r) defined by Notice that the definition of f ⊗ r g does not depend on the particular choice of {e k , k 1}, and that f ⊗ r g is not necessarily symmetric; we denote its symmetrization by f ⊗ r g ∈ H ⊙(p+q−2r) . Moreover, f ⊗ 0 g = f ⊗ g equals the tensor product of f and g while, for p = q, f ⊗ q g = f, g H ⊗q . Contractions appear naturally in the product formula for multiple integrals: if f ∈ H ⊙p and g ∈ H ⊙q , then r! p r q r I p+q−2r (f ⊗ r g). (2.10)

Proof of Theorem 1.3
As we said in Section 2.1, we are going to check the condition (2.7) in our specific framework. For t ∈ R, recall the quantity ∆ n (t) from (2.6); we can write

Now, note that
so that sup |t| r n 2 A1(n,t) n log n < ∞ for all r > 0, by condition (B1) in Theorem 1.3. On the other hand, [16, Proposition 3.1] provides the existence of a universal constant c such that We deduce that , (3.11) implying in turn by condition (B2) that sup |t| r n 2 A2(n,t) n log n < ∞ for all r > 0, and concluding the proof.

Proof of Corollary 1.4
Let (G n ) be given by (1.4) with H ∈ (1 − 1 2q , 1). It is a straightforward exercise to prove that Var(V n ) is equivalent to a constant times n 2−2q(1−H) as n → ∞ (see, e. g. [1] and references therein). Thanks to Lemma 2.2 and because the Hermite distribution G ∞ admits a density (this latter fact is an immediate consequence of the main result of [19] for instance), it is then equivalent to prove an ASLT for (G n ) or for ( G n ) defined as In the sequel, we are thus rather going to prove that ( G n ) satisfies an ASLT.
Our first observation is that G n takes the form of a multiple Wiener-Itô integral with respect to B, after identifying this latter with an isonormal Gaussian process X over the Hilbert space H defined as the closure of the set of step functions with respect to the scalar product 1 In what follows, φ k and φ ∞ denote the characteristic functions of random variables G k and G ∞ respectively.
For (B1), it is a direct consequence of some estimates given in [5]. More precisely, using [5, Proposition 3.1] one infers that, for any (fixed) r > 0, where d W stands for the Wasserstein distance. In fact, the first inequality in above is a direct consequence of the basic fact that function x → e itx is a Lipschitz-continuous function with Lipschitz constant |t|. Also, recall for two real-valued random variables F 1 and F 2 that Let us now turn to (B2). Using the product formula (2.10) for multiple integrals, we can write Thus we are left to check that, for all r ∈ {0, . . . , q − 1}: Let us first focus on the case where 1 r q − 1. We can write, with ρ(a) = 1 2 |a + 1| 2H + |a − 1| 2H − 2|a| 2H and ρ k (a) = |ρ(a)|1 |a| k : Using Young inequality with p = 2q 3r and p ′ = 2q 3(q−r) (so that 1 p + 1 p ′ = 3 2 ), we obtain Thus, for r ∈ {1, . . . , q − 1}, (4.14) Actually, the previous estimate (4.14) is also valid for r = 0. Indeed, we have in this case where the last inequality used the fact that ρ(r) ∼ H(2H − 1)r 2H−2 as |r| → ∞. Finally, (4.14) and (4.15) together imply that (4.13) takes place for any r ∈ {0, . . . , q − 1}, which concludes the proof of Theorem 1.4.

Theorem 1.3 implies Theorem 1.2
To conclude this paper, let us explain how to deduce Theorem 1.2 (taken from [1]) from our Theorem 1.3, even if at first glance conditions (A1)-(A2) and (B1)-(B2) do not seem to be related to each others. Fix q 2, and let (G n ) be a sequence of the form G n = I q (g n ), n 1. Suppose moreover that E[G 2 n ] = q! g n 2 H ⊗q = 1 for all n. In what follows, c q > 0 denotes a constant only depending on q, whose value can change from line to line.