Noncentral limit theorem for the generalized Hermite process

We use techniques of Malliavin calculus to study the convergence in law of a family of generalized Hermite processes Zγ with kernels defined by parameters γ taking values in a tetrahedral region ∆ of R. We prove that, as γ converges to a face of ∆, the process Zγ converges to a compound Gaussian distribution with random variance given by the square of a Hermite process of one lower rank. The convergence in law is shown to be stable. This work generalizes a previous result of Bai and Taqqu, who proved the result in the case q = 2 and without stability.


2
. Two different proofs are given of this result, one based on the method of moments and a second constructive proof based on a discretization argument.
The goal of this paper is to derive this result as an application of a general theorem of convergence in law of multiple stochastic integrals to a mixture of Gaussian distributions (see Theorem 3.2), which is of independent interest.This theorem is proved using a noncentral limit theorem for Skorohod integrals derived by Nourdin and Nualart in [7].This allows us to extend Bai and Taqqu's result in two directions: We can deal with a general Hermite in the qth Wiener chaos, and we can show that the convergence is stable.
On the other hand, using a version of the Fourth Moment Theorem of Nualart and Peccati [12], we show (see Theorem 4.5) that when 2 and γ i > −1 + , 1 ≤ i ≤ q, for a fixed > 0, then the limit is a standard Brownian motion B(t).For q = 2 this was also proved in [2].
2. Preliminaries 2.0.1.Multiple stochastic integrals.We denote by W = {W (x), x ∈ R} a two-sided Brownian motion on the real line defined on some probability space (Ω, F, P ).Then we can define the Wiener integral W (h) = R h(x)dW x for any function h in the Hilbert space H := L 2 (R), and {W (h), h ∈ H} is an isonormal Gaussian process.We recall that this means that this is a centered Gaussian family with covariance given by the scalar product in H: For every integer q ≥ 1, consider the tensor product H ⊗q = L 2 (R q ) and the symmetric tensor product, denoted by H q , formed by the symmetric functions in L 2 (R q ).For any symmetric function f ∈ H q we denote by I q (f ) the multiple Wiener-Itô stochastic integral of f with respect to W , that can be defined as an iterated Itô integral: Then the following isometry formula holds: where σ runs over all the permutations of {1, . . ., q}.Let m, q ≥ 1 two integers.Given a subset I ⊂ {1, . . ., q} of cardinality r = 0, . . ., q ∧ m, a one-to-one mapping ψ : I → {1, . . ., m}, and two functions f ∈ L 2 (R q ) and g ∈ L 2 (R m ), we denote by f ⊗ I,ψ g the element in L 2 (R q+m−2r ) given by That is, f ⊗ I,ψ g is the function in L 2 (R 2q−2r ) obtained by contracting each variable x i , i ∈ I, from f with the corresponding variable y ψ(i) from g.If f and g are symmetric functions, then the contraction f ⊗ I,ψ g only depends on r and is denoted by f ⊗ r g, that is, (f ⊗ r g)(x r+1 , . . ., x q , y r+1 , . . ., y m ) = R r f (x 1 , . . ., x q )g(y 1 , . . ., y m )dx 1 • • • dx r .
When q = m and I = {1, . . ., q}, we simply write f ⊗ ψ g.Then the following product formula for multiple stochastic integrals holds.For any f ∈ L 2 (R q ) and g ∈ L 2 (R m ), (2.1) where the sum runs over all sets I ⊂ {1, . . ., q} of cardinality r and one-to-one mappings ψ : I → {1, . . ., m}.Notice that when f and g are symmetric, this reduces to the well-known formula On the other hand, for any function f ∈ H ⊗q , which is not necessarily symmetric, we have where ψ runs over all bijections of {1, . . ., q}.
Let {F n } be a sequence of random variables, all defined on the probability space (Ω, F, P ) and let F be a random variable defined on some extended probability space (Ω , F , P ).We say that F n converges stably to F , if lim n→∞ E Ze iλFn = E Ze iλF for every λ ∈ R and every bounded F-measurable random variable Z, where E denotes the mathematical expectation in the probability space (Ω , F , P ).2.0.2.Elements of Malliavin calculus.We introduce some basic elements of the Malliavin calculus with respect to the two-sided Brownian motion W .We refer the reader to Bell [?] or Nualart [11] for a more detailed presentation of these notions.Let S be the set of all smooth and cylindrical random variables of the form (2.2) where n ≥ 1, g : R n → R is a infinitely differentiable function with compact support, and h i ∈ H.The Malliavin derivative of F with respect to X is the element of L 2 (Ω; H) defined as By iteration, one can define the qth derivative D q F for every q ≥ 2, which is an element of L 2 (Ω; H q ).For q ≥ 1 and p ≥ 1, D q,p denotes the closure of S with respect to the norm • D q,p , defined by the relation If V is a real separable Hilbert space, we denote by D q,p (V ) the corresponding Sobolev space of V -valued random variables.We denote by δ the adjoint of the operator D, also called the divergence operator.The operator δ is an extension of the Itô integral.It is also called the Skorohod integral because in the case of the Brownian motion it coincides with the anticipating stochastic integral introduced by Skorohod in [13].A random element u ∈ L 2 (Ω; H) belongs to the domain of δ, denoted Domδ, if and only if it satisfies , where c u is a constant depending only on u.If u ∈ Domδ, then the random variable δ(u) is defined by the duality relationship which holds for every F ∈ D 1,2 .The operators D and δ satisfy the following commutation relation: for any u ∈ D 2,2 (H).

Noncentral limit theorems for multiple stochastic integrals
The following result has been proved by Nourdin and Nualart in [7].
Theorem 3.1.Consider a sequence of Skorohod integrals of the form F n = δ(u n ), where u n ∈ D 2,2 (H).Suppose that the sequence {F n , n ≥ 1} is bounded in L 1 (Ω) and the following conditions hold: (i) u n , h H converges to zero in L 1 (Ω) for all elements h ∈ H 0 , where H 0 is a dense subset of H. (ii) u n , DF n H converges in L 1 (Ω) to a nonnegative random variable S 2 .Then F n converges stably to a random variable with conditional Gaussian law N (0, S 2 ) given W .
On the other hand, from Proposition 3.1 of the paper by Nourdin, Nualart and Peccati [8], it follows that for any test function ϕ ∈ C 3 , we have assuming S 2 ∈ D 1,2 , and where η is a N (0, 1)-random variable independent of the process W .This provides a rate of convergence in the previous theorem.Moreover, the in order to show the convergence in law F n ⇒ Sη as n → ∞, it suffices to check the following two conditions: (i) u n , DF n H → S 2 in L 1 (Ω) as n tends to infinity, and (ii) u n , DS 2 H → 0 in L 1 (Ω) as n tends to infinity.
Proof.We can write F n = δ(u n ) where u n (ξ) = I q−1 (f n (ξ, •)).Then, we claim that F n and u n satisfy the conditions of Theorem 3.1.Notice that u n ∈ D 2,2 (H) because u n is a multiple stochastic integral.To show condition (i) of Theorem 3.1, fix h ∈ H.Then, , which converges to zero by condition (i).
It remains to check condition (ii) of Theorem 3.1.Let us first compute the inner product Using the commutation relation (2.5), we can write We claim that u n , G n,i H converges to zero in L 2 (Ω) as n → ∞, for any i = 2, . . ., q.Indeed, we have Then, using the product formula for multiple stochastic integrals (see (2.1)), we can write where the sum is over all sets I ⊂ {1, . . ., q} of cardinality r and one-to-one mappings ψ : I → {1, . . ., q} such that i ∈ I and ψ(1) = i.Because i = 1, by condition (ii) we deduce that u n , G n,i H converges to zero in L 2 (Ω) as n → ∞ for i = 2, . . ., q.
It will have been noted that the proof of Theorem 3.2 depends crucially upon expressing F n as the Skorohod integral of a multiple Wiener integral of rank q − 1, i.e. choosing a kernel f n such that u n (ξ) = I q−1 (f n (ξ)).Obviously, the choice of f n is not unique, e.g. one could equally well choose f n (ξ) = f n (x 1 , x 2 , . . ., , x i−1 , ξ, x i+1 , . . ., x q−1 ), for any 2 ≤ i ≤ q.However, any such choice will lead to the term in the computation of u n , DF n H .This term evidently does not converge in L 2 (Ω) since it can be shown that its L 2 norm converges to a non-zero limit, while the integrand f n (x 1 , x 2 , . . ., , x i−1 , ξ, x i+1 , . . ., x q−1 ) converges pointwise to zero outside of the diagonal in R q−1 .Thus the choice i = 1 is the only one that will work in the argument.On the other hand, if the role of the first coordinated in conditions (i), (ii) and (iii) is played by another coordinate, then the conclusion of Theorem 3.2 still holds if, in the proof, we choose u n accordingly.
The case where the limit is Gaussian is not included in Theorem 3.2.We state this convergence in the next theorem, whose proof would be similar to that of Theorem 3.2.Notice that Theorem 3.3 below is just the Fourth Moment Theorem proved by Nualart and Peccati in [12] (see the reference [9] for extensions and applications of this result).In the version below of the Fourth Moment Theorem we do not require the kernels to be symmetric.Theorem 3.3.Fix q ≥ 2. Let F n be given by where f n ∈ H ⊗q .Suppose that: (i) For any subset I ⊂ {1, . . ., q} of cardinality r = 1, . . ., q − 1 and any one-to-one mapping ψ : I → {1, . . ., q}, we have Then, as n → ∞, F n converges stably to a random variable with Gaussian law N (0, σ 2 ), independent of W .
Notice that under the assumptions of Theorem 3.3, condition (i) of Theorem 3.2 is satisfied because for any h ∈ H, we can write

Generalized Hermite process
We are interested in the asymptotic behavior of the generalized Hermite process Z γ (t) defined in (1.1), when the parameter γ converges to the boundary of the region ∆.Consider first the case when one of the parameters (for simplicity we choose the first one) converges to − 1 2 .We will make use of the following technical lemmas.The first lemma was proved by Bai and Taqqu in [1, Lemma 3 The second lemma concerns the asymptotic behavior of the Beta function (see [2, Lemma 3.8]).Lemma 4.2.As α → 0, we have In the next lemma, we compute the explicit value of the constant A γ .
Lemma 4.3.The constant A γ is given by , where the sum runs over all permutations σ of {1, . . ., n} and we recall that |γ| = q j=1 γ j .Proof.By a scaling argument, we can take t = 1 and we write f γ := f γ,1 .We have , where fγ denotes the symmetrization of f γ .Then By Lemma 4.1, we have Substituting this formula in the above expression for fγ which completes the proof of the lemma.
The following is the main result of this paper.We would like to point out that the role of γ q in this theorem can be taken by any of the other parameters.Theorem 4.4.As γ 1 converges to − 1 2 , the random variable Z γ (t) converges stably to a random variable whose distribution given W is Gaussian with zero mean and variance Z 2 γ 2 ,...,γq (t).
Proof.To simplify, by a scaling argument we can assume that t = 1.Recall that f γ = f γ,1 .
The asymptotic behavior of the constant A γ , when γ 1 → − 1 2 is obtained from Lemma 4.3, taking into account the asymptotic behavior of the Beta function given by Lemma 4.2: (4.1) lim where in the denominator of the second expression, σ runs over all permutations of {2, . . ., q}.The proof will be done in three steps.
Step 1.Let us show condition (i) of Theorem 3.2.We can take h The term b a (s−ξ) γ 1 + dξ is uniformly bounded as γ 1 → − 1 2 and A γ converges to zero.Therefore, the above expression converges to zero in H ⊗(q−1) .
When γ converges to the boundary of ∆ defined by we obtain the following result, that generalizes Theorem 2.1 in [2].In the case γ 1 = • • • = γ q , this theorem provides the asymptotic behavior of the Hermite process when the parameter converges to − 1 2 − 1 2q .
Proof.The proof is an application of Theorem 3.3.We must establish condition (i) of the theorem.To this end, fix a subset I ⊂ {1, . . ., q} of cardinality r = 1, . . ., q − 1 and a one-to-one mapping ψ : I → {1, . . ., q}.We have All the products of Beta functions are uniformly bounded by our hypothesis γ k + q − r.
Remark 1. Functional versions of theorems Theorem 4.4 and Theorem 4.5 in the space C([0, T ]) can be proved by the same arguments as in [2].In fact, using the self-similarity and stationary-increment property of the process Z γ , together with the hypercontractive inequality for multiple stochastic integrals, we can show that for any p ≥ 2, where H = γ 1 + • • • + γ q + q ≥ 1 2 .This leads to the tightness property and the convergence of the finite dimensional distributions is also easy to obtain, using multidimensional versions of Theorems 3.1, 3.2 and 3.3.
Remark 2. We can derive the rate of convergence in Theorem 4.4 using the inequality (3.1).More precisely, it is not difficult to show that sup The same rate was obtained when q = 2 for the Wasserstein distance in [2, Theorem 5.3], using properties of the second order chaos.Concerning Theorem 4.5, using Stein's method and the optimal rate of convergence in the Fourth Moment Theorem derived by Nourdin and Peccati in [10], we can obtain the following rate of convergence for the total variation distance, as in [2, Theorem 5.1]: c 1 (γ + (q + 1)/2) 3 2 ≤ d T V (Z γ , η) ≤ c 2 (γ + (q + 1)/2) 3 2 , where η is a N (0, 1) random variable and γ = γ 1 + • • • + γ q .In this inequaliy γ satisfies γ i > −1 + , 1 ≤ i ≤ q and the distance of γ to the boundary of ∆ defined in (4.5) is less than , for some > 0. To show these inequalities we need to estimate E[Z 3  γ ] using again the product formula for multiple stochastic integrals.We omit the details of this proof.