Central limit theorems for multiple stochastic integrals and Malliavin calculus

Abstract

We give a new characterization for the convergence in distribution to a standard normal law of a sequence of multiple stochastic integrals of a fixed order with variance one, in terms of the Malliavin derivatives of the sequence. We also give a new proof of the main theorem in [7] using techniques of Malliavin calculus. Finally, we extend our result to the multidimensional case and prove a weak convergence result for a sequence of square integrable random variables.

Weak Convergence for Multiple Stochastic Integrals in Skorohod Space

“…Let us recall the following result in [2], which is a collection of some of the results contained in [4] and [6].…”

“…For this, we use central limit theorems of multiple stochastic integrals, being the recent results proved by using the techniques of Malliavin calculus (see [4], [5] and [4]). The purpose of this paper is to study the convergence in distribution of a sequence of random functions of the form F n = I q (f n ), where I q is the multiple stochastic integrals, by using Theorem 3.1.…”

“…It is immediately check that the sequence {F n,2 (t)} converges to zero as n tends to infinity. The result in [4] proves that For every t ∈ [0, 1],…”

Nourdin–Peccati analysis on Wiener and Wiener–Poisson space for general distributions

“…(having unknown distribution). See [13], [16], [20], [21] for results on the convergence of multiple (Wiener) integrals to a standard Normal or Gamma law. [3] and [27] discuss Cramer's theorem for Normal and Gamma distributions applied to multiple integrals.…”

Bipower Variation for Gaussian Processes with Stationary Increments

“…(As discussed in [7], an important early forerunner of that paper is a paper by Guyon and Leon [16], in which quadratic variation limit results for stationary Gaussian processes were derived.) The techniques used there, as well as in the present paper which considers the bipower case, come from very powerful recent results developed in the context of Wiener/Itô/Malliavin calculus, especially by Nualart, Peccati, and coauthors; see [24], [25], and [26] (cf. also [22]).…”