Central limit theorems for multiple stochastic integrals and Malliavin calculus
type: Item
instance of: scholarly article
title: Central limit theorems for multiple stochastic integrals and Malliavin calculus
zbMATH Open document ID: 1142.60015
author: David Nualart
author: Salvador Ortiz-Latorre
published in: Stochastic Processes and their Applications
publication date: 2008-03-18T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0703240
review text: The authors consider a sequence of random variables \(F_{k}\) belonging to the \(n\)th Wiener chaos, \(n \geq 2\), and with unit variance. Nualart and Peccati have proved that this sequence converges in distribution to a normal \(N(0, 1)\) law if and only if one of the following two equivalent conditions holds: \[ \lim_{k \rightarrow \infty} E[F^{4}_{k}] =3,\tag{i} \] \[ \lim_{k \rightarrow \infty} f_{k} \otimes_{l} f_{k} = 0,\text{ for all }1 \leq l \leq n-1, \tag{ii} \] There have been different extensions and applications of these results. The authors 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. The main purpose of this paper is to provide an additional necessary and sufficient condition for the convergence of the sequence \(F_{k}\) to a normal distribution, in terms of the derivatives of \(F_{k}\) in the sense of Malliavin calculus. The authors also discuss the extension of these results to the multidimensional case, and establish a general criterion for the weak convergence of a sequence of centered square integrable random vectors.
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 60H07
zbMATH DE Number: 5250610
zbMATH Keywords: multiple stochastic integrals
zbMATH Keywords: limit theorems
zbMATH Keywords: Gaussian processes
zbMATH Keywords: Malliavin calculus
zbMATH Keywords: weak convergence
reviewed by: Tae Il Jeon
MaRDI profile type: Publication
OpenAlex ID: W2014903152
arXiv ID: math/0703240
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DOI: 10.1016/J.SPA.2007.05.004
description: scientific article