Central limit theorems for multiple stochastic integrals and Malliavin calculus
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.
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