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Central limit theorems for multiple Skorokhod integrals. (English) Zbl 1202.60038
The authors prove a central limit theorem for a sequence of multiple Skorokhod integrals using the techniques of Malliavin calculus. Consider a sequence of random variables \(\{ F_n , n\geq 1 \}\) defined on a complete probability space \(( \Omega, \mathcal{F}, \mathcal{P})\). Suppose that the \(\sigma\)-field \(\mathcal{F}\) is generated by an isonormal Gaussian process \(X=\{ X(h), h \in \mathfrak{H} \}\) on a real separable infinite-dimensional Hilbert space \(\mathfrak{H}\). This means that \(X\) is a central Gaussian family of random variables indexed by the elements in \(\mathfrak{H}\), and such that, for each \(h, g \in \mathfrak{H}\), \[ E[X(h)X(g)]=\langle h, g \rangle_{\mathfrak{H}}. \] Suppose that the sequence \(\{F_n, n \geq 1\}\) is normalized. A natural problem is to find suitable conditions ensuring that \(F_n\) converges in law to a given distribution. When the random variables \(F_n\) belong to the \(q\)th Wiener chaos of \(X\) (for a fixed \(q \geq 2\)), then it turns out that the following conditions are equivalent:
(i) \(F_n\) converges in law to \(N(0,1)\);
(ii) \(\lim_{n\rightarrow \infty}E[F^{4}_{n}]=3 \);
(iii) \(\lim_{n\rightarrow \infty} \| DF_n \|^{2}_{\mathfrak{H}}=q \) in \(L^{2}(\Omega)\).
The purpose of the present paper is to study the convergence in distribution of a sequence of random variables of the form \(F_n=\delta^{q}(u_n)\), where \(u_n\) are random variables with values in \(\mathfrak{H}^{\otimes q}\), and \(\delta^{q}\) denotes the multiple Skorokhod integral, towards a mixture of Gaussian random variables. The main result, which is the Theorem 3.1 in the article, roughly says that under some technical conditions, if \(\langle u_n, D^{q}F_n\rangle_{\mathfrak{H}^{\otimes q}}\) converges stably to a random variable \(F\) with conditional characteristic function \(E(e^{i\lambda F}| X) = E (e^{- \frac{\lambda^2}{2}} S^2)\). If \(u_n\) is deterministic, then \(F_n\) belongs to the \(q\)th Wiener chaos. In particular, if \(S^2\) is also deterministic, condition (iii) above implies the convergence in distribution to the law \(N(0,1)\). The authors develop some particular applications of the main result in the following two directions. First, they consider a sequence of random variables in a fixed Wiener chaos, and they derive new criteria for the convergence to a mixture of Gaussian laws. Secondly, they show the convergence in law of the sequence \(\delta^{q}(u_n)\), where \(q\geq2\), and \(u_n\) is a \(q\)-parameter process of the form \[ u_n = n^{qH-\frac{1}{2}}\sum^{n-1}_{k=0} f(B_{k/n})1_{[(k/n, (k+1)/n)]^q}, \] towards the random variable \(\sigma_{H,q}\int^{1}_{0}f(B_s) d W_s\), where \(B\) is a fractional Brownian motion with Hurst parameter \(H\in (1/4q, 1/2)\), \(W\) is a standard Browinan motion independent of \(B\), and \(\sigma_{H,q}\) denotes some positive constant.

60F05 Central limit and other weak theorems
60H05 Stochastic integrals
60G15 Gaussian processes
60H07 Stochastic calculus of variations and the Malliavin calculus
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