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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.

##### MSC:
 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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