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A Wick functional limit theorem. (English) Zbl 1296.60060

The main result of the article is the weak convergence of arbitrary Wick products of Wick analytic functionals on multivariate Wiener integrals. The proof is based on the representation of arbitrary Wick products (denoted by \(\diamond\)) of Wiener integrals in terms of generalized Hermite polynomials by the formula \[ I(f_1)\diamond\dots\diamond I(f_k)=h^k_\sigma(I(f_1),\dots, I(f_k)), \] where \({k\in\mathbb{N}}, the \) functions \({f_1,\dots, f_k\in L_2([0,1])},\) \(I(f_i)\) are the standard Wiener integrals and the \({h^k_\sigma}\) are generalized Hermite polynomials with set of parameters \({\sigma=\{\sigma_{ij}=\int_0^1f_if_j\,dx: {1\leq i<j\leq k}\}}.\) For the proof, the discrete counterpart of such a representation is used. At the end, the results of the paper are applied to fractional Brownian motion.

MSC:

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