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Stable Convergence of Multiple Wiener-Itô Integrals

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Abstract

We prove sufficient conditions ensuring that a sequence of multiple Wiener-Itô integrals (with respect to a general Gaussian process) converges stably to a mixture of normal distributions. Note that stable convergence is stronger than convergence in distribution. Our key tool is an asymptotic decomposition of contraction kernels, realized by means of increasing families of projection operators. We also use an infinite-dimensional Clark-Ocone formula, as well as a version of the correspondence between “abstract” and “concrete” filtered Wiener spaces, in a spirit similar to that of Üstünel and Zakai (J. Funct. Anal. 143, 10–32, [1997]).

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Authors and Affiliations

  1. Laboratoire de Statistique Théorique et Appliquée, Université Paris VI, Paris, France

    Giovanni Peccati

  2. Department of Mathematics, Boston University, 111 Cummington Road, Boston, MA, USA

    Murad S. Taqqu

Authors
  1. Giovanni Peccati
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  2. Murad S. Taqqu
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Correspondence to Giovanni Peccati.

Additional information

This research was partially supported by the NSF Grant DNS-050547 at Boston University.

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Peccati, G., Taqqu, M.S. Stable Convergence of Multiple Wiener-Itô Integrals. J Theor Probab 21, 527–570 (2008). https://doi.org/10.1007/s10959-008-0154-x

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  • DOI: https://doi.org/10.1007/s10959-008-0154-x

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