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Asymptotic expansion of Gaussian chaos via probabilistic approach

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Abstract

For a centered d-dimensional Gaussian random vector ξ = (ξ 1, … , ξ d ) and a homogeneous function h : ℝd → ℝ we derive asymptotic expansions for the tail of the Gaussian chaos h(ξ) given the function h is sufficiently smooth. Three challenging instances of the Gaussian chaos are the determinant of a Gaussian matrix, the Gaussian orthogonal ensemble and the diameter of random Gaussian clouds. Using a direct probabilistic asymptotic method, we investigate both the asymptotic behaviour of the tail distribution of h(ξ) and its density at infinity and then discuss possible extensions for some general ξ with polar representation.

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Authors and Affiliations

  1. University of Lausanne, (HEC Lausanne), UNIL-Dorigny, 1015, Lausanne, Switzerland

    Enkelejd Hashorva

  2. Lancaster University, Lancaster, UK

    Dmitry Korshunov

  3. Sobolev Institute of Mathematics, Novosibirsk, Russia

    Dmitry Korshunov

  4. Lomonosov Moscow State University, Moscow, Russia

    Vladimir I. Piterbarg

Authors
  1. Enkelejd Hashorva
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  2. Dmitry Korshunov
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  3. Vladimir I. Piterbarg
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Correspondence to Dmitry Korshunov.

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Hashorva, E., Korshunov, D. & Piterbarg, V.I. Asymptotic expansion of Gaussian chaos via probabilistic approach. Extremes 18, 315–347 (2015). https://doi.org/10.1007/s10687-015-0215-3

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  • DOI: https://doi.org/10.1007/s10687-015-0215-3

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