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Seminar on Stochastic Analysis, Random Fields and Applications V pp 363–395Cite as

Some Applications of the Malliavin Calculus to Sub-Gaussian and Non-Sub-Gaussian Random Fields

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Part of the book series: Progress in Probability ((PRPR,volume 59))

Abstract

We introduce a boundedness condition on the Malliavin derivative of a random variable to study sub-Gaussian and other non-Gaussian properties of functionals of random fields, with particular attention to the estimation of suprema. We relate the boundedness of the nth Malliavin derivative to a new class of “sub-nth-Gaussian chaos” processes. An expected supremum estimation, extending the Dudley theorem, is proved for such processes. Sub-nth-Gaussian chaos concentration inequalities for the supremum are obtained, using Malliavin derivative conditions; for n = 1, this generalizes the Borell-Sudakov inequality to a class of sub-Gaussian processes, with a particularly simple and efficient proof; for n = 2 a natural extension to sub-2nd-Gaussian chaos processes is established; for n ≥ 3 a slightly less efficient Malliavin derivative condition is needed.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Purdue University, 150 N. University St., West Lafayette, IN, 47907-2067, USA

    Andrew B. Vizcarra

  2. Department of Statistics and Department of Mathematics, Purdue University, 150 N. University St., West Lafayette, IN, 47907-2067, USA

    Frederi G. Viens

Authors
  1. Andrew B. Vizcarra
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  2. Frederi G. Viens
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Editor information

Editors and Affiliations

  1. Institut de Mathématiques, Ecole Polytechnique Fédérale, CH-1005, Lausanne, Switzerland

    Robert C. Dalang

  2. Département de Mathématiques, Institut Galilée, Université Paris 13, F-95430, Villetaneuse, France

    Francesco Russo

  3. Institut Elie Cartan, Université Henri Poincaré, B.P. 239, F-54506, Vandoeuvre-lès-Nancy Cedex, France

    Marco Dozzi

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© 2007 Birkhäuser Verlag Basel/Switzerland

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Vizcarra, A.B., Viens, F.G. (2007). Some Applications of the Malliavin Calculus to Sub-Gaussian and Non-Sub-Gaussian Random Fields. In: Dalang, R.C., Russo, F., Dozzi, M. (eds) Seminar on Stochastic Analysis, Random Fields and Applications V. Progress in Probability, vol 59. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8458-6_20

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