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Malliavin and Dirichlet structures for independent random variables

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  • Decreusefond, Laurent
  • Halconruy, Hélène

Abstract

On any denumerable product of probability spaces, we construct a Malliavin gradient and then a divergence and a number operator. This yields a Dirichlet structure which can be shown to approach the usual structures for Poisson and Brownian processes. We obtain versions of almost all the classical functional inequalities in discrete settings which show that the Efron–Stein inequality can be interpreted as a Poincaré inequality or that the Hoeffding decomposition of U-statistics can be interpreted as an avatar of the Clark representation formula. Thanks to our framework, we obtain a bound for the distance between the distribution of any functional of independent variables and the Gaussian and Gamma distributions.

Suggested Citation

  • Decreusefond, Laurent & Halconruy, Hélène, 2019. "Malliavin and Dirichlet structures for independent random variables," Stochastic Processes and their Applications, Elsevier, vol. 129(8), pages 2611-2653.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:8:p:2611-2653
    DOI: 10.1016/j.spa.2018.07.019
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    References listed on IDEAS

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    1. Rhee, WanSoo T. & Talagrand, Michel, 1986. "Martingale inequalities and the Jackknife estimate of variance," Statistics & Probability Letters, Elsevier, vol. 4(1), pages 5-6, January.
    2. Nualart, David & Schoutens, Wim, 2000. "Chaotic and predictable representations for Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 90(1), pages 109-122, November.
    3. Arras, Benjamin & Swan, Yvik, 2017. "A stroll along the gamma," Stochastic Processes and their Applications, Elsevier, vol. 127(11), pages 3661-3688.
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    Cited by:

    1. H'el`ene Halconruy, 2021. "The insider problem in the trinomial model: a discrete-time jump process approach," Papers 2106.15208, arXiv.org, revised Sep 2023.

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