Malliavin and Dirichlet structures for independent random variables

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 fu...

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Bibliographic Details
Published in:Stochastic processes and their applications 2019-08, Vol.129 (8), p.2611-2653
Main Authors: Decreusefond, Laurent, Halconruy, Hélène
Format: Article
Language:English
Subjects:
Dirichlet structure
Ewens distribution
Log-Sobolev inequality
Malliavin calculus
Mathematics
Probability
Stein’s method
Talagrand inequality
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Summary: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.
ISSN:0304-4149
1879-209X
DOI:10.1016/j.spa.2018.07.019