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:arXiv.org 2018-07
Main Authors: Decreusefond, Laurent, Halconruy, Hélène
Format: Article
Language:English
Subjects:
Couplings
Decomposition
Dirichlet problem
Divergence
Independent variables
Random variables
Online Access:Get full text
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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 Poincar{é} inequality or that Hoeffding decomposition of U-statistics can be interpreted as a chaos decomposition. We obtain a version of the Lyapounov central limit theorem for independent random variables without resorting to ad-hoc couplings, thus increasing the scope of the Stein method.
ISSN:2331-8422