Malliavin and dirichlet structures for independent random variables

TL;DR

This paper develops a Malliavin calculus framework for independent random variables on product probability spaces, enabling new insights into classical inequalities, chaos decompositions, and central limit theorems in discrete settings.

Contribution

It introduces a Dirichlet structure for independent variables, unifying classical inequalities and chaos decompositions without ad-hoc couplings.

Findings

Reformulation of Efron-Stein as a Poincaré inequality

Interpretation of Hoeffding decomposition as chaos decomposition

A version of the Lyapounov CLT using Stein method

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 Poincar{\'e} 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.