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
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.