  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.
