  In this article, we provide an extension of the Chen-Stein inequality for
Poisson approximation in the total variation distance for sums of independent
Bernoulli random variables in two ways. We prove that we can improve the rate
of convergence (hence the quality of the approximation) by using explicitly
constructed signed or positive probability measures, and that we can extend the
setting to possibly dependent random variables. The framework which allows this
is that of mod-Poisson convergence, and more precisely those mod-Poisson
convergent sequences whose residue functions can be expressed as a
specialization of the generating series of elementary symmetric functions. This
combinatorial reformulation allows us to have a general and unified framework
in which we can fit the classical setting of sums of independent Bernoulli
random variables as well as other examples coming e.g. from probabilistic
number theory and random permutations.
