Non-uniform Bounds in the Poisson Approximation with Applications to Informational Distances. I

Abstract

We explore asymptotically optimal bounds for deviations of Bernoulli convolutions from the Poisson limit in terms of the Shannon relative entropy and the Pearson $\chi^2$-distance. The results are based on proper non-uniform estimates for densities. They deal with models of non-homogeneous, non-degenerate Bernoulli distributions.