A new method for obtaining sharp compound Poisson approximation error estimates for sums of locally dependent random variables

Abstract

Let X1, X2,..., Xn be a sequence of independent or locally dependent random variables taking values in ℤ+. In this paper, we derive sharp bounds, via a new probabilistic method, for the total variation distance between the distribution of the sum Σni=1 Xi and an appropriate Poisson or compound Poisson distribution. These bounds include a factor which depends on the smoothness of the approximating Poisson or compound Poisson distribution. This "smoothness factor" is of order O(σ-2), according to a heuristic argument, where σ-2 denotes the variance of the approximating distribution. In this way, we offer sharp error estimates for a large range of values of the parameters. Finally, specific examples concerning appearances of rare runs in sequences of Bernoulli trials are presented by way of illustration. © 2010 ISI/BS