Poisson approximation of the mixed Poisson distribution with infinitely divisible mixing law

Abstract

In this work, explicit upper bounds are provided for the Kolmogorov and total variation distances between the mixed Poisson distribution with infinitely divisible mixing law and the Poisson distribution. If μ and σ2 are the mean and variance of the mixing distribution respectively, then the bounds provided here are asymptotically equal to σ2 / (2 μ sqrt(2 π e)) and σ2 / (μ sqrt(2 π e)) for the Kolmogorov and the total variation distance respectively when μ → ∞ and σ2 is fixed. Finally, as an application, the Poisson approximation of the negative Binomial distribution is considered. © 2009 Elsevier B.V. All rights reserved