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

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 σ 2 are the mean and variance of the mixing distribution respectively, then the bounds provided here are asymptotically equal to σ 2 / ( 2 μ 2 π e ) and σ 2 / ( μ 2 π e ) for the Kolmogorov and the total variation distance respectively when μ → ∞ and σ 2 is fixed. Finally, as an application, the Poisson approximation of the negative Binomial distribution is considered.