On approximations for two classes of Poisson mixtures

We investigate two related classes of Poisson mixtures which comprise exponential dispersion models. The first one is constructed starting from a law related to Bachelier's work. Certain members of this model emerge in the gambler's ruin problem. The other model is comprised of the distributions employed by Sichel and others for fitting diverse count data. We identify the unit variance function for the former model. The second-order local approximations for both the models, which involve inverse Gaussian laws, are constructed. Our techniques include Poisson mixtures, Laplace's method, and the Esscher transforms.