Bachelier's and Sichel's classes of Poisson mixtures"
We investigate two related classes of Poisson mixtures which comprise additive
exponential dispersion models. The first one is constructed starting from a probability
law considered by Bachelier. It includes distributions which emerge in the gambler's ruin
problem. The other model is comprised of the distributions employed by Sichel and others
for fitting count data. We derive the closed-form expression for the unit variance functions
and construct local approximations for both models
which involve inverse Gaussian laws. Our techniques include a Poisson mixture
representation, Laplace's method, and an analogue of Gnedenko's
method of accompanying infinitely divisible laws.