Vladimir Vinogradov

"On
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.