Local Approximations for Branching Particle Systems

We derive different local approximations along with estimates of the remainders for the total mass processes of two branching particle systems. To this end, we investigate the corresponding classes of integer-valued variables. One of them is comprised of Pólya-Aeppli distributions, while members of the other class are the convolutions of a zero-modified geometric law. We obtain the closed-form representation for the probability function of the latter convolutions and derive its asymptotics. Properties of the limiting Poisson-exponential laws are also described. Our techniques involve a Poisson mixture representation, Laplace’s method and upper estimates in the local Poisson theorem. The parallels with Gnedenko’s method of accompanying infinitely divisible laws are established.