Closed-form solutions for Bernoulli and compound Poisson branching processes in random environments

Abstract

For branching processes, the generating functions for limit distributions of so-called ratios of probabilities of rare events satisfy the Schr\"oder-type integral-functional equations. Excepting limited special cases, the corresponding equations can not be solved analytically. I found a large class of Poisson-type offspring distributions, for which the Schr\"oder-type functional equations can be solved analytically. Moreover, for the asymptotics of limit distributions, the power and constant factor can be written explicitly. As a bonus, Bernoulli branching processes in random environments are treated. The beauty of this example is that the explicit formula for the generating function is unknown. Still, the closed-form expression for the power and constant factor in the asymptotic can be written with the help of some outstanding tricks. Also, the asymptotic expansion contains oscillatory terms absent in the Poisson case. It is shown that at the power $3$ in the Bernoulli binomial kernels short-phase discrete oscillations turn into long-phase continuous ones.