Stochastic Difference Equations and Generalized Gamma Distributions

Abstract

We study the asymptotic growth rates of discrete-time stochastic processes $(X_n)$, where the first two conditional moments of the process depend only on the present state. Such processes satisfy a stochastic difference equation $X_{n + 1} = X_n + g(X_n) + R_{n + 1}$, where $g$ is a positive function and $(R_n)$ is a martingale difference sequence. It is known that a large class of such processes diverges with positive probability, and when properly normalized converges almost surely or converges in distribution to a normal or a lognormal distribution. Here we find a class of processes that when properly normalized converges in distribution to a generalized gamma distribution. Applications of this result to state dependent random walks and population size-dependent branching processes yield new results and reprove some of the known results.