Open Access
January, 1989 Stochastic Difference Equations and Generalized Gamma Distributions
Fima C. Klebaner
Ann. Probab. 17(1): 178-188 (January, 1989). DOI: 10.1214/aop/1176991502


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.


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Fima C. Klebaner. "Stochastic Difference Equations and Generalized Gamma Distributions." Ann. Probab. 17 (1) 178 - 188, January, 1989.


Published: January, 1989
First available in Project Euclid: 19 April 2007

zbMATH: 0674.60077
MathSciNet: MR972779
Digital Object Identifier: 10.1214/aop/1176991502

Primary: 60J80
Secondary: 60J10 , 60J15

Keywords: asymptotic behavior , branching processes , generalized random walks , size dependence , Stochastic growth

Rights: Copyright © 1989 Institute of Mathematical Statistics

Vol.17 • No. 1 • January, 1989
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