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.
Publié le : 1989-01-14
Classification:
Stochastic growth,
asymptotic behavior,
size dependence,
branching processes,
generalized random walks,
60J80,
60J10,
60J15
@article{1176991502,
author = {Klebaner, Fima C.},
title = {Stochastic Difference Equations and Generalized Gamma Distributions},
journal = {Ann. Probab.},
volume = {17},
number = {4},
year = {1989},
pages = { 178-188},
language = {en},
url = {http://dml.mathdoc.fr/item/1176991502}
}
Klebaner, Fima C. Stochastic Difference Equations and Generalized Gamma Distributions. Ann. Probab., Tome 17 (1989) no. 4, pp. 178-188. http://gdmltest.u-ga.fr/item/1176991502/