On the asymptotic behaviour of discrete time stochastic growth processes
The stochastic process \(\{X_ n\}\), defined recursively by \[ X_{n+1}=X_ n+g(X_ n)(1+\xi_{n+1}) \] where \(g(t)=o(t)\) and \(\{\xi_ n\}\) is a zero mean martingale difference sequence, is studied. (The states \(X_ n<0\) are absorbing.) This is a discrete analogue of a stochastic differential equation studied by the authors in an earlier paper [Z. Wahrscheinlichkeitstheor. Verw. Geb., 68, 163-189 (1984; Zbl 0535.60050)]. The main motivation for studying it is however provided by its application in the study of branching processes with state-dependent reproduction, and in this respect this paper is a sequel to that of \textit{P. Küster} [Ann. Probab. 13, 1157-1178 (1985; Zbl 0576.60078)] on state dependent and \(\phi\)-controlled Galton-Watson processes. Results are given when \(\{X_ n\}\) (i) either hits zero or diverges, a.s., (ii) diverges with positive probability, (iii) converges on \(\{X_ n\to \infty \}\) almost surely, in probability or in distribution, when suitably normalized. The conditions needed involve the growth rates of g(t) and the conditional variances of \(\{\xi_ n\}\).
branching processes with state-dependent reproduction
scientific article; zbMATH DE number 3998926