On the Asymptotic Behaviour of Discrete Time Stochastic Growth Processes

Abstract

We study the asymptotic behaviour of the solution of the stochastic difference equation $X_{n+1} = X_n + g(X_n)(1 + \xi_{n+1})$, where $g$ is a positive function, $(\xi_n)$ is a 0-mean, square-integrable martingale difference sequence, and the states $X_n < 0$ are assumed to be absorbing. We clarify, under which conditions $X_n$ diverges with positive probability, satisfies a law of large numbers, and, properly normalized, converges in distribution. Controlled Galton-Watson processes furnish examples for the processes under consideration.