On the asymptotic behaviour of discrete time stochastic growth processes - Keller, Gerhard; Kersting, Götz; Rösler, Uwe (Annals of Probability, 1987)
We study the asymptotic behaviour of the solution of the stochastic difference equation Xn+1=Xn+g(Xn)(1+ξn+1), where g is a positive function, (ξn) is a 0-mean, square-integrable martingale difference sequence, and the states <span class="MathJax" data-mathml="Xn<0" id="MathJax-Element-4-Frame" role="presentation" style="position: relative;" tabindex="0">Xn<0 are assumed to be absorbing. We clarify, under which conditions Xn 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.