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January, 1987 On the Asymptotic Behaviour of Discrete Time Stochastic Growth Processes
G. Keller, G. Kersting, U. Rosler
Ann. Probab. 15(1): 305-343 (January, 1987). DOI: 10.1214/aop/1176992272


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.


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G. Keller. G. Kersting. U. Rosler. "On the Asymptotic Behaviour of Discrete Time Stochastic Growth Processes." Ann. Probab. 15 (1) 305 - 343, January, 1987.


Published: January, 1987
First available in Project Euclid: 19 April 2007

zbMATH: 0616.60079
MathSciNet: MR877606
Digital Object Identifier: 10.1214/aop/1176992272

Primary: 60J80
Secondary: 60G42

Keywords: Asymptotic behaviour , central limit theorem , Controlled branching processes , Discrete time stochastic growth , Martingales , subexponential growth

Rights: Copyright © 1987 Institute of Mathematical Statistics

Vol.15 • No. 1 • January, 1987
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