A central limit theorem for stochastic recursive sequences of topical operators

Glenn Merlet

Published 2006-06-27, updated 2007-10-30Version 3

Let $(A_n)_{n\in\mathbb{N}}$ be a stationary sequence of topical (i.e., isotone and additively homogeneous) operators. Let $x(n,x_0)$ be defined by $x(0,x_0)=x_0$ and $x(n+1,x_0)=A_nx(n,x_0)$. It can model a wide range of systems including train or queuing networks, job-shop, timed digital circuits or parallel processing systems. When $(A_n)_{n\in\mathbb{N}}$ has the memory loss property, $(x(n,x_0))_{n\in\mathbb{N}}$ satisfies a strong law of large numbers. We show that it also satisfies the CLT if $(A_n)_{n\in \mathbb{N}}$ fulfills the same mixing and integrability assumptions that ensure the CLT for a sum of real variables in the results by P. Billingsley and I. Ibragimov.