A central limit theorem for stochastic recursive sequences of topical operators

Glenn Merlet

doi:10.1214/105051607000000168

paper

A central limit theorem for stochastic recursive sequences of topical operators

arXiv:math/0606668 · doi:10.1214/105051607000000168

Abstract

Let $(A_{n})_{n \in N}$ 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 $x (n + 1, x_{0}) = A_{n} x (n, x_{0})$ . It can model a wide range of systems including train or queuing networks, job-shop, timed digital circuits or parallel processing systems. When $(A_{n})_{n \in N}$ has the memory loss property, $(x (n, x_{0}))_{n \in N}$ satisfies a strong law of large numbers. We show that it also satisfies the CLT if $(A_{n})_{n \in N}$ 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.

Published at http://dx.doi.org/10.1214/105051607000000168 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)