A central limit theorem for stochastic recursive sequences of topical operators

Abstract

Let (A_n)_n∈ℕ be a stationary sequence of topical (i.e., isotone and additively homogeneous) operators. Let x(n, x₀) be defined by x(0, x₀)=x₀ and x(n+1, x₀)=A_nx(n, x₀). 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∈ℕ has the memory loss property, (x(n, x₀))_n∈ℕ satisfies a strong law of large numbers. We show that it also satisfies the CLT if (A_n)_n∈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.