A central limit theorem for stochastic recursive sequences of topical operators

Glenn Merlet (1)

1 Mathematics department

Let $\left(A_n\right)_{n\in\N}$ be a sequence of stationary topical (i.e. isotone and additively homogeneous) operators. Let $x(n,x_0)$ be defined by $x(0,x_0)=x_0$ and $x(n+1,x_0)=A_nx(n,x_0)$. This can modelize a wide range of systems including, train or queuing networks, job-shop, timed digital circuits or parallel processing systems. When $\left(A_n\right)_{n\in\N}$ has the memory loss property, $\left(x(n,x_0)\right)_{n\in\N}$ satisfy a strong law of large numbers. We show that it also satisfy the CLT if $\sAn$ satisfy 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.

Mots clés: CLT; central limit theorem; topical functions; max-plus; mixing; stochastic recurrent sequences; products of random matrices

Domaines: Probabilités [math.PR]; Optimisation et contrôle [math.OC]

Glenn Merlet. A central limit theorem for stochastic recursive sequences of topical operators. 2007. ⟨hal-00082113v2⟩