Ju n 20 06 A central limit theorem for stochastic recursive sequences of topical operators . February 17 , 2008 ’ s Version Glenn MERLET

Let (A n) n∈N be a sequence of stationary topical (i.e. isotone and ad-ditively 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). This can modelize 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, x 0)) n∈N satisfy a strong law of large numbers. We show that it also satisfy the CLT if (A n) n∈N 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. This article is based on the work by H. Ishitani for products of random positive matrices. d is called additively homogeneous if it satisfies A(x + a1) = A(x) + a1 for all x ∈ R d and a ∈ R, where 1 is the vector (1, · · · , 1) ′ in R d. It is called isotone if x ≤ y implies A(x) ≤ A(y), where the order is the product order on R d. It is called topical if it is isotone and homogeneous. The set of topical operators on R d will be denoted by T op d. We recall that the action of matrices with entries in the semi-ring, sometimes called (max, +)-algebra, R max = (R ∪ {−∞}, max, +), on R d max is defined by (Ax) i = max j (A ij + x j). When matrix A has no line of −∞, the restriction of this action to R d defines a topical operator, also denoted by A. Such operators are called (max, +) operators and composition of operators corresponds to the product of matrices in the (max, +) semi-ring.