Law of Large Numbers for products of random matrices with coefficients in the max-plus
Glenn Merlet · arXiv (Cornell University) · 2008
We analyze the asymptotic behavior of random variables x(n, x0) defined by x(0, x0) = x0 and x(n+1, x0) = A(n)x(n, x0), where (A(n)) n∈N is a stationary and ergodic sequence of random matrices with entries in the semi-ring R ∪ {−∞} whose addition is the max and whose multiplication is +. Such sequences modelize a large class of discrete event systems, among which timed event graphs, 1-bounded Petri nets, some queuing networks, train or computer networks. We give necessary conditions for 1 x(n, x0) � n∈N to converge almost surely. Then, we prove a general scheme to give partial converse theorems. When maxAij(0)6−∞ |Aij(0)| is integrable, it allows us: - to give a necessary and sufficient condition for the convergence of 1 n x(n,0)