Law of Large Numbers for products of random matrices with coefficients in the max-plus semi-ring.
Résumé
We analyze the asymptotic behavior of random variables $x(n,x_0)$ defined by $x(0,x_0)=x_0$ and $x(n+1,x_0)=A(n)x(n,x_0)$, where $\sAn$ is a stationary and ergodic sequence of random matrices with entries in the semi-ring \mbox{$\R\cup\{-\infty\}$} 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 $\left(\frac{1}{n}x(n,x_0)\right)_{n\in\N}$ to converge almost surely. Then, we prove a general scheme to give partial converse theorems. When $\max_{A_{ij}(0)\neq -\infty}|A_{ij}(0)|$ is integrable, it allows us: - to give a necessary and sufficient condition for the convergence of $\left(\frac{1}{n}x(n,0)\right)_{n\in\N}$ when the sequence $\left(A(n) \right)_{n\in\N}$ is i.i.d., - to prove that, if $\left(A(n) \right)_{n\in\N}$ satisfy a condition of reinforced ergodicity and a condition of fixed structure (i.e. $\P\left(A_{ij}(0)=-\infty\right)\in\{0,1\}$), then $\left(\frac{1}{n}x(n,0)\right)_{n\in\N}$ converges almost-surely, - and to reprove the convergence of $\left(\frac{1}{n}x(n,0)\right)_{n\in\N}$ if the diagonal entries are never $-\infty$.
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