Law of Large Numbers for products of random matrices with coefficients in the max-plus semi-ring.
type: Distribution
access URL: https://hal.science/hal-00085782
format: HTML
issued: 2026-05-09T21:31:41.577647
modified: 2026-05-09T21:31:41.559761

Law of Large Numbers for products of random matrices with coefficients in the max-plus semi-ring.
type: Dataset
contact point: Organization
  fn: CCSD
keyword: discrete-event-systems
keyword: law-of-large-numbers
keyword: products-of-random-matrices
keyword: infoeu-reposemanticspreprint
keyword: msc--60f15-90b15-93c65-93e15
keyword: stochastic-recurrent-sequences
keyword: mathmath-prmathematics-mathprobability-mathpr
keyword: preprints-working-papers-
keyword: max-plus
landing page: https://hal.science/hal-00085782
description: 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 $-\infty$.
issued: 2026-05-09T21:31:41.568727
language: en
modified: 2026-05-09T21:31:41.568733

test_moissonnage_selune
type: Agent