Stable laws and products of positive random matrices

TL;DR

This paper investigates the asymptotic behavior of products of positive random matrices, establishing conditions under which the logarithms of their entries, norms, and spectral radii follow stable laws, extending understanding of multiplicative stochastic processes.

Contribution

It provides new conditions on the distribution of positive random matrices ensuring their product's logarithmic characteristics converge to stable laws, advancing the theory of random matrix products.

Findings

Logarithms of entries, norms, and spectral radii are in the domain of attraction of stable laws.

Conditions on the distribution of the initial matrix ensure stable law convergence.

The contraction property plays a key role in the analysis.

Abstract

Let $S$ be the multiplicative semigroup of $q\times q$ matrices with positive entries such that every row and every column contains a strictly positive element. Denote by $(X_n)_{n\geq1}$ a sequence of independent identically distributed random variables in $S$ and by $X^{(n)} = X_n ... X_1$, $ n\geq 1$, the associated left random walk on $S$. We assume that $(X_n)_{n\geq1}$ verifies the contraction property $\P(\bigcup_{n\geq1}[X^{(n)} \in S^\circ])>0$, where $S^\circ $ is the subset of all matrices which have strictly positive entries. We state conditions on the distribution of the random matrix $X_1$ which ensure that the logarithms of the entries, of the norm, and of the spectral radius of the products $X^{(n)}$, $n\ge 1$, are in the domain of attraction of a stable law.