Stable laws and products of positive random matrices

Details Stable laws and products of positive random matrices Stable laws and products of positive random matrices

2008-01-24

arxiv.org/abs/0801.3780v1

Mathematics

Probability

14 pages. To appear in Journal of Theoretical Probability

Scientific paper

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.

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