The Annals of Probability

Limit theorems for products of positive random matrices

H. Hennion

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Let $S$ be the set of $q \times q$ matrices with positive entries, such that each column and each row contains a strictly positive element, and denote by $S^\circ$ the subset of these matrices, all entries of which are strictly positive. Consider a random ergodic sequence $(X_n)_{n \geq1}$ in $S$. The aim of this paper is to describe the asymptotic behavior of the random products $X^{(n)} =X_n \ldots X _1, n\geq 1$ under the main hypothesis $P(\Bigcup_{n\geq 1}[X^{(n)}\in S^\circ])>0$. We first study the behavior “in direction” of row and column vectors of $X^{(n)}$. Then, adding a moment condition, we prove a law of large numbers for the entries and lengths of these vectors and also for the spectral radius of $X^{(n)}$ . Under the mixing hypotheses that are usual in the case of sums of real random variables, we get a central limit theorem for the previous quantities. The variance of the Gaussian limit law is strictly positive except when $(X^{(n)})_{n\geq 1}$ is tight. This tightness property is fully studied when the $X_n, n\geq 1$, are independent.

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Ann. Probab. Volume 25, Number 4 (1997), 1545-1587.

First available: 7 June 2002

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Primary: 60F99: None of the above, but in this section 60F05: Central limit and other weak theorems

Positive random matrices mixing limit theorems


Hennion, H. Limit theorems for products of positive random matrices. The Annals of Probability 25 (1997), no. 4, 1545--1587. doi:10.1214/aop/1023481103.

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