The Annals of Probability

Limit theorems for products of positive random matrices

H. Hennion

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Abstract

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.

Article information

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

Dates
First available: 7 June 2002

Permanent link to this document
http://projecteuclid.org/euclid.aop/1023481103

Mathematical Reviews number (MathSciNet)
MR1487428

Digital Object Identifier
doi:10.1214/aop/1023481103

Zentralblatt MATH identifier
0903.60027

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

Keywords
Positive random matrices mixing limit theorems

Citation

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. http://projecteuclid.org/euclid.aop/1023481103.


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