## Limit theorems for products of positive random matrices.(English)Zbl 0903.60027

Summary: 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 $$S^0$$ the subset of these matrices, all entries of which are strictly positive. Consider a random ergodic sequence $$(X_n)_{n\geq 1}$$ in $$S$$. The aim of this paper is to describe the asymptotic behavior of the random products $$X^{(n)}= X_n\cdots X_1$$, $$n\geq 1$$, under the main hypothesis $$P\left(\bigcup_{n\geq 1}[X^{(n)}\in S^0]\right)> 0$$. 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.

### MSC:

 60F99 Limit theorems in probability theory 60F05 Central limit and other weak theorems

### Keywords:

central limit theorem; tightness
Full Text:

### References:

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