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

Hennion, H.

Ann. Probab. 25, No. 4, 1545-1587 (1997).

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

References:

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