This article is cited in 32 papers

On the products of random matrices and operators

A. D. Vircer

Moscow

Abstract: Let $\xi_1,\xi_2,\dots$ be a stationary ergodic Markovian process on a measurable space $\Xi$ and $X$ be a measurable mapping of $\Xi$ into the group $SL(m,R)$. We prove that, under some conditions, the norm of the product
$$ X(\xi_1)X(\xi_2)\dots X(\xi_n) $$
of random unimodular matrices grows exponentially with probability 1 (Theorem 1). The proof is based on some facts from the theory of unitary representations of the group $SL(m,R)$ and on the theorem on the exponential decrease of the mean of the product of random unitary operators on a separable Hilbert space (Theorem 2).

Received: 10.01.1977

 English version: Theory of Probability and its Applications, 1979, 24:2, 367–377

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