Ergod. Th. & Dynam. Sys. (1982), 2, 367-382 Printed in Great Britain Perturbations of random matrix products in a reducible case YURI KIFER AND ERIC SLUD Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel; and Department of Mathematics, University of Maryland, College Park, Maryland 20742, USA {Received 5 May 1982 and revised 15 October 1982) Dedicated to the memory of V. M. Alexeyev Abstract. It is known that for any sequence X\, X 2 ..., of identically distributed independent random matrices with a common distribution fj, the limit exists with probability 1. We study some conditions under which A(/^ fc )-» A(/u) provided Hk -* M in the weak sense. 1. Introduction Let Xi, X 2 ,... be a sequence of identically distributed independent random mx.m real matrices with common distribution /JL on the unimodular group SL (m,R). Under the assumption that (1.1) Furstenberg and Kesten [6] showed that A( M )= limn" 1 log||AT n • • • ATiH (1.2) n-»oo exists with probability 1 and is almost surely (a.s.) constant. Because of the applications of random matrix products to physical and to population processes (e.g., see [3] and [8]), it is of interest to understand when A(/i) is stable under perturbations of /x, say, in the weak topology of measures. In the case when the support of /u, is irreducible (in the sense that the minimal closed subgroup of SL (m, R) containing the support of /x leaves no proper subspace of R m invariant) Kifer [9] has applied Furstenberg's formula [7] to show that if ii k converges weakly to fi (fj, k %• n) then A(/x k ) -* A((JL ) as k -> oo. In the present paper, we consider the quite different case of \x supported on a reducible subgroup of SL (m, R) and prove under certain assumptions on n and /x fc that Aik-^/Li implies A(/x fc )-» A(/a). Slud [12] had previously shown A(/u,jc)-» A(/A)