Z. Wahrscheinlichkeitstheorieverw. Gebiete 61, 83-95 (1982) Zeitschrift fur Wahrscheinlichkeitstheorie und verwandte Gebiete 9 Springer-Verlag 1982 Perturbations of Random Matrix Products Yuri Kifer* Institute of Mathematics, Hebrew University,Jerusalem, Israel Summary. If Xt,X2,... are identically distributed independent random matrices with a common distribution # then with the probability 1 the limit A.= lim -lln IIX,... Xtll n~oo F/ exists. The paper treats the problem: is it true that Au~Au if #k--+# in the weak sense? w O. Introduction Let Xt,X2, ... be identically distributed independent random m x m matrices with a common distribution # on the real unimodular group SL(m,R). The theorem of Furstenberg and Kesten (see [3] and [4]) establishes that with the probability 1 there exists A,= lim lln 112.x. 1... 211[ (0.1) n--* oo n provided ~ln [Igl[ #(dg) < oo, (0.2) where A, is a constant depending only on #. Now let {/Zk} be a sequence of probability measures on SL(m, R) satisfying (0.2) such that #k~# in the weak sense as k~oe. (0.3) This paper studies conditions under which (0.3) yields A,,k--+A~, as k--+oo. (0.4) * Supported by a grant from the United States-Israel Binational Sciencc Foundation (BSF), Jerusalem, Israel. AMS Subject classification: 60B15. 0044-3719/82/0061/0083/$02.60