Noncommuting Random Products

HARRY FURSTENBERG(i)Introduction.Let Xy,X2, -,X",---be a sequence of independent real valued random variables with a common distribution function F(x), and consider the sums Xy + X2 + + X".A fundamental theorem of classical probability theory is the strong law of large numbers which asserts that with probability one, Xy + X2+ + Xn ~ n $xdF(x), provided that J"| x\ dF(x) is finite.It is natural to inquire whether there exist laws governing the asymptotic behavior of products X"X"-y ---Xy, where the X are now identically distributed independent random variables with values in an arbitrary group.This type of question arises, for example, in considering solutions to differential or difference equations with random coefficients [13].To illustrate this, consider the problem of determining the asymptotic behavior of a random sequence {<!;"} satisfying n = ut&n-i + ci!-2j where (u",v") forms a sequence of independent identically distributed random vectors.In this case we can write n + 11 -XX X fi1] x -h+1 VJ+1 so the rate of growth of the " is governed by the behavior of the matrix product X"Xn_y--Xy.Bellman, who apparently was the first to consider questions of this kind, studied random products of 2 x 2 matrices with strictly positive entries [2].He showed that in a certain sense the weak law of large numbers holds for the entries of the matrices X"X"_y -Xy.In [9] it was shown that a strong law of large numbers is valid.More precisely, if y is the typical entry of XnX" _y--Xy, then for a certain constant a, n ~1 log y -* a with probability 1, provided the entries of X are positive and bounded away from oo and 0 in an appropriate sense.It should be emphasized that for matrices with arbitrary entries, this type of result does not hold and the analysis of [9] breaks down.In the present investigation we shall consider a noncompact semi-simple Lie group G and independent G-valued random variables {Xn} with a common distribution p on G.The law of large numbers here can be given the following form.We shall exhibit a finite dimensional linear space "VG of functions \/(g)