Limit distributions of random walks on stochastic matrices

Problems similar to Ann. Prob. 22 (1994) 424-430 and J. Appl. Prob. 23 (1986) 1019-1024 are considered here. The limit distribution of the sequence XX ... X, where (X) is a sequence of i.i.d. 2 × 2 stochastic matrices with each X distributed as μ, is identified here in a number of discrete situations. A general method is presented and it covers the cases when the random components C and D (not necessarily independent), (C, D) being the first column of X, have the same (or different) Bernoulli distributions. Thus (C, D) is valued in {0, r}, where r is a positive real number. If for a given positive real r, with 0 < r ≤ 1/2 , rC and rD are each Bernoulli with parameters p and p respectively, 0 < p, p < 1 (which means C ~ p + (1 - p) and D ~ p + (1 - p)), then it is well known that the weak limit λ of the sequence μ exists whose support is contained in the set of all 2 × 2 rank one stochastic matrices. We show that S(λ), the support of λ, consists of the end points of a countable number of disjoint open intervals and we have calculated the λ-measure of each such point. To the best of our knowledge, these results are new.