Limit distributions of random walks on stochastic matrices

Abstract

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 X _n X _n−1⋯X ₁, where (X _n ) _{n ≥ 1} is a sequence of i.i.d. 2 × 2 stochastic matrices with each X _n 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 _n and D _n (not necessarily independent), (C _n , D _n ) being the first column of X _n , have the same (or different) Bernoulli distributions. Thus (C _n , D _n ) is valued in {0, r}², where r is a positive real number. If for a given positive real r, with $0<r\leq \frac {1}{2}$, r ⁻¹ C _n and r ⁻¹ D _n are each Bernoulli with parameters p ₁ and p ₂ respectively, 0 < p ₁, p ₂<1 (which means $C_{n}\sim p_{1}\delta _{\left \{ r\right \} }+(1-p_{1})\delta _{\left \{ 0\right \}}$ and $D_{n}\sim p_{2}\delta _{\left \{ r\right \} }+(1-p_{2})\delta _{\left \{ 0\right \} }$), then it is well known that the weak limit λ of the sequence μ ⁿ 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.