An ergodic result for the limit of a sequence of substochastic products xA₁…A_k

Abstract

Let P and Q be n × n nonnegativc matrices with PQ. Let w be an n-dimensional nonnegaiive vector and set S_k (u) = {uA ₁ … A_k over all substochastic A ₁ with P  A ₁  Q for all i} This paper gives conditions under which the sequence S ₁(w)S ₂(w), has a limit set S. Further, these same conditions are sufficient to guarantee that if u and z are stochastic n-dimensional vectors then the sequences S ₁(w)S ₂(w),… and S ₁(z)S ₂(z),… have the same limit set. Hence, an ergodic type result is obtained for this limit.