Continuous Singularity of the Weak Limit of Convolution Powers of a Discrete Probability Measure on 2 × 2 Stochastic Matrices

Abstract

Let μ be a probability measure on n 2 × 2 stochastic matrices, n an arbitrary positive integer, and λ= (w) lim n→∞ μ n , such that the support of λ consists of 2 × 2 stochastic matrices of rank one, and as such, λ can be regarded as a probability measure on [0, 1]. We present simple sufficient conditions for λ to be continuous singular w.r.t. the Lebesgue measure on [0, 1]. We also determine λ, given μ.