A phase transition in random coin tossing by

Abstract

Suppose that a coin with bias θ is tossed at renewal times of a renewal process, and a fair coin is tossed at all other times. Let μθ be the distribution of the observed sequence of coin tosses, and let un denote the chance of a renewal at time n. Harris and Keane showed that if Σ∞n=1 u2n = ∞, then μθ and μ0 are singular, while if Σ∞n=1 < θc, the measures μθ and μ0 are mutually absolutely continuous, but for |θ| > θC, they are singular. We also prove that when un = O(n-1), the measures μθ for θ ∈ [-1, 1] are all mutually absolutely continuous.