A Phase Transition in Random coin Tossing

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 μ θ be the distribution of the observed sequence of coin tosses, and let u n denote the chance of a renewal at time n. Harris and Keane showed that if Σ∞ n=1 u 2 n = ∞, then μ θ and μ 0 are singular, while if Σ∞ n=1 u 2 n θ c , they are singular. We also prove that when u n = O(n -1 ), the measures μ θ for 0 ∈ [-1,1] are all mutually absolutely continuous.

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