| A phase transition in random coin tossing (2007) | |||||||||||||||
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 in [10] showed that if n=1 u2 n = ∞, then µθ and µ0 are singular, while if � ∞ n=1 u2 n < ∞ and θ is small enough, then µθ is absolutely continuous with respect to µ0. They conjectured that absolute continuity should not depend on θ, but only on the square-summability of {un}. We show that in fact the power law governing the decay of {un} is crucial, and for some renewal sequences {un}, there is a phase transition at a critical parameter θc ∈ (0, 1): for |θ | < 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. | |||||||||||||||
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