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October 2001 A Phase Transition in Random coin Tossing
David A. Levin, Robin Pemantle, Yuval Peres
Ann. Probab. 29(4): 1637-1669 (October 2001). DOI: 10.1214/aop/1015345766

Abstract

Suppose that a coin with bias $\theta$ is tossed at renewal times of a renewal process, and a fair coin is tossed at all other times. Let $\mu_\theta$ be the distribution ofthe observed sequence of coin tosses, and let $u_n$ denote the chance of a renewal at time $n$. Harris and Keane showed that if $\sum_{n=1}^\infty u_n^2 = \infty$ then $\mu_\theta$ and $\mu_0$ are singular, while if $\sum_{n=1}^\infty u_n^2 < \infty$ and $\theta$ is small enough, then $\mu_\theta$ is absolutely continuous with respect to$\mu_0$. They conjectured that absolute continuity should not depend on $\theta$, but only on the square-summability of $\{u_n\}$. We show that in fact the power law governing the decay of $\{u_n\}$ is crucial, and for some renewal sequences $\{u_n\}$, there is a phase transition at a critical parameter $\theta_c \in (0,1):$ for $|\theta| < \theta_c$ the measures $\mu_\theta$ and $\mu_0$ are mutually absolutely continuous, but for $|\theta| > \theta_c$, they are singular. We also prove that when $u_n=O(n ^{-1}), the measures $\mu_\theta$ for $\theta \in [-1,1] are all mutually absolutely continuous.

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David A. Levin. Robin Pemantle. Yuval Peres. "A Phase Transition in Random coin Tossing." Ann. Probab. 29 (4) 1637 - 1669, October 2001. https://doi.org/10.1214/aop/1015345766

Information

Published: October 2001
First available in Project Euclid: 5 March 2002

zbMATH: 1018.60043
MathSciNet: MR1880236
Digital Object Identifier: 10.1214/aop/1015345766

Subjects:
Primary: 60G30
Secondary: 60H35

Keywords: Kakutani’s dichotomy , mutually singular measures , Phase transitions , Random walks , renewal sequences , scenery with noise

Rights: Copyright © 2001 Institute of Mathematical Statistics

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Vol.29 • No. 4 • October 2001
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