A Phase Transition in Random coin Tossing

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.