A phase transition in random coin tossing

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dc.creatorLevin, David A.
dc.creatorPemantle, Robin
dc.creatorPeres, Yuval
dc.date2004-04-05
dc.date.accessioned2026-07-07T05:07:09Z
dc.date.available2026-07-07T05:07:09Z
dc.descriptionSuppose 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_θbe 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 sum_{n=1}^infty u_n^2=\infty, then mu_theta and μ_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.
dc.description48 pages
dc.identifierhttps://arxiv.org/abs/math/0404098
dc.identifierhttp://arxiv.org/abs/math/0404098
dc.identifierAnn. Probab. vol. 29, 1637-1669 (2001)
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/70745
dc.subjectProbability
dc.subject60G30 (Primary) 60K35 (Secondary)
dc.titleA phase transition in random coin tossing
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