2004/04/05 by David A. Levin, Robin Pemantle, Yuval Peres
Mathematics · #math.PR #msc:60G30 #msc:60K35
published as Ann. Probab. vol. 29, 1637-1669 (2001) · 48 pages
arxiv created 2004/04/05 · arxiv updated 2009/12/01
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 muθbe the distribution of the observed sequence of coin tosses, and let un denote the chance of a renewal at time n. Harris and Keane showed that if sumn=1infty un2=∞, then mutheta and μ0 are singular, while if sumn=1infty un2<infty and theta is small enough, then mutheta is absolutely continuous with respect to mu0. They conjectured that absolute continuity should not depend on theta, 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 thetac in (0,1): for |theta|<thetac the measures mutheta and mu0 are mutually absolutely continuous, but for |theta|>thetac, they are singular. We also prove that when un=O(n-1), the measures mutheta for theta in [-1,1] are all mutually absolutely continuous.