Turning a Coin over Instead of Tossing It

Abstract

Given a sequence of numbers (pn)n≥2 in [0, 1], consider the following experiment. First, we flip a fair coin and then, at step n, we turn the coin over to the other side with probability pn, n≥ 2 , independently of the sequence of the previous terms. What can we say about the distribution of the empirical frequency of heads as n→ ∞? We show that a number of phase transitions take place as the turning gets slower (i. e., pn is getting smaller), leading first to the breakdown of the Central Limit Theorem and then to that of the Law of Large Numbers. It turns out that the critical regime is pn= const / n. Among the scaling limits, we obtain uniform, Gaussian, semicircle, and arcsine laws.