Turning a coin over instead of tossing it

Abstract

Given a sequence of numbers $\{p_n\}$ 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 $p_n$, $n\ge 2$. What can we say about the distribution of the empirical frequency of heads as $n\to\infty$? We show that a number of phase transitions take place as the turning gets slower (i.e. $p_n$ 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 $p_n=\text{const}/n$. Among the scaling limits, we obtain Uniform, Gaussian, Semicircle and Arcsine laws.