# Turning a Coin over Instead of Tossing It

Journal of Theoretical Probability, Nov 2016

Given a sequence of numbers $(p_n)_{n\ge 2}$ 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 $p_n$, $n\ge 2$, independently of the sequence of the previous terms. What can we say about the distribution of the empirical frequency of heads as $n\rightarrow \infty$? 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.

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Turning a Coin over Instead of Tossing It, Journal of Theoretical Probability, 2016, DOI: 10.1007/s10959-016-0725-1