Turning a Coin over Instead of Tossing It

Given a sequence of numbers $$(p_n)_{n\ge 2}$$ ( p n ) 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 $$p_n$$ p n , $$n\ge 2$$ n ≥ 2 , independently of the sequence of the previous terms. What can we say about the distribution of the empirical frequency of heads as $$n\rightarrow \infty $$ n → ∞ ? We show that a number of phase transitions take place as the turning gets slower (i. e., $$p_n$$ 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 $$p_n=\text {const}/n$$ p n = const / n . Among the scaling limits, we obtain uniform, Gaussian, semicircle, and arcsine laws.