Description
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 $p_n, n>1$, independently of the sequence of the previous terms. 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 $p_n=const/n$. Among the scaling limits, we obtain Uniform, Gaussian, Semicircle and Arcsine laws.
The critical regime is particularly interesting: when the corresponding random walk is considered, an interesting process emerges as the scaling limit; also, a connection with Polya urns will be mentioned.
This is joint work with S. Volkov (Lund) and Z. Wang (Boulder). See also the paper https://drive.google.com/file/d/0B4ZkCm_J6qB8NXNRcUFPM0hqRUU/view