Turning a coin over instead of tossing it

TL;DR

This paper investigates the asymptotic behavior of empirical head frequencies in a coin-flipping experiment with a variable probability of flipping the coin over, revealing phase transitions and diverse limiting distributions.

Contribution

It identifies phase transitions in the empirical frequency distribution based on the decay rate of the flipping probability sequence, including the critical case p_n=const/n.

Findings

Breakdown of the Central Limit Theorem in certain regimes

Breakdown of the Law of Large Numbers in slower regimes

Various limiting laws such as Uniform, Gaussian, Semicircle, and Arcsine

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.