Title: Coin turning and the walk it generates
Speaker: Janos Englander (University of Colorado Boulder)
Online Transmission: https://meet.google.com/haf-
Abstract: 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 $n\to\infty$?
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.
If time permits I will also discuss the random walk that coin turning generates. Here each step is 1 or -1 according to what the coin shows. In the unlikely case we have even more time, I will discuss the higher dimensional analogs of the walk.
This is joint work with Stas Volkov (Lund).
All the talks are held in English.
More complete information about the seminars can be found at
http://www.dme.ufrj.br/?page_id=3481
Sincerely,
Organizers: Giulio Iacobelli and Maria Eulalia Vares
