Title

Recurrence, transience and anti-concentration of Rademacher walks.

Abstract

Joint work with Satyaki Bhattacharya (Lund PhD student) and Tom Johnston (University of Bristol).

Suppose you start with a fortune X0 = 0 and make a sequence of bets of different (deterministic) sizes a1, a2, a3, ..., on the outcome of a fair coin toss. In the nth bet, your fortune Xn increases by an if the coin shows heads, and it decreases by an if the coin shows tails.  Your fortune can turn negative, but you have an unlimited overdraft so you always can keep betting.  Does your  fortune keep returning close to 0, infinitely often?  In other words, is  lim inf |Xn| finite with probability 1?  The answer depends on the sequence a = (a1, a2, a3,...) in an interesting way.

Surprisingly, this very classical-sounding probability problem has not received much attention. As far as we know, it was first investigated by Stas Volkov and Satyaki Bhattacharya in a 2023 paper. I will describe several new results about this question, just posted at arXiv:2510.24568, and a few open problems.