Rademacher Random Walks
Probability Seminar
5th December 2025, 3:30 pm – 4:30 pm
Fry Building, 2.04
Suppose you start with a fortune X0 = 0 and make a sequence of bets of different (deterministic) sizes a1, a2, a3, ..., on the outcomes of a sequence of fair coin tosses. 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, and for some simple sequences we do not yet know the answer.
Surprisingly, this very classical-seeming 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.
Joint work with Satyaki Bhattacharya (Lund PhD student) and Tom Johnston (University of Bristol).
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