Recurrence, transience and anti-concentration of Rademacher random walks

TL;DR

This paper investigates the recurrence and transience of one-dimensional Rademacher random walks with various step size sequences, establishing conditions under which the walk is recurrent or transient, and demonstrating the sharpness of these conditions.

Contribution

It characterizes the recurrence and transience of Rademacher walks based on step size growth, extending previous work and providing new constructions for different growth rates.

Findings

Bounded step sizes lead to weak recurrence.

Step sizes tending to infinity can still produce weak recurrence.

For step sizes growing faster than n^{1/2}, the walk is transient.

Abstract

The Rademacher random walk associated with a deterministic sequence $(a_n)_{n \geq 1}$ is the walk which starts at zero and, at step $i$, independently steps either up or down by $a_i$ with equal probability. We continue the study begun by Bhattacharya and Volkov in 2023 of the transience or recurrence of one-dimensional Rademacher random walks. In particular, we show that if the sequence of step sizes is bounded, the walk is weakly recurrent, meaning that it returns infinitely often to a random finite interval, while if the step sizes tend to infinity arbitrarily slowly, the walk may be transient. On the other hand, using a construction with integer step sizes, we show that the step sizes may grow arbitrarily fast and still give a weakly recurrent random walk. We also show, using a construction with non-integer step sizes, that the same conclusion holds even if we restrict to strictly increasing step sizes. However, we prove that if $a_n = n^{\alpha + o(1)}$ for some $\alpha > 1/2$, then the walk is transient. We show that the bound on $\alpha$ is tight by giving an example where $a_n = \Theta(n^{1/2})$ and the walk is weakly recurrent.