Recurrence, transience and anti-concentration of Rademacher random walks

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 into 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, we show that the step sizes may grow arbitrarily fast and still give a weakly recurrent random walk, and this is still true even if we restrict to non-decreasing step sizes. However, if $a_n = n^{\alpha + o(1)}$ for some $\alpha > 1/2$, we show that the walk is transient. We also 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.