Recurrence, transience and anti-concentration of Rademacher random walks

TL;DR

The paper investigates when one-dimensional Rademacher walks with deterministic, nonnegative step sizes are recurrent or transient, and how anti-concentration of sums governs these behaviors. It develops sharp anti-concentration bounds for inhomogeneous sums, including modular and scale-merged estimates, and uses these tools to derive transience results for growth rates $a_n=n^{eta+o(1)}$ with $eta>1/2$, while showing tightness via $eta=1/2$ examples that can be weakly recurrent. It also constructs explicit recurrent and topologically recurrent walks with even faster growth, and provides comprehensive results for slowly growing, sparsely supported, and all-covering sequences, including coupling arguments and two-dimensional RW reductions. Altogether, the work clarifies the delicate boundary between recurrence and transience for inhomogeneous Rademacher walks and highlights anti-concentration as a central mechanism. These contributions advance understanding of how step-size growth shapes long-term behavior in inhomogeneous random walks and related anti-concentration phenomena.

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 $a_n = n^{α+ o(1)}$ for some $α> 1/2$, then the walk is transient. We show that the bound on $α$ is tight by giving an example where $a_n = Θ(n^{1/2})$ and the walk is weakly recurrent.

Theorems & Definitions (64)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Corollary 5
• Theorem 6
• proof : Proof of Theorem \ref{['thm:transient']} from Theorem \ref{['thm:anti-concentration']}
• Proposition 6
• Lemma 6