Recurrence and transience of Rademacher series

TL;DR

This paper investigates the recurrence and transience properties of a generalized class of Rademacher series, called a-walks, for various growth sequences of coefficients, providing classifications for polynomial and logarithmic cases.

Contribution

It introduces the concept of a-walks and classifies their recurrence or transience for specific growth sequences of coefficients, extending understanding of Rademacher series behavior.

Findings

Classified recurrence/transience for polynomial sequences a_k=⌊k^β⌋.

Analyzed behavior for logarithmic sequences a_k=⌈log_γ k⌉ and a_k=log_γ k.

Established criteria for recurrence and transience based on sequence growth.

Abstract

We introduce the notion of {\bf a}-walk $S(n)=a_1 X_1+\dots+a_n X_n$, based on a sequence of positive numbers ${\bf a}=(a_1,a_2,\dots)$ and a Rademacher sequence $X_1,X_2,\dots$. We study recurrence/transience (properly defined) of such walks for various sequences of ${\bf a}$. In particular, we establish the classification in the cases where $a_k=\lfloor k^\beta\rfloor$, $\beta>0$, as well as in the case $a_k=\lceil \log_\gamma k \rceil$ or $a_k=\log_\gamma k$ for $\gamma>1$.