Recurrence and transience of Rademacher series

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 $\gamma>1$.