Recurrence and transience of Rademacher series

Abstract

We introduce the notion of {\bf a}-walk $S(n)=a_1{X}_1+\cdots +a_n{X}_n$, based on a sequence of positive numbers $\mathbf{a}=(a_1,a_2,\dots )$ and a Rademacher sequence $X_1,X_2,\dots$. We study recurrence/transience (properly defined) of such walks for various sequences of $\mathbf{a}$. 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 $\gamma >1$.