Convergence and Precise Asymptotics for Series Involving Self-normalized Sums

Abstract

Let $X, X_{1}, X_{2}, \ldots$ be i.i.d. random variables, and set $S_{n}=X_{1}+\cdots +X_{n}$ and $V_{n}^{2}=X_{1}^{2}+\cdots +X_{n}^{2}.$ Without any moment conditions on $X$, assuming that $\{S_{n}/V_{n}\}$ is tight, we establish convergence of series of the type (*) $\sum \nolimits _{n}w_{n}P(\left| S_{n}\right| /V_{n}\ge \varepsilon b_{n}),$ $\varepsilon >0.$ Then, assuming that $X$ is symmetric and belongs to the domain of attraction of a stable law, and choosing $w_{n}$ and $b_{n}$ suitably$,$ we derive the precise asymptotic behavior of the series (*) as $\varepsilon \searrow 0.$