A short note on small deviations of sequences of i.i.d. random variables with exponentially decreasing weights

Frank Aurzada

doi:10.1016/j.spl.2008.02.007

paper

A short note on small deviations of sequences of i.i.d. random variables with exponentially decreasing weights

arXiv:0711.2576 · doi:10.1016/j.spl.2008.02.007

Abstract

We obtain some new results concerning the small deviation problem for $S = \sum_{n} q^{n} X_{n}$ and $M = sup_{n} q^{n} X_{n}$ , where $0 < q < 1$ and $(X_{n})$ are i.i.d. non-negative random variables. In particular, the asymptotics is shown to be the same for $S$ and $M$ in some cases.

Revision, 13p, to appear in: Statist. Probab. Lett