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

TL;DR

This paper investigates the small deviation probabilities for weighted sums and maxima of i.i.d. non-negative random variables with exponentially decreasing weights, revealing cases where their asymptotics coincide.

Contribution

It presents new asymptotic results for small deviations of weighted sums and maxima, highlighting cases of equivalence in their asymptotic behavior.

Findings

Asymptotics for small deviations of $S$ and $M$ are identical in certain cases.

Provides new theoretical insights into weighted sums of i.i.d. variables.

Extends understanding of small deviation problems with exponential weights.

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.