Comparison for upper tail probabilities of random series

TL;DR

This paper compares the asymptotic behavior of upper tail probabilities for infinite series of random variables with different coefficient sequences, establishing equivalence results for Gaussian variables and weaker comparisons for general distributions.

Contribution

It provides new asymptotic comparison results for tail probabilities of infinite series of random variables with different coefficients, including Gaussian and more general cases.

Findings

Gaussian case: tail probabilities are equivalent after scaling.

General case: tail probabilities are comparable at a logarithmic level.

Results extend understanding of tail behavior in infinite series.

Abstract

Let $\{\xi_n\}$ be a sequence of independent and identically distributed random variables. In this paper we study the comparison for two upper tail probabilities $\mathbb{P}\{\sum_{n=1}^{\infty}a_n|\xi_n|^p\geq r\}$ and $\mathbb{P}\{\sum_{n=1}^{\infty}b_n|\xi_n|^p\geq r\}$ as $r\rightarrow\infty$ with two different real series $\{a_n\}$ and $\{b_n\}.$ The first result is for Gaussian random variables $\{\xi_n\},$ and in this case these two probabilities are equivalent after suitable scaling. The second result is for more general random variables, thus a weaker form of equivalence (namely, logarithmic level) is proved.