Comparison for upper tail probabilities of random 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 $\mathbb{P}\{{\sum }_{n=1}^{\infty }{a}_n|{\xi }_n{|}^p\ge r\}$ and $\mathbb{P}\{{\sum }_{n=1}^{\infty }{b}_n|{\xi }_n{|}^p\ge r\}$ as $r\to \infty$ 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.

13 pages