Comparison for upper tail probabilities of random series

Let $\{ξ_n\}$ be a sequence of independent and identically distributed random variables. In this paper we study the comparison for two upper tail probabilities $\mathbb{P}\{\sum_{n=1}^{\infty}a_n|ξ_n|^p\geq r\}$ and $\mathbb{P}\{\sum_{n=1}^{\infty}b_n|ξ_n|^p\geq r\}$ as $r\rightarrow\infty$ with two different real series $\{a_n\}$ and $\{b_n\}.$ The first result is for Gaussian random variables $\{ξ_n\},$ 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.

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