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.
Categories: math.PR