  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.
