Comparison for upper tail probabilities of random series

Let {ξn} be a sequence of independent and identically distributed random variables. In this paper we study the comparison for two upper tail probabilities P{∑n=1∞an∣ξn∣p≥r} and P{∑n=1∞bn∣ξn∣p≥r} as r→∞ with two different real series {an} and {bn}. The first result is for Gaussian random variables {ξn}, 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.