Comparison theorems for small deviations of weighted series

We study comparison theorems for small deviation probabilities of weighted series and obtain more refined versions of the known comparison results. In particular, the following consequence is obtained immediately from Theorem 2.1 of the paper. Let a positive random variable X belong to the domain of attraction of a stable law with an index greater than one and let its distribution function be regularly varying at zero with an exponent β > 0. If {Xn}n­1 are independent copies of X, and {an} and {bn} are positive summable sequences such that..[formula].