On the lower tail probabilities of some random series

The behavior of tail probabilities P{S less than or equal to r}, r --> 0 is investigated, where S is a series S = Sigma lambda(j)Z(j) generated by some sequence of positive numbers {lambda(j)} and by a sequence {Z(j)} of independent copies of a positive random variable Z. We present the exact asymptotic expression for P{S less than or equal to r} by means of Laplace transform Lambda(gamma) = E exp{-gamma S} under weak assumptions on the behavior of the tail probabilities of Z in the vicinity of zero. The bounds of accuracy are also given, and under weak supplementary smoothness conditions the asymptotic properties of the density of S are investigated.