On small deviations of series of weighted random variables

Let xi,xi (1),xi (2),... be positive i.i.d. random variables, S=Sigma (infinity)(j=1)a(j)xi(j) , where the coefficients a(j) >= 0 are such that P(S < infinity)=1. We obtain an explicit form of the asymptotics of -ln P(S < x) as x -> 0 for the following three cases: (i) the sequence {a(j)} is regularly varying with exponent -beta < -1, and -ln P(xi < x) = O(x(-gamma+delta)) as x -> 0 for some delta > 0, where gamma = 1/(beta-1), (ii) -ln P(xi < x) is regularly varying with exponent -gamma < 0 as x -> 0, and a(j)=O(j(-beta-delta)) as j -> infinity for some delta > 0, where gamma = 1/(beta-1), (iii) {a(j)} decreases faster than any power of j, and P(xi < x) is regularly varying with positive exponent as x -> 0.