On Small Deviations of Series of Weighted Random Variables

Let ξ,ξ ₁,ξ ₂,… be positive i.i.d. random variables, S=∑ _j=1^∞ a(j)ξ _j , where the coefficients a(j)≥0 are such that P(S<∞)=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 −β<−1, and −ln P(ξ<x)=O(x −γ+δ ) as x→0 for some δ>0, where γ=1/(β−1),
(ii)	−ln P(ξ<x) is regularly varying with exponent −γ<0 as x→0, and a(j)=O(j −β−δ ) as j→∞ for some δ>0, where γ=1/(β−1),
(iii)	{a(j)} decreases faster than any power of j, and P(ξ<x) is regularly varying with positive exponent as x→0.

The research partially supported by the RFBR grants 05-01-00810 and 06-01-00738, the Russian President’s grant NSh-8980-2006.1, and the INTAS grant 03-51-5018. The second author also supported by the Lavrentiev SB RAS grant for young scientists.