On Small Deviations of Series of Weighted Random Variables

Abstract

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.