Small deviations of series of independent positive random variables with weights close to exponential

Abstract

Let ξ, ξ0, ξ1, ... be independent identically distributed (i.i.d.) positive random variables. The present paper is a continuation of the article [1] in which the asymptotics of probabilities of small deviations of series S = Σ =0 ∞ a(j)ξ j was studied under different assumptions on the rate of decrease of the probability ℙ(ξ < x) as x → 0, as well as of the coefficients a(j) ≥ 0 as j → ∞. We study the asymptotics of ℙ(S < x) as x → 0 under the condition that the coefficients a(j) are close to exponential. In the case when the coefficients a(j) are exponential and ℙ(ξ < x) ∼ bx α as x → 0, b > 0, a > 0, the asymptotics ℙ(S < x) is obtained in an explicit form up to the factor x o(1). Originality of the approach of the present paper consists in employing the theory of delayed differential equations. This approach differs significantly from that in [1].