Small Deviations of Series of Independent Nonnegative Random Variables with Smooth Weights

Abstract

We study the logarithmic small deviations of j 1 λ j X j , where {X j } are independent identically distributed nonnegative random variables and {λ j } is a sequence of positive and nonincreasing numbers satisfying certain conditions of regularity. Introduction. Letwhere {λ j } is a sequence of nonincreasing positive numbers and {X i } are independent copies of a nonnegative random variable X with a distribution function V (x), and P(S < ∞) = 1. We examine the behavior of − log P(S < r) for r → 0. Similar problems were studied by a number of authors (an extensive list of references can be found in [1], [2], [3], [4]).In the present paper we take the results from [1], [2], [3] and [5], [6] as a base. There (see also [9]) an explicit form of asymptotics of − log P(S < r) for r → 0 was obtained under minimal a priori assumptions, provided that the sequence {λ j } satisfies certain additional conditions.We formulate the corresponding conclusions and do so in terms of Laplace transforms of X and S. In our opinion, this formulation is simpler and shorter. The equivalent recalculation in terms of distribution functions is not a problem (see section 6).Set