Small Deviations of the Maximal Element of a Sequence of Weighted Independent Random Variables

Consider a sequence {X-i} of independent copies of a nonnegative random variable X and let M = sup(j) (>=) (1)lambda X-j(j), where {lambda(j)} is a nonincreasing sequence of positive numbers for which P(M < infinity) = 1. The asymptotic behavior of -logP(M < r) as r -> 0 is studied. A similar problem has been considered for weights {lambda(j)} of special form. This paper studies the fairly important and general case in which -logP(M< r) has an explicit asymptotics as r -> 0.