Tail expansions for random record distributions

Abstract

The random record distribution ν associated with a probability distribution μ can be written as a convolution series, ν = [sum ]∞n=1n−1 (n + 1)−1μ*n. Various authors have obtained results on the behaviour of the tails ν((x, ∞)) as x → ∞, using Laplace transforms and the associated Abelian and Tauberian theorems. Here we use Gelfand transforms and the Wiener–Lévy–Gelfand Theorem to obtain expansions of the tails under moment conditions on μ. The results differ notably from those known for other convolution series.