Infinite divisibility of infinite sums of lower records: a simple proof

Abstract

Let {Xn, n = 0, 1,…} be the sequence of the lower records for an arbitrary underlying distribution μ on [0, ∞). We show that is equal in distribution to where {τi, i = 1, 2,…} is a Poisson flow of unit intensity and g is a right-continuous and nonincreasing function defined by μ. This observation allows us to extend results of Bose et al. and simplify their proofs.