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

摘要

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