Ages of records in random walks

Abstract

We consider random walks with continuous and symmetric step distributions. We prove universal asymptotics for the average proportion of the age of the kth longest lasting record for k=1,2,... and for the probability that the record of the kth longest age is broken at step n. Furthermore, we show that the ranked sequence of proportions of ages converges to the Poisson-Dirichlet distribution.Comment: 15 pages, 1 figur