Ages of records in random walks

TL;DR

This paper investigates the asymptotic behavior of record ages in symmetric continuous random walks, revealing universal patterns and convergence to the Poisson-Dirichlet distribution.

Contribution

It establishes universal asymptotics for record ages and proves convergence of age proportions to the Poisson-Dirichlet distribution in symmetric random walks.

Findings

Universal asymptotics for average proportion of record ages

Probability of record age being broken at step n converges asymptotically

Ranked proportions of ages converge to Poisson-Dirichlet distribution

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.