Sparre-Andersen identity and the last passage time

TL;DR

This paper explores the Sparre Andersen identity's implications for last passage times in stochastic processes, revealing new uniform distribution results for certain random times when no positive jumps occur.

Contribution

It uncovers a novel uniform distribution property of six related random times linked to the last passage time in processes without positive jumps, extending classical results.

Findings

Six random times share the same uniform distribution on [0,σ]

The result is new and not previously documented in literature

Based on classical exchangeability observations

Abstract

It is shown that the celebrated result of Sparre Andersen for random walks and L\'evy processes has intriguing consequences when the last time of the process in $(-\infty,0]$, say $\sigma$, is added to the picture. In the case of no positive jumps this leads to six random times, all of which have the same distribution - the uniform distribution on $[0,\sigma]$. Surprisingly, this result does not appear in the literature, even though it is based on some classical observations concerning exchangeable increments.