Longest interval between zeros of the tied-down random walk, the Brownian bridge and related renewal processes

TL;DR

This paper analyzes the distribution of the longest zero-interval in simple random walks and Brownian bridges, extending previous results using renewal theory to unify and revisit past research in the field.

Contribution

It introduces a renewal theory approach to analyze the longest zero-interval, providing new insights and extending prior results for random walks, Brownian bridges, and stable processes.

Findings

Distribution formulas for longest zero-intervals derived

Renewal theory simplifies analysis of zero-intervals

Connections to physics literature revisited

Abstract

The probability distribution of the longest interval between two zeros of a simple random walk starting and ending at the origin, and of its continuum limit, the Brownian bridge, was analysed in the past by Ros\'en and Wendel, then extended by the latter to stable processes. We recover and extend these results using simple concepts of renewal theory, which allows to revisit past or recent works of the physics literature.