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

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 Rosén 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.

28 pages, 3 figures

References in corpus (4)

• Universal Record Statistics of Random Walks and Lévy Flights
• Universal statistics of longest lasting records of random walks and Lévy flights
• Exact extreme value statistics at mixed order transitions
• Ages of records in random walks

Cited by in corpus (9)

• Fluctuation-dominated phase ordering at a mixed order transition
• Extreme value theory for constrained physical systems
• Condensation and extremes for a fluctuating number of independent random variables
• Comment on `Fluctuation-dominated phase ordering at a mixed order transition'
• Rényi Entropy of Zeta-Urns
• On Random Allocation Models in the Thermodynamic Limit
• Fluctuation dominated phase ordering in coarse-grained depth models: Domain wall structures, extreme values and coarsening
• Partition function zeros of zeta-urns
• Big jump principle for heavy-tailed random walks with correlated increments