Maxima of Two Random Walks: Universal Statistics of Lead Changes

TL;DR

This paper studies the universal statistical behavior of lead changes between two random walks, revealing that the average number of lead changes grows logarithmically over time regardless of the jump distribution.

Contribution

It demonstrates the universality of lead change statistics across different types of random walks, including Brownian motion and Levy flights, with explicit asymptotic formulas.

Findings

Average number of lead changes grows as (1/π)ln(t) for large t

Probability of at most n lead changes scales as t^{-1/4} (ln t)^n for Brownian motion

Decay exponent varies with Levy index μ, being 1/4 for μ>2 and continuous for 0<μ<2

Abstract

We investigate statistics of lead changes of the maxima of two discrete-time random walks in one dimension. We show that the average number of lead changes grows as $\pi^{-1}\ln(t)$ in the long-time limit. We present theoretical and numerical evidence that this asymptotic behavior is universal. Specifically, this behavior is independent of the jump distribution: the same asymptotic underlies standard Brownian motion and symmetric Levy flights. We also show that the probability to have at most n lead changes behaves as $t^{-1/4}[\ln t]^n$ for Brownian motion and as $t^{-\beta(\mu)}[\ln t]^n$ for symmetric Levy flights with index $\mu$. The decay exponent $\beta(\mu)$ varies continuously with the Levy index when $0<\mu<2$, while $\beta=1/4$ for $\mu>2$.