First Gap Statistics of Long Random Walks with Bounded Jumps

TL;DR

This paper analyzes the gap statistics between the first two maxima of symmetric bounded jumps random walks, revealing a concentration effect where large gaps are more likely to occur in two successive jumps than over a long walk.

Contribution

It provides exact results for the joint distribution of the gap and time between the first two maxima in bounded jump random walks, highlighting a novel concentration phenomenon.

Findings

Large gaps are more likely to occur in two successive jumps.

The joint distribution reaches a stationary form for large steps or time.

Numerical simulations confirm the concentration effect.

Abstract

We study one-dimensional discrete as well as continuous time random walks, either with a fixed number of steps (for discrete time) $n$ or on a fixed time interval $T$ (for continuous time). In both cases, we focus on symmetric probability distribution functions (PDF) of jumps with a finite support $[-g_{max}, g_{max}]$. For continuous time random walks (CTRWs), the waiting time $\tau$ between two consecutive jumps is a random variable whose probability distribution (PDF) has a power law tail $\Psi(\tau) \propto \tau^{-1-\gamma}$, with $0<\gamma<1$. We obtain exact results for the joint statistics of the gap between the first two maximal positions of the random walk and the time elapsed between them. We show that for large $n$ (or large time $T$ for CTRW), this joint PDF reaches a stationary joint distribution which exhibits an interesting concentration effect in the sense that a gap close to its maximum possible value, $g\approx g_{max}$, is much more likely to be achieved by two successive jumps rather than by a long walk between the first two maxima. Our numerical simulations confirm this concentration effect.