Extreme Value Statistics of Jump Processes

TL;DR

This paper develops a comprehensive theoretical framework for analyzing the extreme value statistics of symmetric jump processes, revealing universal behaviors and providing exact formulas for key probabilistic quantities.

Contribution

It introduces the semi-infinite propagator as a central tool and derives novel universal asymptotics for extremes in jump processes, including bounded and unbounded cases.

Findings

Exact expressions for the semi-infinite propagator and strip probability.

Universal asymptotic behaviors of extreme value distributions.

New insights into the joint distributions of extremes and their occurrence times.

Abstract

We investigate extreme value statistics (EVS) of general discrete time and continuous space symmetric jump processes. We first show that for unbounded jump processes, the semi-infinite propagator $G_0(x,n)$, defined as the probability for a particle issued from $0$ to be at position $x$ after $n$ steps whilst staying positive, is the key ingredient needed to derive a variety of joint distributions of extremes and times at which they are reached. Along with exact expressions, we extract novel universal asymptotic behaviors of such quantities. For bounded, semi-infinite jump processes killed upon first crossing of zero, we introduce the \textit{strip probability} $\mu_{0,\underline{x}}(n)$, defined as the probability that a particle issued from 0 remains positive and reaches its maximum $x$ on its $n^{\rm th}$ step exactly. We show that $\mu_{0,\underline{x}}(n)$ is the essential building block to address EVS of semi-infinite jump processes, and obtain exact expressions and universal asymptotic behaviors of various joint distributions.