An excursion approach to maxima of the Brownian Bridge

TL;DR

This paper derives the distributions of extrema of the Brownian bridge using excursion theory, providing insights into key statistical tests like Kolmogorov-Smirnov and Kuiper, with a focus on the Poisson nature of excursions.

Contribution

It introduces a novel derivation method for Brownian bridge extrema distributions based solely on excursion theory and Poisson processes.

Findings

Distribution formulas for maxima of the Brownian bridge

Applications to Kolmogorov-Smirnov and Kuiper statistics

Insights into the ratio of positive to negative maxima

Abstract

Functionals of Brownian bridge arise as limiting distributions in nonparametric statistics. In this paper we will give a derivation of distributions of extrema of the Brownian bridge based on excursion theory for Brownian motion. Only the Poisson character of the excursion process will be used. Particular cases of calculations include the distributions of the Kolmogorov-Smirnov statistic, the Kuiper statistic, and the ratio of the maximum positive ordinate to the minumum negative ordinate.