Tail Asymptotic Behavior of the supremum of a class of chi-square processes

TL;DR

This paper investigates the tail behavior of a multivariate chi-square process's supremum, providing exact asymptotics and examples involving Brownian bridge and fractional Brownian motion.

Contribution

It derives the boundedness and exact tail asymptotics of a multivariate weighted chi-square process over non-compact intervals, extending previous univariate results.

Findings

Derived the tail asymptotics for the supremum of the process.

Established boundedness conditions for the process.

Illustrated results with Brownian bridge and fractional Brownian motion examples.

Abstract

In this paper, we analyze a multivariate counterpart of the generalized weighted Kolmogorov-Smirnov statistic, which is the supremum of weighted locally stationary chi-square process over non-compact interval. The boundedness and the exact tail asymptotic behavior of the statistics are derived. We illustrate our findings by two examples where the statistic is defined by Brownian bridge and fractional Brownian motion respectively.