
Statistics Papers
Document Type
Journal Article
Date of this Version
1986
Publication Source
The Annals of Probability
Volume
14
Issue
1
Start Page
1
Last Page
30
DOI
10.1214/aop/1176992616
Abstract
We consider multivariate empirical processes Xn(t):=√n(Fn(t)−F(t)), where Fn is an empirical distribution function based on i.i.d. variables with distribution function F and t ∈ Rk. For XF the weak limit of Xn, it is shown that
c(F,k)λ2(k−1)e−2λ^2≤ P{suptXF(t) > λ} ≤ C(k) λ2(k−1)e−2λ^2
for large λ and appropriate constants c,C. When k = 2 these constants can be identified, thus permitting the development of Kolmogorov--Smirnov tests for bivariate problems. For general k the bound can be used to obtain sharp upper-lower class results for the growth of suptXn(t) with n.
Keywords
tail behaviour of suprema, empirical processes, Kolmogorov-Smirnov tests, Gaussian random fields
Recommended Citation
Adler, R. J., & Brown, L. D. (1986). Tail Behaviour for Suprema of Empirical Processes. The Annals of Probability, 14 (1), 1-30. http://dx.doi.org/10.1214/aop/1176992616
Date Posted: 27 November 2017
This document has been peer reviewed.