ScholarlyCommons

Title

Tail Behaviour for Suprema of Empirical Processes

Author(s)

Robert J. Adler
Lawrence D. Brown, University of Pennsylvania

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 X_n(t):=√n(F_n(t)−F(t)), where F_n is an empirical distribution function based on i.i.d. variables with distribution function F and t ∈ R^k. For X_F the weak limit of X_n, it is shown that

c(F,k)λ^2(k−1)e^{−2λ^2}≤ P{sup_tX_F(t) > λ} ≤ C(k) λ^2(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 sup_tX_n(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