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January, 1986 Tail Behaviour for Suprema of Empirical Processes
Robert J. Adler, Lawrence D. Brown
Ann. Probab. 14(1): 1-30 (January, 1986). DOI: 10.1214/aop/1176992616


We consider multivariate empirical processes $X_n(t) := \sqrt n (F_n(t) - F(t))$, where $F_n$ is an empirical distribution function based on i.i.d. variables with distribution function $F$ and $t \in \mathbb{R}^k$. For $X_F$ the weak limit of $X_n$, it is shown that $c(F, k)\lambda^{2(k-1)}e^{-2\lambda^2} \leq P\big\{\sup_t X_F(t) > \lambda\big\} \leq C(k)\lambda^{2(k-1)}e^{-2\lambda^2}$ for large $\lambda$ 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 $n$.


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Robert J. Adler. Lawrence D. Brown. "Tail Behaviour for Suprema of Empirical Processes." Ann. Probab. 14 (1) 1 - 30, January, 1986.


Published: January, 1986
First available in Project Euclid: 19 April 2007

zbMATH: 0596.62053
MathSciNet: MR815959
Digital Object Identifier: 10.1214/aop/1176992616

Primary: 62G30
Secondary: 60F10 , 60F15 , 62E20

Keywords: Empirical processes , Gaussian random fields , Kolmogorov-Smirnov tests , Tail behaviour of suprema

Rights: Copyright © 1986 Institute of Mathematical Statistics


Vol.14 • No. 1 • January, 1986
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