The Annals of Probability

Asymptotic Tail Behavior of Uniform Multivariate Empirical Processes

Miklos Csorgo and Lajos Horvath

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Abstract

Let $\alpha_n$ be the empirical process of independent uniformly distributed random vectors on the unit square $I^2$. We study the asymptotic distribution of the random variable $\sup|\alpha_n(s,t)|/(s^\nu t^\mu L(s)G(s))$ when $\sup$ is taken over various subintervals of $I^2$. We show that in the case of $-\infty < \mu, \nu < 1/2$ the limit is given in terms of a two-time parameter Wiener process, and for $1/2 < \mu, \nu < \infty$ it is determined by a Poisson process.

Article information

Source
Ann. Probab., Volume 18, Number 4 (1990), 1723-1738.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176990643

Digital Object Identifier
doi:10.1214/aop/1176990643

Mathematical Reviews number (MathSciNet)
MR1071820

Zentralblatt MATH identifier
0718.60017

JSTOR
links.jstor.org

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60F17: Functional limit theorems; invariance principles

Keywords
Multivariate empirical process two-time parameter Wiener and Poisson processes weak convergence weighted processes tail behavior

Citation

Csorgo, Miklos; Horvath, Lajos. Asymptotic Tail Behavior of Uniform Multivariate Empirical Processes. Ann. Probab. 18 (1990), no. 4, 1723--1738. doi:10.1214/aop/1176990643. https://projecteuclid.org/euclid.aop/1176990643


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