## The Annals of Probability

### Asymptotic Tail Behavior of Uniform Multivariate Empirical Processes

#### 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

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