On the supremum and its location of the standardized uniform empirical process

TL;DR

This paper proves the convergence in distribution of the supremum and the point of maximum of the standardized uniform empirical process, revealing that the limit involves independent components with specific distributions.

Contribution

It establishes the joint convergence of the supremum and its location for the standardized uniform empirical process, with explicit limit distributions.

Findings

The supremum converges in distribution to a Gumbel distribution.

The location of the maximum converges to a variable taking values 0 and 1 with equal probability.

The limit components are independent, with one being Bernoulli and the other Gumbel.

Abstract

We show that the maximizing point and the supremum of the standardized uniform empirical process converge in distribution. Here, the limit variable (Z, Y ) has independent components. Moreover, Z attains the values zero and one with equal probability one half and Y follows the Gumbel-distribution.