## The Annals of Probability

- Ann. Probab.
- Volume 20, Number 2 (1992), 681-695.

### The A.S. Behavior of the Weighted Empirical Process and the LIL for the Weighted Tail Empirical Process

#### Abstract

The tail empirical process is defined to be for each $n \in \mathbb{N}, w_n(t) = (n/k_n)^{1/2}\alpha_n(tk_n/n), 0 \leq t \leq 1$, where $\alpha_n$ is the empirical process based on the first $n$ of a sequence of independent uniform (0,1) random variables and $\{k_n\}^\infty_{n=1}$ is a sequence of positive numbers with $k_n/n \rightarrow 0$ and $k_n \rightarrow \infty$. In this paper a complete description of the almost sure behavior of the weighted empirical process $a_n\alpha_n/q$, where $q$ is a weight function and $\{a_n\}^\infty_{n=1}$ is a sequence of positive numbers, is established as well as a characterization of the law of the iterated logarithm behavior of the weighted tail empirical process $w_n/q$, provided $k_n/\log\log n \rightarrow \infty$. These results unify and generalize several results in the literature. Moreover, a characterization of the central limit theorem behavior of $w_n/q$ is presented. That result is applied to the construction of asymptotic confidence bands for intermediate quantiles from an arbitrary continuous distribution.

#### Article information

**Source**

Ann. Probab., Volume 20, Number 2 (1992), 681-695.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176989800

**Digital Object Identifier**

doi:10.1214/aop/1176989800

**Mathematical Reviews number (MathSciNet)**

MR1159568

**Zentralblatt MATH identifier**

0754.60028

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F15: Strong theorems

Secondary: 60F05: Central limit and other weak theorems 62G15: Tolerance and confidence regions 62G30: Order statistics; empirical distribution functions

**Keywords**

Confidence band empirical process intermediate quantiles strong and weak limit theorems tail empirical process weight-function

#### Citation

Einmahl, John H. J. The A.S. Behavior of the Weighted Empirical Process and the LIL for the Weighted Tail Empirical Process. Ann. Probab. 20 (1992), no. 2, 681--695. doi:10.1214/aop/1176989800. https://projecteuclid.org/euclid.aop/1176989800