## The Annals of Probability

- Ann. Probab.
- Volume 18, Number 4 (1990), 1693-1722.

### Nonstandard Functional Laws of the Iterated Logarithm for Tail Empirical and Quantile Processes

Paul Deheuvels and David M. Mason

#### Abstract

Let $\{\alpha_n(t), 0 \leq t \leq 1\}$ and $\{\beta_n(s), 0 \leq s \leq 1\}$ denote the uniform empirical and quantile processes. We show that, for suitable sequences $A(n, \kappa_n)$ and $B(n, l_n)$, the tail empirical process $\{A(n, \kappa_n)\alpha_n(n^{-1}\kappa_nt), 0 \leq t \leq 1\}$ and the tail quantile process $\{B(n, l_n)\beta_n(n^{-1}l_n s), 0 \leq s \leq 1\}$ are almost surely relatively compact in appropriate topological spaces, where $0 \leq \kappa_n \leq n$ and $0 \leq l_n \leq n$ are sequences such that $\kappa_n$ and $l_n$ are $O(\log \log n)$ as $n \rightarrow \infty$. The limit sets of functions are defined through integral conditions and differ from the usual Strassen set obtained when $\kappa_n$ and $l_n$ are $\infty(\log \log n)$ as $n \rightarrow \infty$. Our results enable us to describe the strong limiting behavior of classical statistics based on the top extreme order statistics of a sample or on the empirical distribution function considered in the tails.

#### Article information

**Source**

Ann. Probab., Volume 18, Number 4 (1990), 1693-1722.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176990642

**Digital Object Identifier**

doi:10.1214/aop/1176990642

**Mathematical Reviews number (MathSciNet)**

MR1071819

**Zentralblatt MATH identifier**

0719.60030

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F15: Strong theorems

Secondary: 60F05: Central limit and other weak theorems 62G30: Order statistics; empirical distribution functions 60F17: Functional limit theorems; invariance principles

**Keywords**

Functional laws of the iterated logarithm empirical and quantile processes order statistics extreme values large deviations strong laws

#### Citation

Deheuvels, Paul; Mason, David M. Nonstandard Functional Laws of the Iterated Logarithm for Tail Empirical and Quantile Processes. Ann. Probab. 18 (1990), no. 4, 1693--1722. doi:10.1214/aop/1176990642. https://projecteuclid.org/euclid.aop/1176990642