## The Annals of Probability

- Ann. Probab.
- Volume 15, Number 3 (1987), 932-953.

### A Law of the Iterated Logarithm for Sums of Extreme Values from a Distribution with a Regularly Varying Upper Tail

Erich Haeusler and David M. Mason

#### Abstract

Let $X_1, X_2,\ldots$ be independent observations from a distribution with a regularly varying upper tail with index $a$ greater than 2. For each $n \geq 1$, let $X_{1,n} \leq \cdots \leq X_{n,n}$ denote the order statistics based on $X_1,\ldots, X_n$. Choose any sequence of integers $(k_n)_{n\geq 1}$ such that $1 \leq k_n \leq n, k_n \rightarrow \infty$, and $k_n/n \rightarrow 0$. It has been recently shown by S. Csorgo and Mason (1986) that the sum of the extreme values $X_{n,n} + \cdots + X_{n-k_n,n}$, when properly centered and normalized, converges in distribution to a standard normal random variable. In this paper, we completely characterize such sequences $(k_n)_{n\geq 1}$ for which the corresponding law of the iterated logarithm holds.

#### Article information

**Source**

Ann. Probab., Volume 15, Number 3 (1987), 932-953.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176992074

**Digital Object Identifier**

doi:10.1214/aop/1176992074

**Mathematical Reviews number (MathSciNet)**

MR893907

**Zentralblatt MATH identifier**

0646.60034

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62G30: Order statistics; empirical distribution functions

Secondary: 60F15: Strong theorems

**Keywords**

Law of the iterated logarithm sums of extreme values order statistics empirical processes

#### Citation

Haeusler, Erich; Mason, David M. A Law of the Iterated Logarithm for Sums of Extreme Values from a Distribution with a Regularly Varying Upper Tail. Ann. Probab. 15 (1987), no. 3, 932--953. doi:10.1214/aop/1176992074. https://projecteuclid.org/euclid.aop/1176992074