## The Annals of Probability

- Ann. Probab.
- Volume 10, Number 3 (1982), 863-868.

### A Log Log Law for Maximal Uniform Spacings

#### Abstract

Let $X_1, X_2, \cdots$ be a sequence of independent uniformly distributed random variables on $\lbrack 0, 1\rbrack$ and $K_n$ be the $k$th largest spacing induced by $X_1, \cdots, X_n$. We show that $P(K_n \leq (\log n - \log_3n - \log 2)/n$ i.o.) = 1 where $\log_j$ is the $j$ times iterated logarithm. This settles a question left open in Devroye (1981). Thus, we have $\lim \inf(nK_n - \log n + \log_3n) = -\log 2 \text{almost surely},$ and $\lim \sup(nK_n - \log n)/2 \log_2n = 1/k \text{almost surely}$.

#### Article information

**Source**

Ann. Probab., Volume 10, Number 3 (1982), 863-868.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993799

**Digital Object Identifier**

doi:10.1214/aop/1176993799

**Mathematical Reviews number (MathSciNet)**

MR659558

**Zentralblatt MATH identifier**

0491.60030

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F15: Strong theorems

**Keywords**

Law of the iterated logarithm uniform spacings strong laws almost sure convergence order statistics

#### Citation

Devroye, Luc. A Log Log Law for Maximal Uniform Spacings. Ann. Probab. 10 (1982), no. 3, 863--868. doi:10.1214/aop/1176993799. https://projecteuclid.org/euclid.aop/1176993799