## The Annals of Probability

### Laws of the Iterated Logarithm for Order Statistics of Uniform Spacings

Luc Devroye

#### Abstract

Let $X_1, X_2, \cdots$ be a sequence of independent uniformly distributed random variables on $\lbrack 0, 1\rbrack$, and let $K_n$ be the $k$th largest spacing induced by the order statistics of $X_1, \cdots, X_{n - 1}$. We show that $\lim \sup(nK_n - \log n)/2 \log_2n = 1/k \quad\text{almost surely},$ and $\lim \inf(nK_n - \log n + \log_3n) = c \quad\text{almost surely},$ where $-\log 2 \leq c \leq 0$, and $\log_j$ is the $j$ times iterated logarithm.

#### Article information

Source
Ann. Probab., Volume 9, Number 5 (1981), 860-867.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176994313

Digital Object Identifier
doi:10.1214/aop/1176994313

Mathematical Reviews number (MathSciNet)
MR628878

Zentralblatt MATH identifier
0465.60038

JSTOR