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October, 1981 Laws of the Iterated Logarithm for Order Statistics of Uniform Spacings
Luc Devroye
Ann. Probab. 9(5): 860-867 (October, 1981). DOI: 10.1214/aop/1176994313

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.

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Luc Devroye. "Laws of the Iterated Logarithm for Order Statistics of Uniform Spacings." Ann. Probab. 9 (5) 860 - 867, October, 1981. https://doi.org/10.1214/aop/1176994313

Information

Published: October, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0465.60038
MathSciNet: MR628878
Digital Object Identifier: 10.1214/aop/1176994313

Subjects:
Primary: 60F15

Keywords: Almost sure convergence , Law of the iterated logarithm , order statistics , spacings , strong laws

Rights: Copyright © 1981 Institute of Mathematical Statistics

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Vol.9 • No. 5 • October, 1981
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