The Annals of Probability

Laws of the Iterated Logarithm for Order Statistics of Uniform Spacings

Luc Devroye

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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

Permanent link to this document
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
links.jstor.org

Subjects
Primary: 60F15: Strong theorems

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

Citation

Devroye, Luc. Laws of the Iterated Logarithm for Order Statistics of Uniform Spacings. Ann. Probab. 9 (1981), no. 5, 860--867. doi:10.1214/aop/1176994313. https://projecteuclid.org/euclid.aop/1176994313


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