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November, 1982 Strong Limiting Bounds for Maximal Uniform Spacings
Paul Deheuvels
Ann. Probab. 10(4): 1058-1065 (November, 1982). DOI: 10.1214/aop/1176993728

Abstract

Let $U_1, U_2 \cdots$ be a sequence of independent uniformly distributed random variables on (0, 1) and $M_n$ be the largest spacing induced by $U_1, \cdots, U_n$. We show that $P(M_n \geq (\log n + 2 \log_2n + \log_3n + \cdots + \log_jn)/n \text{i.o.}) = 1$, where $\log_j$ is the $j$ times iterated logarithm, and $j \geq 4$. If $1 = N_1 < N_2 < \cdots < N_k < \cdots$ is the sequence of the successive times $n$ where $M_n < M_{n-1}$, we derive strong limiting bounds for $\{N_k, k \geq 1\}$.

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Paul Deheuvels. "Strong Limiting Bounds for Maximal Uniform Spacings." Ann. Probab. 10 (4) 1058 - 1065, November, 1982. https://doi.org/10.1214/aop/1176993728

Information

Published: November, 1982
First available in Project Euclid: 19 April 2007

zbMATH: 0505.60033
MathSciNet: MR672307
Digital Object Identifier: 10.1214/aop/1176993728

Subjects:
Primary: 60F15

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

Rights: Copyright © 1982 Institute of Mathematical Statistics

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Vol.10 • No. 4 • November, 1982
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