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February, 1973 Probabilities for a $k$th Nearest Neighbor Problem on the Line
Sylvan R. Wallenstein, Joseph I. Naus
Ann. Probab. 1(1): 188-190 (February, 1973). DOI: 10.1214/aop/1176997037


Given $N$ points distributed at random on $\lbrack 0, 1)$, let $p_n$ be the size of the smallest interval that contains $n$ points. Previous work finds $\operatorname{Pr}(p_n \leqq p)$, for $n > N/2$, and for $n \leqq N/2, p = 1/L, L$ an integer. This paper finds the distribution of $p_n$, for all $n, N,$ and $p$.


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Sylvan R. Wallenstein. Joseph I. Naus. "Probabilities for a $k$th Nearest Neighbor Problem on the Line." Ann. Probab. 1 (1) 188 - 190, February, 1973.


Published: February, 1973
First available in Project Euclid: 19 April 2007

zbMATH: 0263.60005
MathSciNet: MR346877
Digital Object Identifier: 10.1214/aop/1176997037

Keywords: 60 , Clusters , Coincidences , E05 , maximum clusters , Nearest neighbor distances , smallest intervals

Rights: Copyright © 1973 Institute of Mathematical Statistics

Vol.1 • No. 1 • February, 1973
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