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February, 1980 The Asymptotic Behavior of Spacings Under Kakutani's Model for Interval Subdivision
Ronald Pyke
Ann. Probab. 8(1): 157-163 (February, 1980). DOI: 10.1214/aop/1176994832

Abstract

If $X_1, X_2, \cdots$ are random variables with values in (0, 1), let $D_{n1}, \cdots, D_{n,n+1}$ denote the $n + 1$ spacings given by the first $n$ observations, $X_1, \cdots, X_n$. If $G^\ast_n$ denotes the empirical distribution function of the normalized spacings $\{(n + 1)D_{ni}\}$, it is proved in this paper that under the Kakutani model in which $X_m$ is a uniform random variable over the largest spacing determined by $X_1, \cdots, X_{m-1}$, with probability one $G^\ast_n \rightarrow G$ uniformly, where $G$ is the uniform distribution function on (0, 2). This is in sharp contrast to the known exponential limiting distribution when the $X_i$ are independent uniform random variables on (0, 1).

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Ronald Pyke. "The Asymptotic Behavior of Spacings Under Kakutani's Model for Interval Subdivision." Ann. Probab. 8 (1) 157 - 163, February, 1980. https://doi.org/10.1214/aop/1176994832

Information

Published: February, 1980
First available in Project Euclid: 19 April 2007

zbMATH: 0426.60031
MathSciNet: MR556422
Digital Object Identifier: 10.1214/aop/1176994832

Subjects:
Primary: 60F15
Secondary: 60K99

Keywords: Empirical distribution function , Glivenko-Cantelli theorem , Kakutani model , normalized spacings , spacings

Rights: Copyright © 1980 Institute of Mathematical Statistics

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Vol.8 • No. 1 • February, 1980
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