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January 2004 Weak convergence results for the Kakutani interval splitting procedure
Ronald Pyke, Willem R. van Zwet
Ann. Probab. 32(1A): 380-423 (January 2004). DOI: 10.1214/aop/1078415840


This paper obtains the weak convergence of the empirical processes of both the division points and the spacings that result from the Kakutani interval splitting model. In both cases, the limit processes are Gaussian. For the division points themselves, the empirical processes converge to a Brownian bridge as they do for the usual uniform splitting model, but with the striking difference that its standard deviations are about one-half as large. This result gives a clear measure of the degree of greater uniformity produced by the Kakutani model. The limit of the empirical process of the normalized spacings is more complex, but its covariance function is explicitly determined. The method of attack for both problems is to obtain first the analogous results for more tractable continuous parameter processes that are related through random time changes. A key tool in their analysis is an approximate Poissonian characterization that obtains for cumulants of a family of random variables that satisfy a specific functional equation central to the K-model.


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Ronald Pyke. Willem R. van Zwet. "Weak convergence results for the Kakutani interval splitting procedure." Ann. Probab. 32 (1A) 380 - 423, January 2004.


Published: January 2004
First available in Project Euclid: 4 March 2004

zbMATH: 1049.60025
MathSciNet: MR2040787
Digital Object Identifier: 10.1214/aop/1078415840

Primary: 60F17
Secondary: 60G99 , 62G30

Keywords: Cumulants , Empirical processes , Kakutani interval splitting , self-similarity , spacings , weak convergence

Rights: Copyright © 2004 Institute of Mathematical Statistics


Vol.32 • No. 1A • January 2004
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