The Annals of Probability

Weak convergence results for the Kakutani interval splitting procedure

Ronald Pyke and Willem R. van Zwet

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Abstract

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.

Article information

Source
Ann. Probab., Volume 32, Number 1A (2004), 380-423.

Dates
First available in Project Euclid: 4 March 2004

Permanent link to this document
https://projecteuclid.org/euclid.aop/1078415840

Digital Object Identifier
doi:10.1214/aop/1078415840

Mathematical Reviews number (MathSciNet)
MR2040787

Zentralblatt MATH identifier
1049.60025

Subjects
Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60G99: None of the above, but in this section 62G30: Order statistics; empirical distribution functions

Keywords
Empirical processes Kakutani interval splitting spacings weak convergence cumulants self-similarity

Citation

Pyke, Ronald; van Zwet, Willem R. Weak convergence results for the Kakutani interval splitting procedure. Ann. Probab. 32 (2004), no. 1A, 380--423. doi:10.1214/aop/1078415840. https://projecteuclid.org/euclid.aop/1078415840


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