Weak Convergence Results for the Kakutani Interval Splitting Procedure

  • Ronald Pyke
  • Willem R. van Zwet
Open Access
Part of the Selected Works in Probability and Statistics book series (SWPS)


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.

Key words and phrases

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


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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Ronald Pyke
    • 1
  • Willem R. van Zwet
    • 2
  1. 1.University of WashingtonWashingtonUSA
  2. 2.University of LeidenLeidenThe Netherlands

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