Weak Convergence Results for the Kakutani Interval Splitting Procedure

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

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.

Key words and phrases

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

Reference

  1. ANSCOMBE, P. (1952). Large-sample theory of sequential estimation. Proc. Cambridge Philos. Soc. 45 600-607.Google Scholar
  2. KAKUTANI INTERVAL SPLITTING 423Google Scholar
  3. BILLINGSLEY, P. (1968). Convergence of Probability Measures. Wiley, New York.Google Scholar
  4. BRENNAN, M. D. and DURRETT, R. (1987). Splitting intervals. II. Limit laws for lengths. Probab. Theory Related Fields 75 109- 127.Google Scholar
  5. CsbRGO, S. (1974). On weak convergence of the empirical process with random sample size. Acta Sci. Math. Szeged 36 17-25.Google Scholar
  6. DONSKER, M. (1952). Justification and extension of Doob's heuristic approach to the KolmogorovSmirnov theorems. Ann. Math. Statist. 23 277-281.CrossRefMATHMathSciNetGoogle Scholar
  7. GIHMAN, I. I. and SKOROHOD, A. V. (1974). The Theory of Stochastic Processes I. Springer, New York.MATHGoogle Scholar
  8. KAKUTANI, S. (1975). A problem of equidistribution on the unit interval [0, 1]. Proceedings of ObelWolfach Conference on Measure Theory. Lecture Notes in Math. 541 369-376. Springer, Berlin.Google Scholar
  9. KLAASSEN, C. A. J. and WELLNER, J. A. (1992). Kac empirical processes and the bootstrap. In Proceedings of the Eighth International Conference on Probability in Banach Spaces (M. Hahn and J. Kuebs, eds.) 411-429. Birkhauser, Boston.Google Scholar
  10. KOMAKI, F. and ITOH, Y. (1992). A unified model for Kakutani's interval splitting and Renyi's random packing. Adv. in Appl. Probab. 24 502-505.Google Scholar
  11. LOOTGIETER, J. C. (1977). Sur Ia repartition des suites de Kakutani (I). Ann. lnst. H. Poincare Ser. B 13 385-410.Google Scholar
  12. PYKE, R. (1965). Spacings. J. Roy. Statist. Soc. Ser. B 27 395-449.Google Scholar
  13. PYKE, R. (1968). The weak convergence of the empirical process with random sample size. Proc. Cambridge Philos. Soc. 64 155-160.Google Scholar
  14. PYKE, R. (1980). The asymptotic behavior of spacings under Kakutani's model for interval subdivision. Ann. Probab. 8 157-163.CrossRefMATHMathSciNetGoogle Scholar
  15. SIBUYA, M. and ITOH, Y. (1987). Random sequential bisection and its associated binary tree. Ann. Inst. Statist. Math. 39 69-84.CrossRefMATHMathSciNetGoogle Scholar
  16. VAN ZWET, W. R. (1978). A proof of Kakutani's conjecture on random subdivision of longest intervals. Ann. Probab. 6 133-137.CrossRefMATHGoogle Scholar
  17. WEISS, L. (1955). The stochastic convergence of a function of sample successive differences. Ann. Math. Statist. 26 532-536.CrossRefMATHMathSciNetGoogle Scholar

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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