The Annals of Probability

Weak convergence results for the Kakutani interval splitting procedure

Ronald Pyke and Willem R. van Zwet

Full-text: Open access


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

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

First available in Project Euclid: 4 March 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

Export citation


  • Anscombe, P. (1952). Large-sample theory of sequential estimation. Proc. Cambridge Philos. Soc. 45 600--607.
  • Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • Brennan, M. D. and Durrett, R. (1987). Splitting intervals. II. Limit laws for lengths. Probab. Theory Related Fields 75 109--127.
  • Csörgő, S. (1974). On weak convergence of the empirical process with random sample size. Acta Sci. Math. Szeged 36 17--25.
  • Donsker, M. (1952). Justification and extension of Doob's heuristic approach to the Kolmogorov--Smirnov theorems. Ann. Math. Statist. 23 277--281.
  • Gihman, I. I. and Skorohod, A. V. (1974). The Theory of Stochastic Processes I. Springer, New York.
  • Kakutani, S. (1975). A problem of equidistribution on the unit interval $[0, 1]$. Proceedings of Oberwolfach Conference on Measure Theory. Lecture Notes in Math. 541 369--376. Springer, Berlin.
  • 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. Birkhäuser, Boston.
  • 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.
  • Lootgieter, J. C. (1977). Sur la répartition des suites de Kakutani (I). Ann. Inst. H. Poincaré Ser. B 13 385--410.
  • Pyke, R. (1965). Spacings. J. Roy. Statist. Soc. Ser. B 27 395--449.
  • Pyke, R. (1968). The weak convergence of the empirical process with random sample size. Proc. Cambridge Philos. Soc. 64 155--160.
  • Pyke, R. (1980). The asymptotic behavior of spacings under Kakutani's model for interval subdivision. Ann. Probab. 8 157--163.
  • Sibuya, M. and Itoh, Y. (1987). Random sequential bisection and its associated binary tree. Ann. Inst. Statist. Math. 39 69--84.
  • van Zwet, W. R. (1978). A proof of Kakutani's conjecture on random subdivision of longest intervals. Ann. Probab. 6 133--137.
  • Weiss, L. (1955). The stochastic convergence of a function of sample successive differences. Ann. Math. Statist. 26 532--536.