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Threshold Results for U-Statistics of Dependent Binary Variables

Author

Listed:
  • Niels G. Becker

    (Australian National University)

  • Sergey Utev

    (Australian National University)

Abstract

Large sample results for certain U-statistics, and related statistics, of binary dependent random variables are studied. The class of U-statistics include partial sums and polynomials of partial sums of a sequence of random variables. A very wide range of limit results are found. The form of the limit result can depend substantially on the magnitude of the appropriate normalizing sequence for the sum. Unexpectedly, the nature of the limit result also depends significantly on whether the degree of the U-statistic is even or odd. It is shown that dependence is a major factor contributing to this result. The limit results are illustrated with reference to a simple dynamic sequence of binary variables and a reinforced random walk.

Suggested Citation

  • Niels G. Becker & Sergey Utev, 2001. "Threshold Results for U-Statistics of Dependent Binary Variables," Journal of Theoretical Probability, Springer, vol. 14(1), pages 97-114, January.
    • Handle: RePEc:spr:jotpro:v:14:y:2001:i:1:d:10.1023_a:1007821131604
    DOI: 10.1023/A:1007821131604
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