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.
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REFERENCES
Aaronson, J., Burton, R., Dehling, H., Gilat, D., Hill, T., and Weiss, B. (1997). Strong laws for L-and U-statistics. Trans. Amer. Math. Soc. 348, 2845–2866.
Bradley, R. C. (1986). Basic properties of strong mixing conditions. In Eberlein, E., and Taqqu, M. (eds.), Progress in Prob. and Stat., Dependence in Prob. and Stat., Vol. 11, Birkhäuser, pp. 165–192.
Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution. Ann. Math. Statist. 19, 293–325.
Mauldwin, R. D., Monticino, M., and von Weizsacker, H. (1996). Directionally reinforced random walks. Advances in Mathematics 117, 239–252.
Peligrad, M., and Utev, S. (1997). Central limit theorem for stationary linear processes. The Annals of Probability 25, 443–456.
Rubin, H., and Vitale, R. A. (1980). Asymptotic distribution of symmetric statistics. Ann. Statist. 8, 165–180.
Teicher, H., and Zhang, C.-H. (1996). A decomposition for some U-type statistics, J. Theor. Prob. 9, 160–170.
Utev, S. A. (1990). On the central limit theorem for φ-mixing triangular arrays of random variables. Theory Probability and Its Application 35, 131–139.
Utev, S. A. (1999). A weak limit theorem for a sum of certain 2-state Markov chains. Technical Report, Institute of Mathematics, Novosibirsk University, Russia.
Utev, S., and Becker, N. G. (1998). A limit theorem for U-statistics of binary variables. J. Theor. Prob. 11, 853–856.
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Becker, N.G., Utev, S. Threshold Results for U-Statistics of Dependent Binary Variables. Journal of Theoretical Probability 14, 97–114 (2001). https://doi.org/10.1023/A:1007821131604
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DOI: https://doi.org/10.1023/A:1007821131604