Kybernetika 53 no. 2, 220-230, 2017

On weighted U-statistics for stationary random fields

Jana KlicnarováDOI: 10.14736/kyb-2017-2-0220

Abstract:

The aim of this paper is to introduce a central limit theorem and an invariance principle for weighted U-statistics based on stationary random fields. Hsing and Wu (2004) in their paper introduced %, see \cite{HsiWu:04}, some asymptotic results for weighted U-statistics based on stationary processes. We show that it is possible also to extend their results for weighted $U$-statistics based on stationary random fields.

Keywords:

limit theorem, U-statistics, random fields

Classification:

60F05

paper.pdf

References:

  1. P. J. Bickel and M. J. Wichura: Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Statist. 42 (1971), 5, 1656-1670.   DOI:10.1214/aoms/1177693164
  2. E. Bolthausen: On the central limit theorem for stationary mixing random fields. Ann. Probab. 10 (1982), 4, 1047-1050.   DOI:10.1214/aop/1176993726
  3. S. Borovkova, R. Burton and H. Dehling: Limit theorems for functionals of mixing processes with applications to U-statistics and dimension estimation. Trans. Amer. Math. Soc. 353) (2001), 11, 4261-4318.   DOI:10.1090/s0002-9947-01-02819-7
  4. M. Denker and M. Gordin: Limit theorems for von Mises statistics of a measure preserving transformation. Probab. Theory Related Fields, \mi{160} (2014), 1-2, 1-45.   DOI:10.1007/s00440-013-0522-z
  5. M. Denker and G. Keller: On U-statistics and von Mises statistics for weakly dependent processes. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 64 (1983), 4, 505-522.   DOI:10.1007/bf00534953
  6. I. Dewan and B. P. Rao: Asymptotic normality for U-statistics of associated random variables. J. Statist. Planning Inference 97 (2001), 2, 201-225.   DOI:10.1016/s0378-3758(00)00226-3
  7. M. El Machkouri, D. Volný and W. B. Wu: A central limit theorem for stationary random fields. Stochast. Processes Appl. 123 (2013), 1, 1-14.   DOI:10.1016/j.spa.2012.08.014
  8. L. Heinrich: Asymptotic behaviour of an empirical nearest neighbour distance function for stationary Poisson cluster processes. Math. Nachrichten 136 (1988), 1, 131-148.   DOI:10.1002/mana.19881360109
  9. W. Hoeffding: A class of statistics with asymptotically normal distribution. Ann. Math. Statist. 19 (1948), 3, 293-325.   DOI:10.1214/aoms/1177730196
  10. T. Hsing and W. B. Wu: On Weighted U-statistics for stationary processes. Ann. Probab. 32 (2004), 2, 1600-1631.   DOI:10.1214/009117904000000333
  11. D. Khoshnevisan: Multiparameter Processes: An Introduction to Random Fields. Springer, 2002.   CrossRef
  12. V. S. Koroljuk and Y. V. Borovskich: Theory of $U $-statistics. Springer, 1994.   CrossRef
  13. J. Lee: U-statistics. Theory and Practice. Academic Press, 1990.   CrossRef
  14. A. Leucht and M. H. Neumann: Degenerate U-and V-statistics under ergodicity: Asymptotics, bootstrap and applications in statistics. Ann. Inst. of Statist. Math. 65 (2013), 2, 349-386.   DOI:10.1007/s10463-012-0374-9
  15. D. Volný and Y. Wang: An invariance principle for stationary random fields under Hannan's condition. Stoch. Process. Appl. 124 (12) (2014), 12, 4012-4029.   DOI:10.1016/j.spa.2014.07.015
  16. Y. Wang and M. Woodroofe: A new condition on invariance principles for stationary random fields. Statist. Sinica 23 (2013), 4, 1673-1696.   DOI:10.5705/ss.2012.114s