On weighted U-statistics for stationary processes

Annals of Probability

Volumn 32, Issue 2, 2004, Pages 1600-1631

  Hsing, Tailen ^a   Wu, Wei Biao ^b  

^a TEXAS A AND M UNIVERSITY   (United States)

^b UNIVERSITY OF CHICAGO   (United States)

Author keywords

Limit theorem; Linear processes; Nonlinear time series; U statistics

Indexed keywords

EID: 3042541174     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/009117904000000333     Document Type: Article

References (38)

1. AARONSON, J., BURTON, R., DEHLING, H., GILAT, D., HILL, T. and WEISS, B. (1996). Strong laws for L- and U-statistics. Trans. Amer. Math. Soc. 348 2845-2866.
2. BERAN, J. (1994). Statistics for Long-Memory Processes. Chapman and Hall, New York.
3. BOROVKOVA, S., BURTON, R. and DEHLING, H. (1999). Consistency of the Takens estimator for the correlation dimension. Ann. Appl. Probab. 9 376-390.
4. BOROVKOVA, S., BURTON, R. and DEHLING, H. (2001). Limit theorems for functionals of mixing processes with applications to U-statistics and dimension estimation. Trans. Amer. Math. Soc. 353 4261-4318.
5. BOROVKOVA, S., BURTON, R. and DEHLING, H. (2002). From dimension estimation to asymptotics of dependent U-processes. In Limit Theorems in Probability and Statistics I (I. Berkes, E. Csàki and M. Csörgo, eds.) 201-234. János Bolyai Mathematical Society, Budapest.
6. DEHLING, H. and TAQQU, M. S. (1989). The empirical processes of some long-range dependent sequences with an application to U-statistics. Ann. Statist. 17 1767-1783.
7. DEHLING, H. and TAQQU, M. S. (1991). Bivariate symmetric statistics of long-range dependent observations. J. Statist. Plann. Inference 28 153-165.
8. DENKER, M. and KELLER, G. (1983). On U-statistics and von Mises' statistic for weakly dependent processes. Z. Wahrsch. Verw. Gebiete 64 505-552.
9. DENKER, M. and KELLER, G. (1986). Rigorous statistical procedures for data from dynamical systems. J. Statist. Phys. 44 67-93.
10. DIACONIS, P. and FREEDMAN. D. (1999). Iterated random functions. SIAM Rev. 41 41-76.
11. ENGLE, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50 987-1007.
12. GIRAITIS, L. and SURGAILIS, D. (1999). Central limit theorem for the empirical process of a linear sequence with long memory. J. Statist. Plann. Inference 80 81-93.
13. GIRAITIS, L. and TAQQU, M. S. (1997). Limit theorems for bivariate Appell polynomials. I. Central limit theorems. Probab. Theory Related Fields 107 359-381.
14. GIRAITIS, L., TAQQU, M. S. and TERRIN, N. (1998). Limit theorems for bivariate Appell polynomials. II. Non-central limit theorems. Probab. Theory Related Fields 110 333-367.
15. GRANGER, C. W. and JOYEUX, R. (1980). An introduction to long-memory time series models and fractional differencing. J. Time Ser. Anal. 1 15-29.
16. HAGGAN, V. and OZAKI, T. (1981). Modelling nonlinear random vibrations using an amplitude-dependent autoregressive time series model. Biometrika 68 189-196.
17. HO, H.-C. and HSING, T. (1996). On the asymptotic expansion of the empirical process of long-memory moving averages. Ann. Statist. 24 992-1024.
18. HO, H.-C. and HSING, T. (1997). Limit theorems for functional of moving averages. Ann. Probab. 25 1636-1669.
19. HOEFFDING, W. (1961). The strong law of large numbers for U-statistics. Mimeograph Series No. 302, Dept. of Statistics, Univ. North Carolina, Chapel Hill.
20. KOUL, H. L. and SURGAILIS, D. (1997). Asymptotic expansion of M-estimators with long memory error. Ann. Statist. 25 818-850.
21. LEE, A. J. (1990). U-Statistics, Theory and Practice. Dekker, New York.
22. MAJOR, P. (1980). Multiple Wiener-Ito Integrals. Springer, Berlin.
23. MAJOR, P. (1994). Asymptotic distributions for weighted U-statistics. Ann. Probab. 22 1514-1535.
24. NICHOLLS, D. F. and QUINN, B. G. (1982). Random Coefficient Autoregressive Models: An Introduction. Springer, New York.
25. O'NEIL, K. A. and REDNER, R. A. (1993). Asymptotic distributions of weighted U-statistics of degree 2. Ann. Probab. 21 1159-1169.
26. RANDLES, R. H. and WOLFE, D. A. (1979). Introduction to the Theory of Nonparametric Statistics. Wiley, New York.
27. RIFI, M. and UTZET, F. (2000). On the asymptotic behavior of weighted U-statistics J. Theoret. Probab. 13 141-167.
28. SERFLING, R. J. (1980). Approximation Theorems of Mathematical Statistics. Wiley, New York.
29. SERINKO, R. J. (1994). A consistent approach to least squares estimation of correlation dimension in weak Bernoulli dynamical systems. Ann. Appl. Probab. 4 1234-1254.
30. SURGAILIS, D. (1982). Zones of attraction of self-similar multiple integrals. Lithuanian Math. J. 22 327-340.
31. TONG, H. (1990). Non-Linear Time Series: A Dynamical System Approach. Oxford University Press.
32. WOLFF, R. C. L. (1990). A note on the behaviour of the correlation integral in the presence of a time series. Biometrika 77 689-697.
33. WOODROOFE, M. (1992). A central limit theorem for functions of a Markov chain with applications to shifts. Stochastic Process. Appl. 41 33-44.
34. WU, W. B. (2003). Empirical processes of long-memory sequences. Bernoulli 9 809-831.
35. WU, W. B. and MIELNICZUK, J. (2002). Kernel density estimation for linear processes. Ann. Statist. 30 1441-1459.
36. WU, W. B. and WOODROOFE, M. (2000). A central limit theorem for iterated random functions. J. Appl. Probab. 37 748-755.
37. WU, W. B. and WOODROOFE, M. (2004). Martingale approximations for sums of stationary processes. Ann. Probab. 32 1674-1690.
38. YOSHIHARA, K. (1976). Limiting behavior of U-statistics for stationary, absolutely regular processes. Z. Wahrsch. Verw. Gebiete 35 237-252.