Central limit theorems for empirical andU-processes of stationary mixing sequences

Miguel A. Arcones1, Bin Yu2
1Department of Mathematics, University of Utah, Salt Lake City
2Statistics Department, University of Wisconsin, Madison

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Tài liệu tham khảo

Andersen, N. T. and Dobrić, V. (1987). The central limit theorem for stochastic processes.Ann. Prob. 15, 164–177.

Andrews, D. W. K. (1991). An empirical process central limit theorem for dependent nonidentically distributed random variables.J. Mult. Anal. 38, 187–203.

Andrews, D. W. K. and Pollard, D. (1990). An introduction to functional central limit theorems for dependent processes. Preprint.

Arcones, M. A. and Giné, E. (1993). Limit theorems forU-processes.Ann. Prob. 23, 1494–1542.

Bernstein, S. N. (1927). Sur l'extension du théorème limite du calcul des probabilités aux sommes de quantités dépendantes.Math. Ann. 97, 1–59.

Bonami, A. (1968). Ensembles λ(p) dans le dual deD ∞.Ann. Inst. Four. 18, 193–204.

Bradley, R. C. (1985). On the central limit question under absolute regularity.Ann. Prob. 13, 1314–1325.

Doukhan, P., Massart, P., Rio, E. (1992a). The functional central limit theorem for strongly mixing processes. Tech. Rept. Univ. de Paris-Sud, France.

Doukhan, P., Massart, P., and Rio, E. (1992b). Invariance principles for absolute regular empirical processes. Tech. Rept. Univ. de Paris-Sud, France.

Dudley, R. M. (1967). The sizes of compact subsets of Hilbert space and continuity of Gaussian processes.J. Funct. Ann. 1, 290–330.

Dudley, R. M. (1978). Central limit theorem for empirical processes.Ann. Prob. 6, 899–929.

Dudley, R. M. (1984). A course on empirical processes.Lect. Not. in Math. 1097, 1–142, Springer, New York.

Eberlein, E. (1984). Weak rates of convergence of partial sums of absolute regular sequences.Statist. Prob. Lett. 2, 291–293.

Giné, E. and Zinn, J. (1984). Some limit theorems for empirical processes.Ann. Prob. 12, 929–989.

Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution.Ann. Math. Statist. 19, 293–325.

Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables.J. Amer. Statist. Assos. 58, 13–30.

Hoffmann-Jørgensen, J. (1984).Convergence of Stochastic Processes on Polish Spaces. Unpublished.

Ibragimov, I. A. (1962). Some limit theorems for stationary processes.Th. Prob. Appl. 7, 349–382.

Ibragimov, I. A. and Linnik, Yu. V. (1971).Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff Publishing, Groningen, The Netherlands.

de Jong, R. M. (1993). Stochastic equicontinuity for unbounded mixing processes. Tech. Report. Free University of Amsterdam, The Netherlands.

Kolmogorov, A. M. and Tikhomirov, V. M. (1961). The ε-entropy and ε-capacity of sets in functional spaces.Amer. Math. Soc. Transl. 17, 277–364.

Lee, A. J. (1990).U-Statistics, Theory and Practice. Marcel Dekker, Inc., New York.

Massart, P. (1988). Invariance principles for empirical processes: the weakly dependent case. Chapter 1B of Ph.D. dissertation. Univ. of Paris.

Nobel, A. and Dembo, A. (1990). On uniform convergence for dependent processes. Preprint.

Nolan, D. and Pollard, D. (1987).U-processes: rates of convergence.Ann. Statist. 15, 780–799.

Nolan, D. and Pollard, D. (1988). Functional limit theorems forU-processes.Ann. Prob. 16, 1291–1298.

de la Peña, V. (1992). Decoupling and Khintchine's inequalities forU-statistics.Ann. Prob. 20, 1877–1892.

Le Cam, L. (1984). A remark on empirical measures. In Bickel, P., Doksum, K., and Hodges, J. (eds.),Festschrift for E. L. Lehmann, pp. 305–327. Belmont, CA: Wadsworth.

Philipp, W. (1982). Invariance principles for sums of mixing random elements and the multivariate empirical processes.Colloquia Mathematica Societatis János Bolyai 36, 843–873.

Pisier, G. (1983). Some applications of the metric entropy condition to harmonic analysis.Lect. Not. in Math. 995, 123–154, Springer, New York.

Pollard, D. (1984).Convergence of Stochastic Processes. Springer, New York.

Rio, E. (1992). Covariance inequalities for strongly mixing processes. Tech. Rept. Univ. de Paris-Sud, France.

Serfling, R. J. (1980).Approximation theorems of Mathematical Statistics. Wiley, New York.

Volkonskii, V. A. and Rozanov, Y. A. (1959). Some limit theorems for random functions I.Theor. Prob. Appl. 4, 178–197.

Volkonskii, V. A. and Rozanov, Y. A. (1961). Some limit theorems for random functions I.Theor. Prob. Appl. 6, 186–198.

Yoshihara, K. (1976). Limiting behavior ofU-statistics for stationary, absolute regular processes.Z. Wahrsch. verw. Geb. 35, 237–252.

Yu, B. (1990). Rates of convergence and central limit theorems for empirical processes of stationary mixing sequences. Tech. Rep. 260, Stat. Dept., Univ. of California, Berkeley.

Yu, B. (1993). Rates of convergence of empirical processes for stationary mixing sequences.Ann. Prob. (to appear).

Yukich, J.E. (1986). Rates of convergence for classes of functions: the non-i.i.d. case.J. Mult. Anal. 20, 175–189.