zbMATH — the first resource for mathematics

Weak convergence for weighted empirical processes of dependent sequences. (English) Zbl 0874.60006
Weak convergence theorems for weighted empirical processes based on strictly stationary observations under strong mixing, \(\rho\)-mixing and associated dependence assumptions are established. Weak convergence of integral type functionals of empirical processes and of mean residual life processes in reliability theory are obtained. Two Rosenthal-type inequalities for \(\alpha\)-mixing and associated sequences are proved. These inequalities are also of independent interest.

60B10 Convergence of probability measures
60F17 Functional limit theorems; invariance principles
60F15 Strong limit theorems
60E15 Inequalities; stochastic orderings
Full Text: DOI
[1] Barlow, R. E. and Proschan, F. (1981). Statistical Theory of Reliability and Life Testing: Probability Models. To Begin with Publisher, Silver Spring, MD. · Zbl 0379.62080
[2] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York. Birkel, T. (1988a). Moment bounds for associated sequences. Ann. Probab. 16 1184-1193. Birkel, T. (1988b). On the convergence rate in the central limit theorem for associated processes. Ann. Probab. 16 1685-1698. · Zbl 0172.21201
[3] Birkel, T. (1989). A note on the strong law of large numbers for positively dependent random variables. Statist. Probab. Lett. 7 17-20. · Zbl 0661.60048
[4] Burke, M. D., Cs örg o, S. and Horváth, L. (1981). Strong approximations of some biometric estimates under random censorship. Z. Wahrsch. Verw. Gebiete 56 87-112. · Zbl 0439.60012
[5] Burkholder, D. L. (1973). Distribution function inequalities for martingales. Ann. Probab. 1 19-42. · Zbl 0301.60035
[6] Chibisov, D. (1964). Some theorems on the limiting behaviour of empirical distribution functions. Selected Transl. Math. Statist. Probab. 9 147-156. · Zbl 0192.26204
[7] Cs örg o, M., Cs örg o, S. and Horváth, L. (1986). An Asy mptotic Theory for Empirical Reliability and Concentration Processes. Lecture Notes in Statist. 33. Springer, New York. · Zbl 0605.62105
[8] Cs örg o, M. and Horváth, L. (1986). Approximations of weighted empirical and quantile processes. Statist. Probab. Lett. 4 275-280. · Zbl 0676.60042
[9] Cs örg o, M. and Horváth, L. (1993). Weighted Approximations in Probability and Statistics. Wiley, Chichester. · Zbl 0770.60038
[10] Davy dov, Yu. A. (1970). The invariance principle for stationary processes. Theory Probab. Appl. 15 487-498. · Zbl 0219.60030
[11] Esary, J., Proschan, F. and Walkup, D. (1967). Association of random variables with applications. Ann. Math. Statist. 38 1466-1474. · Zbl 0183.21502
[12] Hardy, G. H., Littlewood, J. E. and P óly a, G. (1959). Inequalities, 2nd ed. Cambridge Univ. Press. · Zbl 0047.05302
[13] H öeffding, W. (1973). On the centering of a simple linear rank statistic. Ann. Statist. 1 54-66. · Zbl 0255.62015
[14] M óricz, F. (1982). A general moment inequality for the maximum of partial sums of single series. Acta Sci. Math. 44 67-75. · Zbl 0487.60025
[15] O’Reilly, N. (1974). On the weak convergence of empirical processes in sup-norm metrics. Ann. Probab. 2 642-651. · Zbl 0301.60007
[16] Peligrad, M. (1994). Convergence of stopped sums of weakly dependent random variables. · Zbl 0931.60008
[17] Rény i, A. (1953). On the theory of order statistics. Acta Math. Sci. Hung. 4 191-232. · Zbl 0052.14202
[18] Rio, E. (1993). Covariance inequalities for strongly mixing processes. Ann. Inst. H. Poincaré 29 587-597. · Zbl 0798.60027
[19] Shao, Q.-M. (1986). Weak convergence of multidimensional empirical processes for strong mixing sequences. Chinese Ann. Math. Ser. A 7 547-552. · Zbl 0652.60032
[20] Shao, Q.-M. (1995). Maximal inequality for partial sums of -mixing sequences. Ann. Probab. 23 948-965. · Zbl 0831.60028
[21] Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York. · Zbl 1170.62365
[22] Yang, G. L. (1978). Estimation of a biometric function. Ann. Statist. 6 112-116. · Zbl 0371.62055
[23] Yokoy ama, R. (1980). Moment bounds for stationary mixing sequences. Z. Wahrsch. Verw. Gebiete 52 45-57. Yu, H. (1993a). Weak convergence for empirical and quantile processes of associated sequences with applications to reliability and economics. Ph.D. dissertation, Carleton Univ., Ottawa. Yu, H. (1993b). A Glivenko-Cantelli lemma and weak convergence for empirical processes of associated sequences. Probab. Theory Related Fields 95 357-370. · Zbl 0407.60002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.