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A triangular central limit theorem under a new weak dependence condition. (English) Zbl 0956.60006

The central limit theorem is proved for triangular arrays under a new weak dependence condition which is a variation of that from P. Doukhan and S. Louhichi [Stochastic Processes Appl. 84, 313-342 (1999)]. The definition of such a weak dependence extends on strong mixing and includes non-mixing Markov processes and associated or Gaussian sequences. The theorems proved in this paper apply for linear arrays and standard kernel density estimates under weak dependence and lead to an extension of results of M. Peligrad and S. Utev [Ann. Probab. 25, No. 1, 443-456 (1997; Zbl 0876.60013)]. The method of proof is a variation of Lindeberg method after E. Rio [ESAIM, Probab. Stat. 1, 35-61 (1997; Zbl 0869.60021) and Probab. Theory Relat. Fields 104, No. 2, 255-282 (1996; Zbl 0838.60017)].

MSC:

60F05 Central limit and other weak theorems
60F17 Functional limit theorems; invariance principles
60G10 Stationary stochastic processes
60G99 Stochastic processes
60E15 Inequalities; stochastic orderings
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[1] Ango Nze, P.; Doukhan, P., Non-parametric Minimax estimation in a weakly dependent framework Is: Quadratic properties, Math. Meth. Statist., 5, 4, 404-423 (1996) · Zbl 0893.62024
[2] Chanda, K. C.; Ruymgart, F. H., General linear processes: a property of the empirical process applied to density and mode estimation, J. Time Ser. Anal., 11, 3, 185-199 (1990) · Zbl 0719.62049
[3] Doukhan, P., 1994. Mixing: Properties and Examples, Lecture Notes in Statistics, Vol. 85. Springer, Berlin.; Doukhan, P., 1994. Mixing: Properties and Examples, Lecture Notes in Statistics, Vol. 85. Springer, Berlin. · Zbl 0801.60027
[4] Doukhan, P., Louhichi, S., 1997. Functional estimation of a density under a new weak dependence condition. Université de Cergy Pontoise, Prépublication 10.97 (a short version of this appeared in C.R.A.S. Paris (1998) 327, 989-992).; Doukhan, P., Louhichi, S., 1997. Functional estimation of a density under a new weak dependence condition. Université de Cergy Pontoise, Prépublication 10.97 (a short version of this appeared in C.R.A.S. Paris (1998) 327, 989-992). · Zbl 0919.62029
[5] Doukhan, P., Louhichi, S., 1999. A new weak dependence condition and applications to moment inequalities. Stoch. Process. Appl. 84, 313-342.; Doukhan, P., Louhichi, S., 1999. A new weak dependence condition and applications to moment inequalities. Stoch. Process. Appl. 84, 313-342. · Zbl 0996.60020
[6] Esary, J.; Proschan, F.; Walkup, D., Association of random variables with applications, Ann. Math. Statist., 38, 1466-1476 (1967) · Zbl 0183.21502
[7] Isha, B.; Prakasa, R., Kernel-type density and failure rate estimation for associated sequences, Ann. Inst. Statist. Math., 47, 253-266 (1995) · Zbl 0833.62036
[8] Peligrad, M.; Utev, S., Central limit theorem for linear processes, Ann. Probab., 25, 1, 443-456 (1997) · Zbl 0876.60013
[9] Rio, E., About the Lindeberg method for strongly mixing sequences, ESAIM, Probab. Statist., 1, 35-61 (1995) · Zbl 0869.60021
[10] Rio, E., Sur le théorème de Berry-Esseen pour les suites faiblement dépendantes, Probab. Theory Related Fields, 104, 255-282 (1996) · Zbl 0838.60017
[11] Robinson, P. M., Non parametric estimators for time series, J. Time Ser. Anal., 4, 3, 185-207 (1983) · Zbl 0544.62082
[12] Rosenblatt, M., 1991. Stochastic curve estimation. NSF-CBMS Regional Conference Series in Probability and Statistics, Vol. 3.; Rosenblatt, M., 1991. Stochastic curve estimation. NSF-CBMS Regional Conference Series in Probability and Statistics, Vol. 3.
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