On the Central Limit Theorem and Its Weak Invariance Principle for Strongly Mixing Sequences with Values in a Hilbert Space via Martingale Approximation

Journal of Theoretical Probability

Volumn 16, Issue 3, 2003, Pages 625-653

  Merlevède, Florence ^a  

^a SORBONNE UNIVERSITÉ   (France)

Author keywords

Central limit theorem; Hilbert space; Martingale approximation; Strong mixing sequences; Weak invariance principle

Indexed keywords

EID: 0142014448     PISSN: 08949840     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1025668415566     Document Type: Article

References (33)

1. Berbee, H. P. C. (1979). Random walks with stationary increments and renewal theory. In Math. Center Tracts, Vol. 112, Amsterdam.
2. Billingsley, P. (1968). Convergence of Probability Measures, Wiley, New York.
3. Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed., Wiley, New York.
4. Bingham, M. S. (1986). A central limit theorem for approximate martingale arrays with values in a locally abelian group. Math. Z. 192, 409-419.
5. Bingham, M. S. (2000). Approximate martingale central limit theorems on Hilbert space. C. R. Math. Rep. Acad. Sci. Canada 22, 111-117.
6. Bolthausen, E. (1982). On the central limit theorem for stationary mixing random fields. Ann. Probab. 10, 1047-1050.
7. Bosq, D. (2000). Linear processes in function spaces. Theory and applications. In Lecture Notes in Statistics, Vol. 149, Springer.
8. Bradley, R. C. (1997). On quantiles and the central limit question for strongly mixing sequences. J. Theor. Probab. 10, 507-555.
9. Bryc, W. (1982). On the approximation theorem of Berkes and Philipp. Demonstratio Matematica 15, No. 3, 807-816.
10. Chen, X., and White, H. (1998). Central limit and functional central limit theorems for Hilbert-valued dependent heterogeneous arrays with applications. Econometric Theory 14, 260-284.
11. Dedecker, J., and Merlevède, F. (2002). The conditional central limit theorem in Hilbert spaces. Submitted.
12. Dehling, H. (1983). Limit theorems for sums of weakly dependent Banach space valued random variables. Z. Wahr. Verw. Gebiete 63, 393-432.
13. Dehling, H. (1983). A note on a theorem of Berkes and Philipp. Z. Wahr. Verw. Gebiete 62, 39-42.
14. Doukhan, P., Massart, P., and Rio, E. (1994). The functional central limit theorem for strongly mixing processes. Ann. Inst. H. Poincaré Probab. Statist. 30, 63-82.
15. Hall, P., and Heyde, C. C. (1980). Martingale Limit Theory and Its Application, Academic Press.
16. Ibragimov, I. A. (1962). Some limit theorems for stationary processes. Tear. Verojatnost. i Primenen. 7, 361-392.
17. Ibragimov, I. A., and Linnik, Yu. V. (1971). Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff, Groningen.
18. Jakubowski, A. (1980). On limit theorems for sums of dependent Hilbert space valued random variables. In Lecture Notes in Statistics, Vol. 2, Springer-Verlag, New York, pp. 178-187.
19. Kuelbs, J. (1973). The invariance principle for Banach space valued random variables. J. Multivariate Anal 3, 161-172.
20. Merlevède, F. (1995). Sur l'inversibilité des processus linéaires à valeurs dans un espace de Hilbert. C. R. Acad. Sci. Paris Sér. I 321, 477-480.
21. Merlevède, F. (1996). Processus linéaires hilbertiens: Inversibilité, théorèmes limites, estimation et prévision, Ph.D. thesis, University of Paris VI.
22. Merlevède, F., Peligrad, M., and Utev, S. (1997). Sharp conditions for the CLT of linear processes in a Hilbert space. J. Theoret. Probab. 10, 681-693.
23. Merlevède, F., and Peligrad, M. (2000). The functional central limit theorem for strong mixing sequences of random variables. Ann. Probab. 28, No. 3, 1336-1352.
24. Merlevède, F., and Peligrad, M. (2002). On the coupling of dependent random variables and applications. To appear in Empirical Process Techniques for Dependent Data, Birkhäuser.
25. Mourid, T. (1995). Contribution à la statistique des processus à temps continu, Thèse d'état, University of Paris VI.
26. Philipp, W. (1986). Invariance principles for independent and weakly dependent random variables. In Eberlein, E., and Taqqu, M. S. (eds.), Dependence in Probability and Statistics. A Survey of Recent Results, Birkhäuser, Boston.
27. Pinelis, I. (1994). Optimum bounds for the distributions of martingales in Banach spaces. Ann. Probab. 22, No. 4, 1679-1706.
28. Rényi, A. (1963). On stable sequences of events. Sankhya Ser. A 25, 293-302.
29. Rio, E. (1993). Covariance inequalities for strongly mixing processes. Ann. Inst. H. Poincaré Probab. Statist. 29, 587-597.
30. Rio, E. (1995). The functional law of the iterated logarithm for stationary strongly mixing sequences. Ann. Probab 23, 1188-1203.
31. Rio, E. (2000). Théorie asymptotique des processus aléatoires faiblement dépendants. Mathématiques et applications de la SMAI, Vol. 31, Springer.
32. Rosenblatt, M. (1956). A central limit theorem and a strong mixing condition. Proc. Nat. Acad. Sci. 42, 43-47.
33. Walk, H. (1977). An invariance principle for the Robbins-Monro process in a Hilbert space. Z. Wahr. Verw. Gebiete 39, 135-150.