Coupling for τ-dependent sequences and applications

Journal of Theoretical Probability

Volumn 17, Issue 4, 2004, Pages 861-885

  Dedecker, J ^a   Prieur, C ^b  

^a SORBONNE UNIVERSITÉ   (France)

^b INSTITUT DE MATHÉMATIQUES DE TOULOUSE   (France)

Author keywords

Bernoulli shifts; Coupling; Dependent sequences; Exponential inequalities; Law of the iterated logarithm; Markov chains; Strong invariance principle

Indexed keywords

EID: 23944434888     PISSN: 08949840     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10959-004-0578-x     Document Type: Article

References (35)

1. Andrews, D. W. K. (1984). Nonstrong mixing autoregressive processes. J. Appl. Probab. 21, 930-934.
2. Berbee, H. C. P. (1979). Random walks with stationary increments and renewal theory. Math. Cent. Tracts. Amsterdam.
3. Billingsley, P. (1968). Convergence of probability measures. Wiley, New York.
4. Billingsley, P. (1995). Probability and Measure, 3rd ed., Wiley, New York.
5. Bosq, D. (1993). Bernstein's type large deviation inequalities for partial sums of strong mixing process. Statistics 24, 59-70.
6. Bradley, R. C. (1983). Approximations theorems for strongly mixing random variables. Michigan Math. J. 30, 69-81.
7. Bradley, R. C. (2002). Introduction to Strong Mixing Conditions, Vol 1. Technical Report, Department of Mathematics, I. U. Bloomington.
8. Broise, A. (1996). Transformations dilatantes de l'intervalle et théorèmes limites. Études spectrales d'opérateurs de transfert et applications. Astérisque 238, 1-109.
9. Bryc, W (1982). On the approximation theorem of Berkes and Philipp, Demonstratio Mathematics 15, 807-816.
10. Coulon-Prieur, C., and Doukhan, P. (2000). A triangular CLT for weakly dependent sequences. Statist. Probab. Lett. 47, 61-68.
11. Dedecker, J., and Doukhan, P. (2002). A new covariance inequality and applications, Stochastic Process. Appl. 106, 63-80.
12. Dedecker, J., and Rio, E. (2000). On the functional central limit theorem for stationary processes. Ann. Fnst. H. Poincaré Probab. Statist. 36, 1-34.
13. Dehling, H., and Philipp, W (1982). Almost sure invariance principles for weakly dependent vector-valued random variables, Ann. Probab. 10, 689-701.
14. Doukhan, P. (1994). Mixing: properties and examples. Lecture Notes in Statist. 85, Springer-Verlag.
15. Doukhan, P., and Louhichi, S. (1999). A new weak dependence condition and applications to moment inequalities. Stochastic Process. Appl. 84, 313-342.
16. 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.
17. Dudley, R. M., (1989). Real analysis and probability. Wadworsth Inc., Belmont, California.
18. Fuk, D. K., and Nagaev, S. V. (1971). Probability inequalities for sums of independent random variables. Theory Probab. Appl. 16, 643-660.
19. Goldstein, S. (1979). Maximal coupling. Z. Wahrsch. Verw. Gebiete 46, 193-204.
20. Gordin, M. I. (1969). The central limit theorem for stationary processes. Dokl. Akad. Nauk SSSR 188, 739-741.
21. Gordin, M. I. (1973). Abstracts of Communication, T.1: A-K, International Conference on Probability Theory, Vilnius.
22. Heyde, C. C., and Scott, D. J. (1973). Invariance principles for the law of the iterated logarithm for martingales and processes with stationary increments. Ann. Probab. 1, 428-436.
23. Heyde, C. C. (1975). On the central limit theorem and iterated logarithm law for stationary processes. Bull. Austral. Math. Soc. 12, 1-8.
24. Major, P. (1978). On the invariance principle for sums of identically distributed random variables. J. Multivariate Anal. 8, 487-517.
25. Merlevède, F., and Peligrad, M. (2002). On the coupling of dependent random variables and applications. Empirical process techniques for dependent data. 171-193. Birkhäuser.
26. McShane, E. J. (1947). Integration. Princeton university press, London.
27. Oodaira, H., and Yoshihara, K. I. (1971). Note on the law of the iterated logarithm for stationary processes satisfying mixing conditions. Kodai Math. Sem. Rep. 23, 335-342.
28. Peligrad, M. (2001). A note on the uniform laws for dependent processes via coupling. J. Theoret. Probab. 14, 979-988.
29. Peligrad, M. (2002). Some remarks on coupling of dependent random variables. Stat. Prob. Letters 60, 201-209.
30. Philipp, W. (1986). Invariance principles for independent and weakly dependent random variables. Dependence in probability and statistics (Oberwolfach, 1985), 225-268. Prog. Probab. Statist. 11, Birkhäuser, Boston.
31. Rio, E. (1995). The functional law of the iterated logarithm for stationary strongly mixing sequences, Ann. Probab. 23, 1188-1203.
32. Rio, E. (2000). Théorie asymptotique des processus aléatoires faiblement dépendants. Collection Mathématiques & Applications 31. Springer, Berlin.
33. Rosenblatt, M. (1956). A central limit theorem and a strong mixing condition, Proc. Nat. Acad. Sci. U.S.A. 42, 43-47.
34. Shao, Q-.M. (1993). Almost sure invariance principle for mixing sequences of random variables. Stochastic Process. Appl. 48, 319-334.
35. Strassen, V. (1964). An invariance principle for the law of the iterated logarithm. Z. Wahrsch. Verw. Gebiete 3, 211-226.