Exact convergence rates in the central limit theorem for a class of martingales

Bernoulli

Volumn 13, Issue 4, 2007, Pages 981-999

  El Machkouri, Mohamed ^a   Ouchti, Lahcen ^b  

^a UNIVERSITÉ LILLE 1   (France)

^b UNIVERSITÉ DE ROUEN   (France)

Author keywords

Central limit theorem; Lindeberg's decomposition; Martingale difference sequence; Rate of convergence

Indexed keywords

EID: 47249105908     PISSN: 13507265     EISSN: None     Source Type: Journal    
DOI: 10.3150/07-BEJ6116     Document Type: Article

References (24)

1. Berry, A.C. (1941). The accuracy of the Gaussian approximation to the sum of independent variates. Trans. Amer. Math. Soc. 49 122-136.
2. Bolthausen, E. (1980). The Berry-Esseen theorem for functionals of discrete Markov chains. Probab. Theory Related Fields 54 59-73.
3. Bolthausen, E. (1982). Exact convergence rates in some martingale central limit theorems. Ann. Probab. 10 672-688.
4. Brown, B.M. (1971). Martingale central limit theorems. Ann. Math. Statist. 42 59-66.
5. Chow, Y.S. and Teicher, H. (1978). Probability Theory: Independence, Interchangeability, Martingales. Berlin: Springer.
6. Dvoretzky, A. (1970). Asymptotic normality for sums of dependent random variables. Proc. Sixth Berkeley Symp. Math. Statist. Probab. 2 513-535. Berkeley: Univ. California Press.
7. El Machkouri, M. and Volný, D. (2004). On the local and central limit theorems for martingale difference sequences. Stochastics and Dynamics 4 1-21.
8. Esseen, C.-G. (1942). On the Liapunov limit of error in the theory of probability. Ark. Mat. Astr. Fys. 28A 1-19.
9. Gordin, M.I. (1969). The central limit theorem for stationary processes. Soviet Math. Dokl. 1174-1176.
10. Haeusler, E. (1984). A note on the rate of convergence in the martingale central limit theorem. Ann. Probab. 12 635-639.
11. Hall, P. and Heyde, C.C. (1980). Martingale Limit Theory and Its Application. New York: Academic Press.
12. Ibragimov, I.A. (1963). A central limit theorem for a class of dependent random variables. Theory Probab. Appl. 8 83-89.
13. Jan, C. (2000). Vitesse de convergence dans le TCL pour des chaînes de Markov et certains processus associés à des systèmes dynamiques. C. R. Acad. Sci. Paris Sér. I 331 395-398.
14. Kato, Y (1978). Rates of convergences in central limit theorem for martingale differences. Bull. Math. Statist. 18 1-8.
15. Landers, D. and Rogge, L. (1976). On the rate of convergence in the central limit theorem for Markov chains. Z. Wahrsch. Verw. Gebiete 35 57-63.
16. Leborgne, S. and Pène, F. (2005). Vitesse dans le théorème limite central pour certains systèmes dynamiques quasi-hyperboliques. Bull. Soc. Math. France 133 395-417.
17. Lindeberg, J.W. (1922). Eine neue Herleitung des Exponentialgezetzes in der Wahrscheinlichkeitsrechnung. Math. Z. 15 211-225.
18. Ouchti, L. (2005). On the rate of convergence in the central limit theorem for martingale difference sequences. Ann. Inst. H. Poincaré Probab. Statist. 41 35-43.
19. Philipp, W. (1969). The remainder in the central limit theorem for mixing stochastic processes. Ann. Math. Statist. 40 601-609.
20. Rinott, Y. and Rotar, V. (1998). Some bounds on the rate of convergence in the CLT for martingales. I. Theory Probab. Appl. 43 604-619.
21. Rinott, Y. and Rotar, V. (1999). Some bounds on the rate of convergence in the CLT for martingales. II. Theory Probab. Appl. 44 523-536.
22. Rio, E. (1996). Sur le théorème de Berry-Esseen pour les suites faiblement dépendantes. Probab. Theory Related Fields 104 255-282.
23. Sunklodas, J. (1977). An estimation of the convergence rate in the central limit theorem for weakly dependent random variables. Litovsk. Mat. Sb. 17 41-51.
24. Volný, D. (1993). Approximating martingales and the central limit theorem for strictly stationary processes. Stochastic Process. Appl. 44 41-74.