Bernoulli

  • Bernoulli
  • Volume 13, Number 4 (2007), 981-999.

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

M. El Machkouri and L. Ouchti

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Abstract

We give optimal convergence rates in the central limit theorem for a large class of martingale difference sequences with bounded third moments. The rates depend on the behaviour of the conditional variances and, for stationary sequences, the rate n−1/2log n is reached.

Article information

Source
Bernoulli, Volume 13, Number 4 (2007), 981-999.

Dates
First available in Project Euclid: 9 November 2007

Permanent link to this document
https://projecteuclid.org/euclid.bj/1194625599

Digital Object Identifier
doi:10.3150/07-BEJ6116

Mathematical Reviews number (MathSciNet)
MR2364223

Zentralblatt MATH identifier
1139.60309

Keywords
central limit theorem Lindeberg’s decomposition martingale difference sequence rate of convergence

Citation

Machkouri, M. El; Ouchti, L. Exact convergence rates in the central limit theorem for a class of martingales. Bernoulli 13 (2007), no. 4, 981--999. doi:10.3150/07-BEJ6116. https://projecteuclid.org/euclid.bj/1194625599


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References

  • Berry, A.C. (1941). The accuracy of the Gaussian approximation to the sum of independent variates., Trans. Amer. Math. Soc. 49 122--136.
  • Bolthausen, E. (1980). The Berry--Esseen theorem for functionals of discrete Markov chains., Probab. Theory Related Fields 54 59--73.
  • Bolthausen, E. (1982). Exact convergence rates in some martingale central limit theorems., Ann. Probab. 10 672--688.
  • Brown, B.M. (1971). Martingale central limit theorems., Ann. Math. Statist. 42 59--66.
  • Chow, Y.S. and Teicher, H. (1978)., Probability Theory: Independence, Interchangeability, Martingales. Berlin: Springer.
  • 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.
  • El Machkouri, M. and Volný, D. (2004). On the local and central limit theorems for martingale difference sequences., Stochastics and Dynamics 4 1--21.
  • Esseen, C.-G. (1942). On the Liapunov limit of error in the theory of probability., Ark. Mat. Astr. Fys. 28A 1--19.
  • Gordin, M.I. (1969). The central limit theorem for stationary processes., Soviet Math. Dokl. 1174--1176.
  • Haeusler, E. (1984). A note on the rate of convergence in the martingale central limit theorem., Ann. Probab. 12 635--639.
  • Hall, P. and Heyde, C.C. (1980)., Martingale Limit Theory and Its Application. New York: Academic Press.
  • Ibragimov, I.A. (1963). A central limit theorem for a class of dependent random variables., Theory Probab. Appl. 8 83--89.
  • Jan, C. (2000). Vitesse de convergence dans le TCL pour des chaî nes de Markov et certains processus associés à des systèmes dynamiques., C. R. Acad. Sci. Paris Sér. I 331 395--398.
  • Kato, Y. (1978). Rates of convergences in central limit theorem for martingale differences., Bull. Math. Statist. 18 1--8.
  • 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.
  • 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.
  • Lindeberg, J.W. (1922). Eine neue Herleitung des Exponentialgezetzes in der Wahrscheinlichkeitsrechnung., Math. Z. 15 211--225.
  • 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.
  • Philipp, W. (1969). The remainder in the central limit theorem for mixing stochastic processes., Ann. Math. Statist. 40 601--609.
  • 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.
  • 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.
  • Rio, E. (1996). Sur le théorème de Berry--Esseen pour les suites faiblement dépendantes., Probab. Theory Related Fields 104 255--282.
  • 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.
  • Volný, D. (1993). Approximating martingales and the central limit theorem for strictly stationary processes., Stochastic Process. Appl. 44 41--74.