Literature Cited
Y. S. Chow and H. Teicher, Probability Theory, Independence, Interchangeability, Martingales, Springer-Verlag, New York (1978).
Yu. A. Davydov, “Convergence of distributions generated by stationary stochastic processes,” Teor. Veroyatn. Primen.,13, No. 4, 730–737 (1968).
T. M. Zuparov, “Moment inequalities and estimates of the error term in the central limit theorem for a sequence of weakly dependent random variables,” in: Limit Theorems for Stochastic Processes and Statistical Inference [in Russian], Tashkent (1981), pp. 69–87.
Ch. Chipp, “Convergence rates of the strong law for stationary mixing sequences,” Z. Wahr. Verw. Geb.,49, No. 1, 49–62 (1979).
W. Hoeffding and H. Robbins, “The central limit theorem for dependent random variables,” Duke Math. J.,15, 773–780 (1948).
I. A. Ibragimov and Yu. V. Linnik, Independent and Stationary Connected Variables [in Russian], Nauka, Moscow (1965).
H. Negishi, “The rate of convergence to normality for strong mixing sequences of random variables,” Sci. Repts. Yokohama Nat. Univ. Sec. 1, No. 144, 17–25 (1977).
W. Philipp, “The remainder in the central limit theorem for mixing stochastic processes,” Ann. Math. Stat.,40, No. 2, 601–609 (1969).
V. V. Petrov, “Central limit theorem for m-dependent random variables,” in: Memoirs of the All-Union Conference on Probability Theory and Mathematical Statistics [in Russian], Izd. Akad. Nauk ArmSSR, Erevan (1960), pp. 38–44.
V. V. Petrov, Sums of Independent Random Variables, Springer-Verlag (1975).
M. Rosenblatt, “A central limit theorem and a strong mixing condition,” Proc. Nat. Acad. Sci. USA,42, No. 1, 43–47 (1956).
E. Schneider, “On the speed of convergence in the random central limit theorem for ϕ-mixing processes,” Z. Wahr. Verw. Geb.,58, No. 1, 125–138 (1981).
V. Statulevicius, “On limit theorems for dependent random variables,” in: Abstracts of Commun. in the Second Vilnius Confernece on Probability Theory and Math. Stat., June 28–July 3, 1977, Vol. 3, Vilnius, pp. 212–215.
Ch. Stein, “A bound for the error in the normal approximation to the distribution of a sum of dependent random variables,” in: Proc. of the Sixth Berkeley Symp. on Math. Stat. and Probab., Vol. 2, Univ. California Press (1972), pp. 583–602.
I. Sunklodas, “Estimate of the rate of convergence in the central limit theorem for weakly dependent random variables,” Liet. Mat. Rinkinys,17, No. 3, 41–51 (1977).
H. Takahata, “L∞-bound for asymptotic normality of weakly dependent summands using Stein's result,” Ann. Probab.,9, No. 4, 676–683 (1981).
A. N. Tikhomirov, “Rate of convergence in the central limit theorem for weakly dependent variables,” Vestn. Leningr. Gos. Univ., Ser. Mat., Mekh., Astron.,2, No. 7, 158–159 (1976).
A. N. Tikhomirov, “Rate of convergence in the central limit theorem for weakly dependent variables,” Teor. Veroyatn. Primen.,25, No. 4, 800–818 (1980).
V. V. Shergin, “Rate of convergence in the central limit theorem for m-dependent random variables,” Teor. Veroyatn. Primen.,24, No. 4, 781–794 (1979).
M. D. Yudin, “Rate of convergence of the distribution of a sum of f(n)-dependent random variables to a normal law,” Izv. Akad. Nauk BSSR, Ser. Fiz.-Mat. Nauk, No. 3, 57–60 (1981).
K. Yoshihara, “Moment inequalities for mixing sequences,” Kodai Math. J.,1, No. 2, 316–328 (1978).
Additional information
Institute of Mathematics and Cybernetics, Academy of Sciences of the Lithuanian SSR. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 24, No. 2, pp. 174–185, April–June, 1984.
Rights and permissions
About this article
Cite this article
Sunklodas, J. Rate of convergence in the central limit theorem for random variables with strong mixing. Lith Math J 24, 182–190 (1984). https://doi.org/10.1007/BF00970405
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00970405