Abstract
Let ξ n be a φ-mixing sequence of real random variables such that \( \mathbb{E}{\xi_n} = 0 \), and let Y be a standard normal random variable. Write S n = ξ 1 + · · · + ξ n and consider the normalized sums Z n = S n /B n , where \( B_n^2 = \mathbb{E}S_n^2 \). Assume that a thrice differentiable function \( h:\mathbb{R} \to \mathbb{R} \) satisfies \( {\sup_{x \in \mathbb{R}}}\left| {{h^s}(x)} \right| < \infty \). We obtain upper bounds for \( {\Delta_n} = \left| {\mathbb{E}h\left( {{Z_n}} \right) - \mathbb{E}h(Y)} \right| \) in terms of Lyapunov fractions with explicit constants (see Theorem 1). In a particular case, the obtained upper bound of Δ n is of order O(n −1/2).
We note that the φ-mixing coefficients φ(r) are defined between the “past” and “future.” To prove the results, we apply the Bentkus approach.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. Bentkus, A new method for approximation in probability and operator theories, Lith. Math. J., 43(4):367–388, 2003.
V. Bentkus, A Lyapunov type bound in \( {\mathbb{R}^d} \), Teor. Veroyatn. Primen., 49(2):400–410, 2004.
V. Bentkus and J. Sunklodas, On normal approximations to strongly mixing random fields, Publ. Math., 70(3–4):253–270, 2007.
A.V. Bulinski, Limit Theorems Under Conditions of Weak Dependency, Moscow State Univ. Press, Moscow, 1989 (in Russian).
C. Cealera, Contributions to the study of dependent random variables, Stud. Cercet. Mat., 44(1):13–35, 1992 (in Romanian).
L.H.Y. Chen, Poisson approximation for dependent trials, Ann. Probab., 3(3):534–545, 1975.
R.V. Erickson, L 1 bounds for asymptotic normality of m-dependent sums using Stein’s technique, Ann. Probab., 2(3):522–529, 1974.
P. Hall and C.C. Heyde, Martingale Limit Theory and Its Applications, Academic Press, New York, London, Toronto, Sydney, San Francisco, 1980.
L. Heinrich, A method of the derivation of limit theorems for sums ofm-dependent random variables, Z.Wahrscheinlichkeitstheor. Verw. Geb., 60(4):501–515, 1982.
I.A. Ibragimov, Some limit theorems for stationary processes, Theory Probab. Appl., 7:349–382, 1962.
B.S. Nahapetian, An approach to limit theorems for dependent random variables, Teor. Veroyatn. Primen., 32(3):589–594, 1987 (in Russian).
E. Rio, About the Lindeberg method for strongly mixing sequences, ESAIM, Probab. Stat., 1:35–61, 1995.
E. Rio, Sur le théorème de Berry–Esseen pour les suites faiblement dépendantes, Probab. Theory Relat. Fields, 104(2):255–282, 1996.
V.V. Shergin, Rate of convergence in the central limit theorem for m-dependent random variables, Teor. Veroyatn. Primen., 24(4):781–794, 1979 (in Russian).
Ch. Stein, A bound for the error in the normal approximation to the distribution of a sum of dependent random variables, in Proceedings of the 6th Berkeley Symposium on Mathematical Statistics and Probability, Vol. 2, Univ. California Press, Berkeley, CA, 1972, pp. 583–602.
J. Sunklodas, Approximation of distributions of sums of weakly dependent random variables by the normal distribution, in Yu.V. Prokhorov and V. Statulevičius (Eds.), Limit Theorems of Probability Theory, Springer-Verlag, Berlin, Heidelberg, New York, 2000, pp. 113–165.
J. Sunklodas, On normal approximation for strongly mixing random variables, Acta Appl. Math., 97(1–3):251–260, 2007.
J. Sunklodas, On normal approximation for strongly mixing random fields, Teor. Veroyatn. Primen., 52(1):60–68, 2007 (in Russian).
J. Sunklodas, On L 1 bounds for asymptotic normality of some weakly dependent random variables, Acta Appl. Math., 102(1):87–98, 2008.
J. Sunklodas, Some estimates of the normal approximation for independent non-identically distributed random variables, Lith. Math. J., 51(1):66–74, 2011.
M. Talagrand, Mean Field Models for Spin Glasses. Vol. I: Basic Examples, Springer-Verlag, Berlin, Heidelberg, New York, 2010.
A.N. Tikhomirov, On the rate of convergence in the central limit theorem for weakly dependent variables, Theory Probab. Appl., 25(4):790–809, 1980.
S.A. Utev, Sums of weakly dependent random variables, Sib. Math. J., 32(4):675–690, 1991.
T.M. Zuparov, On the rate of convergence in the central limit theorem for weakly dependent random variables, Theory Probab. Appl., 36(4):783–792, 1991.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kazys Sunklodas, J. Some estimates of the normal approximation for φ-mixing random variables. Lith Math J 51, 260–273 (2011). https://doi.org/10.1007/s10986-011-9124-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10986-011-9124-6