Abstract
We present upper bounds of the integral \( {\int}_{-\infty}^{\infty }{\left|x\right|}^l\left|\mathrm{P}\left\{{Z}_N<x\right\}-\varPhi (x)\right|\mathrm{d}x \) for 0 ⩽ l ⩽ 1 + δ, where 0 < δ ⩽ 1, Φ(x) is a standard normal distribution function, and \( {Z}_N={S}_N/\sqrt{\mathrm{E}{S}_N^2} \) is the normalized random sum with \( \mathrm{E}{S}_N^2>0\left({S}_N{X}_1+\dots +{X}_N\right) \) of centered random variables X1,X2, . . . satisfying the uniformly strong mixing condition. The number of summands N is a nonnegative integer-valued random variable independent of X1,X2, . . . .
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A.V. Bulinskii, Limit Theorems under Weak Dependence Conditions, Moscow State Univ. Press, Moscow, 1989 (in Russian).
V. ˇCekanaviˇcius, Approximation Methods in Probability Theory, Universitext, Springer, Switzerland, 2016.
P. Hall and S.S. Heyde, Martingale Limit Theory and Its Application, Academic Press, New York, 1980.
I.A. Ibragimov, Some limit theorems for stationary in the weak sense random processes, Dokl. Akad. Nauk SSSR, 125(4):711–714, 1959 (in Russian).
I.A. Ibragimov, Some limit theorems for stationary processes, Teor. Veroyatn. Primen., 7(4):361–392, 1962 (in Russian). English transl.: Theory Probab. Appl., 7(4):349–382, 1962.
U. Islak, Asymptotic normality of random sums of m-dependent random variables, Stat. Probab. Lett., 109:22–29, 2016.
Z. Lin and C. Lu, Limit Theory for Mixing Dependent Random Variables, Math. Appl., Dordr., Vol. 378, Kluwer Academic/Science Press, Dordrecht/Beijing, 1996.
B.L.S. Prakasa Rao, On the rate of convergence in the random central limit theorem for martingales, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys., 22(12):1255–1260, 1974.
B.L.S. Prakasa Rao, Remarks on the rate of convergence in the random central limit theorem for mixing sequences, Z. Wahrscheinlichkeitstheor. Verw. Geb., 31:157–160, 1975.
B.L.S. Prakasa Rao and M. Sreehari, On the order of approximation in the random central limit theorem for m-dependent random variables, Probab. Math. Stat., 36(1):47–57, 2016.
Y. Shang, A martingale central limit theorem with random indices, Azerb. J. Math., 1(2):109–114, 2011.
Y. Shang, A central limit theorem for randomly indexed m-dependent random variables, Filomat, 26(4):713–717, 2012.
Ch. Stein, A bound for the error in the normal approximation to the distribution of a sum of dependent random variables, in L.M. Le Cam, J. Neyman, and E.L. Scott (Eds.), Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability. Vol. 2: Probability Theory, 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. Statuleviˇcius (Eds.), Limit Theorems of Probability Theory, Springer, Berlin, Heidelberg, New York, 113–165, p. 2000.
J. Sunklodas, On the global central limit theorem for ϕ-mixing random variables, Lith. Math. J., 35(2):185–196, 1995.
J. Sunklodas, On the rate of convergence in the global central limit theorem for random sums of independent random variables, Lith. Math. J., 57(2):244–258, 2017.
J. Sunklodas, On the rate of convergence in the central limit theorem for random sums of strongly mixing random variables, Lith. Math. J., 58(2):219–234, 2018.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Sunklodas, J.K. On the rate of convergence in the global central limit theorem for random sums of uniformly strong mixing random variables. Lith Math J 60, 410–423 (2020). https://doi.org/10.1007/s10986-020-09483-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10986-020-09483-9
Keywords
- global central limit theorem
- random sum
- normal approximation
- uniformly strong mixing random variables
- τ-shifted distributions