Abstract
We present upper bounds for supx ∈ ℝ|P{Z N < x} − Φ(x)|, where Φ(x) is the standard normal distribution function, for random sums \( {Z}_N={S}_N/\sqrt{\mathbf{V}{S}_N} \) with variances VS N > 0 (S N = X1 + ⋯ + X N ) of centered strongly mixing or uniformly strongly mixing random variables X1, X2, . . . . Here the number of summands N is a nonnegative integer-valued random variable independent of X1,X2, . . . .
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References
R.N. Bhattacharya and R. Ranga Rao, Normal Approximation and Asymptotic Expansions, Krieger Publishing Co., Malabar, FL, 1986.
P. Billingsley, Convergence of Probability Measures, John Willey & Sons, New York, 1968.
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 strict sense stochastic 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.
V.V. Petrov, Sums of Independent Random Variables, Springer Verlag, Berlin, Heidelberg, New York, 1975.
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.
E. Rio, Sur le théorème de Berry–Esseen pour les suites faiblement dépendantes, Probab. Theory Relat. Fields, 104(2):255–282, 1996.
M. Rosenblatt, A central limit theorem and a strong mixing condition, Proc. Natl. Acad. Sci. USA, 42(1):43–47, 1956.
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.
J. Sunklodas, Approximation of distributions of sums of weakly dependent random variables by the normal distribution, in Probability Theory – 6. Limit Theorems in Probability Theory, R.V. Gamkrelidze et al. (Eds.), Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya, Vol. 81, VINITI, Moscow, 1991, pp. 140–199 (in Russian). Engl. transl.: Limit Theorems of Probability Theory, Yu.V. Prokhorov and V. Statulevičius (Eds.), Springer-Verlag, Berlin, Heidelberg, New York, 2000, pp. 113–165.
J.K. 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.
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Sunklodas, J.K. On the rate of convergence in the central limit theorem for random sums of strongly mixing random variables. Lith Math J 58, 219–234 (2018). https://doi.org/10.1007/s10986-018-9391-6
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DOI: https://doi.org/10.1007/s10986-018-9391-6