Lithuanian Mathematical Journal

, Volume 58, Issue 2, pp 219–234 | Cite as

On the rate of convergence in the central limit theorem for random sums of strongly mixing random variables

  • Jonas Kazys Sunklodas


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, . . . .


central limit theorem random sum normal approximation strongly mixing random variables uniformly strongly mixing random variables τ-shifted distributions 




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  1. 1.
    R.N. Bhattacharya and R. Ranga Rao, Normal Approximation and Asymptotic Expansions, Krieger Publishing Co., Malabar, FL, 1986.Google Scholar
  2. 2.
    P. Billingsley, Convergence of Probability Measures, John Willey & Sons, New York, 1968.zbMATHGoogle Scholar
  3. 3.
    P. Hall and S.S. Heyde, Martingale Limit Theory and Its Application, Academic Press, New York, 1980.zbMATHGoogle Scholar
  4. 4.
    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).MathSciNetzbMATHGoogle Scholar
  5. 5.
    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.Google Scholar
  6. 6.
    U. Islak, Asymptotic normality of random sums of m-dependent random variables, Stat. Probab. Lett., 109:22–29, 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    V.V. Petrov, Sums of Independent Random Variables, Springer Verlag, Berlin, Heidelberg, New York, 1975.CrossRefzbMATHGoogle Scholar
  8. 8.
    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.MathSciNetzbMATHGoogle Scholar
  9. 9.
    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.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    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.MathSciNetzbMATHGoogle Scholar
  11. 11.
    E. Rio, Sur le théorème de Berry–Esseen pour les suites faiblement dépendantes, Probab. Theory Relat. Fields, 104(2):255–282, 1996.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    M. Rosenblatt, A central limit theorem and a strong mixing condition, Proc. Natl. Acad. Sci. USA, 42(1):43–47, 1956.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Y. Shang, A martingale central limit theorem with random indices, Azerb. J. Math., 1(2):109–114, 2011.MathSciNetzbMATHGoogle Scholar
  14. 14.
    Y. Shang, A central limit theorem for randomly indexed m-dependent random variables, Filomat, 26(4):713–717, 2012.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    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.Google Scholar
  16. 16.
    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.MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Institute of Data Science and Digital TechnologiesVilnius UniversityVilniusLithuania

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