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



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.MATHGoogle Scholar
  3. 3.
    P. Hall and S.S. Heyde, Martingale Limit Theory and Its Application, Academic Press, New York, 1980.MATHGoogle 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).MathSciNetMATHGoogle 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.MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    V.V. Petrov, Sums of Independent Random Variables, Springer Verlag, Berlin, Heidelberg, New York, 1975.CrossRefMATHGoogle 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.MathSciNetMATHGoogle 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.MathSciNetCrossRefMATHGoogle 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.MathSciNetMATHGoogle 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.MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    M. Rosenblatt, A central limit theorem and a strong mixing condition, Proc. Natl. Acad. Sci. USA, 42(1):43–47, 1956.MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Y. Shang, A martingale central limit theorem with random indices, Azerb. J. Math., 1(2):109–114, 2011.MathSciNetMATHGoogle Scholar
  14. 14.
    Y. Shang, A central limit theorem for randomly indexed m-dependent random variables, Filomat, 26(4):713–717, 2012.MathSciNetCrossRefMATHGoogle 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.MathSciNetCrossRefMATHGoogle Scholar

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Authors and Affiliations

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

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