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Lithuanian Mathematical Journal

, Volume 52, Issue 4, pp 435–443 | Cite as

On the normal approximation of a sum of a random number of independent random variables

  • Jonas Kazys Sunklodas
Article

Abstract

In this paper, we extend the results obtained in [J. Sunklodas, Some estimates of normal approximation for the distribution of a sum of a random number of independent random variables, Lith. Math. J., 52(3):326–333, 2012] for a thrice-differentiable function h : ℝ ℝ to the case of hBL(ℝ); namely, we estimate the quantity | E h(Z N ) E h(Y)| where h is a real bounded Lipschitz function, \( {Z_N}={{{\left( {{S_N}-\mathrm{E}{S_N}} \right)}} \left/ {{\sqrt{{\mathrm{D}{S_N}}}}} \right.} \), S N = X 1 + · · · + X N , X 1 , X 2 , . . . are independent, not necessarily identically distributed, real random variables, N is a positive integer-valued r.v. independent of X 1 , X 2 , . . . , and Y is a standard normal random variable.

Keywords

central limit theorem random sum normal approximation Lipschitz condition Stein’s method 

MSC

60F05 

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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Institute of Mathematics and InformaticsVilnius UniversityVilniusLithuania
  2. 2.Vilnius Gediminas Technical UniversityVilniusLithuania

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