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On the inequalities of Berry-Esseen and V.M. Zolotarev

Part of the Lecture Notes in Mathematics book series (LNM,volume 1233)

Abstract

This paper contains a connection between the well-known Berry Esseen inequality for the uniform distance between two distributions (theorem A) and Zolotarev's inequality for the Levy distance between two distributions ([3]), theorem B). More precisely we give an inequality (theorem 1), from which both inequalities follow, as well as Fainleib's inequality (theorem C). Our estmations use the second modulus of smoothness of the distributions.

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References

  1. Berry, A.C. The accuracy of the Gaussian approximation to the sums of independent variates, TAMS,49 (1941) 122–136.

    CrossRef  MATH  Google Scholar 

  2. Esseen, C.G. Fourier analysis of distribution functions. A mathematical study of Laplace Gaussian law, Acta Math. 77(1945) 1–125.

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. Zolotarev, V.M. Estimations of different distributions in the Levy metric, Trudi Math.Inst. V.A.Steklova, 112 (1970) 224–231. (In Russian)

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  4. Sendov, Bl. Some problems of approximation theory of functions and sets in Hausdorff metric, Uspehi Mat.Nauk,24(1969) 141–178 (In Russian)

    MathSciNet  Google Scholar 

  5. Sendov Bl. Hausdorff approximations. Sofia, 1979 (in Russian)

    Google Scholar 

  6. Hausdorff F. Set theory, Moskva, 1936 (in Russian)

    Google Scholar 

  7. Sendov Bl., B. Penkov. E-entropy and E-capacity of continuous curves, Vestnik MGU, Mech-mat. 3(1962) 20–23 (in Russian)

    MathSciNet  Google Scholar 

  8. Levy,P. Theorie de l'addition des variables aleatoires, Paris, 1937

    Google Scholar 

  9. Hengartner,W., R.Theodorescu. Concentration functions, New York, 1973

    Google Scholar 

  10. Sendov, Bl., V.A. Popov. On a generalization of Jackson's theorem for best approximation, J.Approx. Theory, 9 (1973) 102–111

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  11. Fainleib,A.S. A generalization of the Esseen inequality and its application in the probability number theory. Isvestia Acad. Nauk SSSR, Seria mat., 32

    Google Scholar 

  12. Postnikov,A.G. Introduction in analytic number theory, Moscow,1971 (in Russian)

    Google Scholar 

  13. Timan,A.F. Theory of approximation of functions of real variable, Moskva, 1960 (in Russian)

    Google Scholar 

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© 1987 Springer-Verlag

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Popov, V.A. (1987). On the inequalities of Berry-Esseen and V.M. Zolotarev. In: Kalashnikov, V.V., Penkov, B., Zolotarev, V.M. (eds) Stability Problems for Stochastic Models. Lecture Notes in Mathematics, vol 1233. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072717

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  • DOI: https://doi.org/10.1007/BFb0072717

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-17204-8

  • Online ISBN: 978-3-540-47394-7

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