Edgeworth Corrections in Randomized Central Limit Theorems

  • Sergey G. BobkovEmail author
Part of the Lecture Notes in Mathematics book series (LNM, volume 2256)


We consider rates of approximation of distributions of weighted sums of independent, identically distributed random variables by the Edgeworth correction of the 4-th order.


Central limit theorem Edgeworth correction Rates of approximation 


  1. 1.
    A. Bikjalis, Remainder terms in asymptotic expansions for characteristic functions and their derivatives. (Russian) Litovsk. Mat. Sb. 7, 571–582 (1967). Selected Transl. Math. Stat. Probab. 11, 149–162 (1973)Google Scholar
  2. 2.
    S.G. Bobkov, Closeness of probability distributions in terms of Fourier–Stieltjes transforms. (Russian) Uspekhi Mat. Nauk 71(6), 37–98 (2016); translation in Russian Math. Surv. 71(6), 1021–1079 (2016)Google Scholar
  3. 3.
    S.G. Bobkov, Asymptotic expansions for products of characteristic functions under moment assumptions of non-integer orders, in Convexity and Concentration. The IMA Volumes in Mathematics and its Applications, vol. 161 (2017), pp. 297–357Google Scholar
  4. 4.
    S. Bobkov, M. Ledoux, Weighted Poincaré-type inequalities for Cauchy and other convex measures. Ann. Probab. 37(2), 403–427 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    S.G. Bobkov, G.P. Chistyakov, Götze, F. Berry–Esseen bounds for typical weighted sums. Electron. J. Probab. 23(92), 22 (2018)Google Scholar
  6. 6.
    A.M. Kagan, Yu.V. Linnik, C.R. Rao, Characterization Problems in Mathematical Statistics. Translated from the Russian by B. Ramachandran. Wiley Series in Probability and Mathematical Statistics. (Wiley, London 1973), pp. xii+499Google Scholar
  7. 7.
    B. Klartag, S. Sodin, Variations on the Berry–Esseen theorem. (Russian summary) Teor. Veroyatn. Primen. 56(3), 514–533 (2011); reprinted in Theory Probab. Appl. 56(3), 403–419 (2012)Google Scholar
  8. 8.
    M. Ledoux, Concentration of measure and logarithmic Sobolev inequalities, in Séminaire de Probabilités XXXIII. Lecture Notes in Mathematics, vol. 1709 (Springer, Berlin, 1999), pp. 120–216Google Scholar
  9. 9.
    M. Ledoux, The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs, vol. 89 (American Mathematical Society, Providence, 2001), pp. x+ 181Google Scholar
  10. 10.
    C.E. Mueller, F.B. Weissler, Hypercontractivity for the heat semigroup for ultraspherical polynomials and on the n-sphere. J. Funct. Anal. 48(2), 252–283 (1982)MathSciNetCrossRefGoogle Scholar
  11. 11.
    V.V. Petrov, Sums of Independent Random Variables (Springer, Berlin, 1975), pp. x +  345Google Scholar
  12. 12.
    G. Pólya, Herleitung des Gaußschen Fehlergesetzes aus einer Funktionalgleichung (German). Math. Z. 18(1), 96–108 (1923)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  2. 2.HSE UniversityMoscowRussia

Personalised recommendations