Chris Heyde’s Work on Rates of Convergence in the Central Limit Theorem

Chapter
First Online:
Part of the Selected Works in Probability and Statistics book series (SWPS)

Abstract

The best-known result on rates of convergence in the central limit theorem is undoubtedly that of A.C. Berry and C.-G. Esseen, which describes the rate in the case of finite third moments. In particular, if X,X 1,X 2, … are independent and identically distributed random variables for which
$$E\, |\, X\, |^3 <\, \infty,$$
(1)
\({\rm if}\ E(X)=0\ {\rm and}\ E(X^2)=1,\ {\rm and\ if\ we\ define}\ S_n=\sum\limits_{i = 1}^n{X_i},\) then
$$\mathop{\rm s\ up}\limits_{-\infty < x < \infty} {} |\,P(S_n \leq n^{1/2} x) - \Phi (x)| \leq A\,E\,|\,X\,|^3 \,n^{-1/2},$$
(2)
where Φ is the standard normal distribution function and A denotes an absolute constant.
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References

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    Egorov, V.A. (1973). The rate of convergence to the normal law that is equivalent to the existence of the second moment. (In Russian.) Teor. Verojatnost. i Primenen. 18, 180–185.MathSciNetGoogle Scholar
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    Ibragimov, I.A. (1966). On the accuracy of approximation by the normal distribution of distribution functions of sums of independent random variables. (In Russian.) Teor. Verojatnost. i Primenen 11, 632–655.MathSciNetGoogle Scholar
  3. 3.
    Osipov, L.V. and Petrov, V.V. (1967). On the estimation of the remainder term in the central limit theorem. (In Russian.) Teor. Verojatnost. i Primenen 12, 322–329.MATHMathSciNetGoogle Scholar
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    Rozovskiĭ, L.V. (1978). The accuracy of an estimate of the remainder term in the central limit theorem. (In Russian.) Teor. Veroyatnost. i Primenen. 23, 744–761.MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsThe University of MelbourneMelbourneAustralia

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