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On non-uniform and global descriptions of the rate of convergence of Asymptotic expansions in the central limit theorem

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  09 April 2009

Peter Hall  and
T. Nakata
Peter Hall
Affiliation:
Department of Statistics, Faculty of Economics and Commerce, Australian National University, G.P.O. Box 4 Canberra, A.C.T. 2601, Australia
T. Nakata
Affiliation:
Department of Mathematics, Chukyo University, 101-2 Yagota Honmachi, Showa-Ku, Nagoya 468, Japan
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Abstract

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The leading term approach to rates of convergence is employed to derive non-uniform and global descriptions of the rate of convergence in the central limit theorem. Both upper and lower bounds are obtained, being of the same order of magnitude, modulo terms of order n-r. We are able to derive general results by considering only those expansions with an odd number of terms.

MSC classification

Secondary: 60F05: Central limit and other weak theorems 60G50: Sums of independent random variables; random walks
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 41 , Issue 3 , December 1986 , pp. 326 - 335
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Hall, P., Rates of convergence in the central limit theorem (Pitman, London, 1982).Google Scholar
[2]Hall, P., ‘A leading term approach to asymptotic expansions in the central limit theorem’, Proc. London Math. Soc. 49 (1984), 423–444.Google Scholar
[3]Heyde, C. C. and Nakata, T., ‘On the asymptotic equivalence of Lp metrics for convergence to normality’, Z. Wahrsch. Verw. Gebiete 68 (1984), 97–106.Google Scholar
[4]Petrov, V. V., Sums of independent random variables (Springer-Verlag, Berlin, 1975).Google Scholar