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On the rate of convergence of moments in the central limit theorem

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  09 April 2009

Peter Hall
Peter Hall
Affiliation:
Department of Statistics University of Melbourne Parkville, Victoria 3052Australia
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Abstract

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An early extension of Lindeberg's central limit theorem was Bernstein's (1939) discovery of necessary and sufficient conditions for the convergence of moments in the central limit theorem. Von Bahr (1965) made a study of some asymptotic expansions in the central limit theorem, and obtained rates of convergence for moments. However, his results do not in general imply that the moments converge. Some better rates have been obtained by Bhattacharya and Rao for moments between the second and third. In this paper we give improved rates of convergence for absolute moments between the third and fourth.

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 25 , Issue 2 , March 1978 , pp. 250 - 256
Copyright
Copyright © Australian Mathematical Society 1978

References

REFERENCES

von Bahr, B. (1965), “On the convergence of moments in the central limit theorem”, Ann. Math. Statist. 36, 808–818.CrossRefGoogle Scholar
Bernstein, S. (1939), “Quelques remarques sur le théorème limite Liapounoff”, Dokl. Akad. Nauk SSR (Comptes rendus) 24, 3–8.Google Scholar
Bhattacharya, R. N. and Rao, R. R. (1976), Normal Approximation and Asymptotic Expansions (Wiley, New York).Google Scholar
Brown, B. M. (1969), “Moments of a stopping rule related to the central limit theorem”, Ann. Math. Statist. 40, 1236–1249.CrossRefGoogle Scholar
Brown, B. M. (1970), “Characteristic functions, moments and the central limit theorem”, Ann. Math. Statist. 41, 658–664.CrossRefGoogle Scholar
Loève, M. (1960), Probability Theory (Van Nostrand, Princeton).Google Scholar
Michel, R. (1976), “Nonuniform central limit bounds with applications to probabilities of deviations”, Ann. Probability 4, 102–106.CrossRefGoogle Scholar
Pitman, E. J. G. (1968), “On the behaviour of the characteristic function of a probability distribution in the neighbourhood of the origin”, J. Austral. Math. Soc. 8, 423–443.CrossRefGoogle Scholar