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Rates of convergence to normality for samples from a finite set of random variables

Part of: Limit theorems

Published online by Cambridge University Press:  09 April 2009

R. D. John  and
J. Robinson
R. D. John
Affiliation:
Statistical Consulting Group Department of MathematicsUniversity of Western Australia Nedlands, WA 6009, Australia
J. Robinson
Affiliation:
School of Mathematics and StatisticsUniversity of SydneyNSW 2006Australia e-mail: robinson_j@maths.su.oz.au
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Abstract

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Rates of convergence to normality of O(N-½) are obtained for a standardized sum of m random variables selected at random from a finite set of N random variables in two cases. In the first case, the sum is randomly normed and the variables are not restricted to being independent. The second case is an alternative proof of a result due to von Bahr, which deals with independent variables. Both results derive from a rate obtained by Höglund in the case of sampling from a finite population.

Keywords

Berry-Esseen lemmafinite sampling.

MSC classification

Secondary: 60F05: Central limit and other weak theorems

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 60 , Issue 3 , June 1996 , pp. 355 - 362
Copyright
Copyright © Australian Mathematical Society 1996

References

[1]von Bahr, B., ‘On sampling from a finite set of independent random variables’, Z. Wahrsch. Verw. Gebiete. 24 (1972), 279–286.CrossRefGoogle Scholar
[2]Chernoff, H. and Teicher, H., ‘A central limit theorem for sums of interchangeable random variables’, Ann. Math. Statist. 29 (1958), 118–130.CrossRefGoogle Scholar
[3]Chung, K. L., ‘The strong law of large numbers’, in: Proc. 2nd Berkeley Symp. on Math. Statist. and Prob. (ed. Neyman, J.) (University of California, Berkeley, 1951), 341–352.Google Scholar
[4]Erdös, P. and Rényi, A., ‘On the central limit theorem for samples from a finite population’, Magyar Tud. Akad. Mat. Kutató Intéz. Közl. 4 (1959), 49–61.Google Scholar
[5]Feller, W., An introduction to probability theory and its applications, vol. II, 2nd edition (Wiley, New York, 1971).Google Scholar
[6]Höglund, T., ‘Sampling from a finite population. A remainder term estimate’, Scand. J. Statist. 5 (1978), 69–71.Google Scholar
[7]Neymann, J., Iwaszkiewicz, K. and St., Kolodziejczyk, ‘Statistical problems in agricultural experimentation’, Roy. Stat. Soc. Jour. Suppl. 2 (1935), 107–180.CrossRefGoogle Scholar