Abstract
Various asymptotically correct bounds on the uniform metric for distance between distribution functions in the central limit theorem for sums of independent and identically distributed random variables have previously been given. It is shown in the present paper that corresponding nonuniform bounds can be given for the difference between distribution functions. These results have much wider applicability, such as for obtaining probabilities of moderate deviation or for dealing with L p metrics, 1 ā¤ pā¤ ā.
Received November 4, 1974; revised March 16, 1975.
AMS 1970 subject classification. Primary 60F05, 60G50; Secondary 60F10.
Chapter PDF
Similar content being viewed by others
Key words and phrases
References
Davis, J. A. (1968). Convergence rates for probabilities of moderate deviations. Ann. Math. Statist. 39 2016ā2028.
Heyde, C. C (1967). On the influence of moments on the rate of convergence to the normal distribution. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 8 12ā18.
Heyde, C. C. (1969). Some properties of metrics in a study on convergence to normality. Z. Wahrscheinlichkietstheorie und Verw. Gebiete 11 181ā192.
Heyde, C. C (1973). On the uniform metric in the context of convergence to normality. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 25 83ā95.
Ibragimov, I. A. (1966). On the accuracy of the Gaussian approximation to the distribution function of sums of independent random variables. Theor. Probability Appl. 11 559ā 576.
Nagaev, S. V. (1965). Some limit theorems for large deviations. Theor. Probability Appl. 10214ā235.
Osipov, L. V. and Petrov, V. V. (1967). On an estimate of the remainder in the central limit theorem. Theor. Probability Appl. 12 281ā286.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
Ā© 2010 Springer New York
About this chapter
Cite this chapter
Heyde, C.C. (2010). A Nonuniform Bound on Convergence to Normality. In: Maller, R., Basawa, I., Hall, P., Seneta, E. (eds) Selected Works of C.C. Heyde. Selected Works in Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-5823-5_37
Download citation
DOI: https://doi.org/10.1007/978-1-4419-5823-5_37
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-5822-8
Online ISBN: 978-1-4419-5823-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)