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Martingale central limit theorems without uniform asymptotic negligibility

Published online by Cambridge University Press:  17 April 2009

R.J. Adler  and
D.J. Scott
R.J. Adler
Affiliation:
School of Mathematics, University of New South Wales, Kensington, New South Wales;
D.J. Scott
Affiliation:
Department of Probability and Statistics, University of Sheffield, Sheffield, England.
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Abstract

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Central limit theorems are obtained for martingale arrays without the requirement of uniform asymptotic negligibility. The results obtained generalise the sufficiency part of Zolotarev's extension of the classical Lindeberg-Feller central limit theorem [V.M. Zolotarev, Theor. Probability Appl. 12 (1967), 608–618] and also the main martingale central limit theorem (not functional central limit theorem however) of D.L. McLeish [Ann. Probability 2 (1974), 620–628.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 13 , Issue 1 , August 1975 , pp. 45 - 55
Copyright
Copyright © Australian Mathematical Society 1975

References

[1]Brown, B.M., “Martingale central limit theorems”, Ann. Math. Statist. 42 (1971), 59–66.CrossRefGoogle Scholar
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[5]McLeish, D.L., “Dependent central limit theorems and invariance principles”, Ann. Probability 2 (1974), 620–628.CrossRefGoogle Scholar
[6]Scott, D.J., “Central limit theorems for martingales and for processes with stationary increments using a Skorokhod representation approach”, Adv. in Appl. Probability 5 (1973), 119–137.CrossRefGoogle Scholar
[7]Zolotarev, V.M., “A generalization of the Lindeberg-Feller theorem”, Theor. Probability Appl. 12 (1967), 608–618.CrossRefGoogle Scholar