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A martingale central limit theorem without negligibility conditions

Published online by Cambridge University Press:  17 April 2009

Robert J. Adler
Robert J. Adler
Affiliation:
Commonwealth Scientific and Industrial Research Organization, Division of Mathematics and Statistics, Sydney, New South Wales.
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Abstract

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We obtain sufficient conditions for the convergence of martingale triangular arrays to infinitely divisible laws with finite variances, without making the usual assumptions of uniform asymptotic negligibility. Our results generalise known results for both the martingale case under a negligibility assumption and the classical (independence) case without such assumptions.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 18 , Issue 1 , February 1978 , pp. 13 - 19
Copyright
Copyright © Australian Mathematical Society 1978

References

[1]Adler, R.J. and Scott, D.J., “Martingale central limit theorems without uniform asymptotic negligibility”, Bull. Austral. Math. Soc. 13 (1975), 45–55.CrossRefGoogle Scholar
[2]Billingsley, Patrick, Convergence of probability measures (John Wiley & Sons, New York, London, Sydney, Toronto, 1968).Google Scholar
[3]Brown, B.M. and Eagleson, G.K., “Martingale convergence to infinitely divisible laws with finite variances”, Trans. Amer. Math. Soc. 162 (1971), 449–453.CrossRefGoogle Scholar
[4]Dvoretzky, Aryeh, “Central limit theorems for dependent random variables”, Actes du Congrès International des Mathématiciens, 1970, Volume 2, 565–570 (Gauthier-Villars, Paris, 1971).Google Scholar
[5]Gnedenko, B.V. and Kolmogorov, A.N., Limit distributions for sums of independent random variables (translated by Chung, K.L.. Addison-Wesley, Cambridge, Massachusetts, 1954).Google Scholar
[6]Machis, Yu Yu, “Non-classical limit theorems”, Theor. Probability Appl. 16 (1971), 175–182.CrossRefGoogle Scholar