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On central limit and iterated logarithm supplements to the martingale convergence theorem

Published online by Cambridge University Press:  14 July 2016

C. C. Heyde
C. C. Heyde*
Affiliation:
C.S.I.R.O. Division of Mathematics and Statistics, Canberra
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Abstract

Let {Sn, n ≧ 1} be a zero, mean square integrable martingale for which so that Sn → S∞ a.s., say, by the martingale convergence theorem. The paper is principally concerned with obtaining central limit and iterated logarithm results for Bn(Sn – S∞) where the multipliers Bn ↑ ∞ a.s. An example on the Pólya urn scheme is given to illustrate the results.

Keywords

RATES OF CONVERGENCE: MARTINGALE CONVERGENCE THEOREMCENTRAL LIMIT THEOREMITERATED LOGARITHM LAWPÓLYA URN SCHEME
Type
Research Papers
Information
Journal of Applied Probability , Volume 14 , Issue 4 , December 1977 , pp. 758 - 775
Copyright
Copyright © Applied Probability Trust 1977 

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References

Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer-Verlag, Berlin.Google Scholar
Barbour, A. D. (1974) Tail sums of convergent series of independent random variables. Proc. Camb. Phil. Soc. 75, 361–364.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Chow, Y. S. and Teicher, H. (1973) Iterated logarithm laws for weighted averages. Z. Wahrscheinlichkeitsth. 26, 87–94.Google Scholar
Feller, W. (1966) An Introduction to Probability Theory and its Applications, Vol. II. Wiley, New York.Google Scholar
Hall, P. G. (1977) Martingale invariance principles. Ann. Prob. To appear.Google Scholar
Hall, P. G. and Heyde, C. C. (1976) On a unified approach to the law of the iterated logarithm for martingales. Bull. Austral. Math. Soc. 14, 435–447.Google Scholar
Heyde, C. C. (1968) On almost sure convergence for sums of independent random variables. Sankhya A30, 353–358.Google Scholar
McLeish, D. L. (1974) Dependent central limit theorems and invariance principles. Ann. Prob. 2, 620–628.Google Scholar
Müller, D. W. (1968) Verteilungs-Invarianzprinzipien für das starke Gesetz der grossen Zahl. Z. Wahrscheinlichkeitsth. 10, 173–192.Google Scholar
Neveu, J. (1970) Bases Mathématiques du Calcul des Probabilités, 2nd edn. Masson, Paris.Google Scholar
Vervaat, W. (1972) Success Epochs in Bernoulli Trials. Math. Centre Tracts 42, Mathematisch Centrum, Amsterdam.Google Scholar
Whitt, W. (1972) Stochastic Abelian and Tauberian theorems. Z. Wahrscheinlichkeitsth. 22, 251–267.Google Scholar