Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-02T20:30:48.873Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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
[2]Dvoretzky, Aryeh, “Central limit theorems for dependent random variables and some applications”, Abstract 81, Ann. Math. Statist. 40 (1969), 1871.Google Scholar
[3]Peller, William, An introduction to probability theory and its applications, Volume I, 3rd ed. (John Wiley & Sons, New York, London, Sydney, 1968).Google Scholar
[4]Loève, Michel, Probability theory, 3rd ed. (Van Nostrand, Princeton, New Jersey; Toronto, Ontario; London; 1963).Google Scholar
[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