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Convergence Rates in the Law of Large Numbers for Arrays of Banach Valued Martingale Differences
Author
Source
Issue
Vol. 2013, Issue 2013 (31 Dec. 2013), pp.1-26, 26 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2013-11-21
Country of Publication
Egypt
No. of Pages
26
Main Subjects
Abstract EN
We study the convergence rates in the law of large numbers for arrays of Banach valued martingale differences.
Under a simple moment condition, we show sufficient conditions about the complete convergence for arrays of Banach valued martingale differences; we also give a criterion about the convergence for arrays of Banach valued martingale differences.
In the special case where the array of Banach valued martingale differences is the sequence of independent and identically distributed real valued random variables, our result contains the theorems of Hsu-Robbins-Erdös (1947, 1949, and 1950), Spitzer (1956), and Baum and Katz (1965).
In the real valued single martingale case, it generalizes the results of Alsmeyer (1990).
The consideration of Banach valued martingale arrays (rather than a Banach valued single martingale) makes the results very adapted in the study of weighted sums of identically distributed Banach valued random variables, for which we prove new theorems about the rates of convergence in the law of large numbers.
The results are established in a more general setting for sums of infinite many Banach valued martingale differences.
The obtained results improve and extend those of Ghosal and Chandra (1998).
American Psychological Association (APA)
Hao, Shunli. 2013. Convergence Rates in the Law of Large Numbers for Arrays of Banach Valued Martingale Differences. Abstract and Applied Analysis،Vol. 2013, no. 2013, pp.1-26.
https://search.emarefa.net/detail/BIM-492763
Modern Language Association (MLA)
Hao, Shunli. Convergence Rates in the Law of Large Numbers for Arrays of Banach Valued Martingale Differences. Abstract and Applied Analysis No. 2013 (2013), pp.1-26.
https://search.emarefa.net/detail/BIM-492763
American Medical Association (AMA)
Hao, Shunli. Convergence Rates in the Law of Large Numbers for Arrays of Banach Valued Martingale Differences. Abstract and Applied Analysis. 2013. Vol. 2013, no. 2013, pp.1-26.
https://search.emarefa.net/detail/BIM-492763
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-492763