Convergence Rates in the Law of Large Numbers for Arrays of Banach Valued Martingale Differences
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 converge...
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| Format: | Article |
| Language: | English |
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Wiley
2013-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2013/715054 |
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| author | Shunli Hao |
| author_facet | Shunli Hao |
| author_sort | Shunli Hao |
| collection | DOAJ |
| description | 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). |
| format | Article |
| id | doaj-art-8ec7ceb291294ea0af9eca007e275970 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-8ec7ceb291294ea0af9eca007e2759702025-02-03T06:12:46ZengWileyAbstract and Applied Analysis1085-33751687-04092013-01-01201310.1155/2013/715054715054Convergence Rates in the Law of Large Numbers for Arrays of Banach Valued Martingale DifferencesShunli Hao0Trade and Event Management, School of Economics, Beijing International Studies University, Beijing 100024, ChinaWe 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).http://dx.doi.org/10.1155/2013/715054 |
| spellingShingle | Shunli Hao Convergence Rates in the Law of Large Numbers for Arrays of Banach Valued Martingale Differences Abstract and Applied Analysis |
| title | Convergence Rates in the Law of Large Numbers for Arrays of Banach Valued Martingale Differences |
| title_full | Convergence Rates in the Law of Large Numbers for Arrays of Banach Valued Martingale Differences |
| title_fullStr | Convergence Rates in the Law of Large Numbers for Arrays of Banach Valued Martingale Differences |
| title_full_unstemmed | Convergence Rates in the Law of Large Numbers for Arrays of Banach Valued Martingale Differences |
| title_short | Convergence Rates in the Law of Large Numbers for Arrays of Banach Valued Martingale Differences |
| title_sort | convergence rates in the law of large numbers for arrays of banach valued martingale differences |
| url | http://dx.doi.org/10.1155/2013/715054 |
| work_keys_str_mv | AT shunlihao convergenceratesinthelawoflargenumbersforarraysofbanachvaluedmartingaledifferences |