## Abstract and Applied Analysis

### Convergence Rates in the Law of Large Numbers for Arrays of Banach Valued Martingale Differences

Shunli Hao

#### Abstract

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).

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 715054, 26 pages.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393512156

Digital Object Identifier
doi:10.1155/2013/715054

Mathematical Reviews number (MathSciNet)
MR3132544

Zentralblatt MATH identifier
07095265

#### Citation

Hao, Shunli. Convergence Rates in the Law of Large Numbers for Arrays of Banach Valued Martingale Differences. Abstr. Appl. Anal. 2013 (2013), Article ID 715054, 26 pages. doi:10.1155/2013/715054. https://projecteuclid.org/euclid.aaa/1393512156

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