Convergence rates in the law of large numbers for B -valued random elements

2001 ◽  
Vol 21 (2) ◽  
pp. 229-236 ◽  
Author(s):  
Hanying Liang ◽  
Li Wang
Keyword(s):  
Law Of Large Numbers ◽  
Convergence Rates ◽  
Large Numbers ◽  
Random Elements ◽  
The Law

scholarly journals Convergence rates in the law of large numbers for Banach-valued dependent variables

10.4213/tvp78 ◽  
2007 ◽  
Vol 52 (3) ◽  
pp. 562-587 ◽  
Author(s):  
Jerome Dedecker ◽  
Jerome Dedecker ◽  
Florence Merlevede ◽  
Florence Merlevede
Keyword(s):  
Law Of Large Numbers ◽  
Convergence Rates ◽  
Large Numbers ◽  
Dependent Variables ◽  
The Law

The Bochner Mean Square Deviation and Law of Large Numbers for Squares of Random Elements in Banach Lattices

2002 ◽  
Vol 9 (1) ◽  
pp. 137-148
Author(s):  
Ivan Matsak ◽  
Anatolij Plichko
Keyword(s):  
Law Of Large Numbers ◽  
Function Spaces ◽  
Banach Lattices ◽  
Mean Square ◽  
Mean Square Deviation ◽  
Large Numbers ◽  
Random Elements ◽  
The Law

Abstract A Bochner mean square deviation for random elements of 2-convex Banach lattices is introduced and investigated. Results, analogous to the law of large numbers for squares of random elements are proved in some classes of Köthe function spaces.

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

2013 ◽  
Vol 2013 ◽  
pp. 1-26 ◽  
Author(s):  
Shunli Hao
Keyword(s):  
Law Of Large Numbers ◽  
Convergence Rates ◽  
Sufficient Conditions ◽  
Random Variables ◽  
General Setting ◽  
Weighted Sums ◽  
Moment Condition ◽  
Martingale Differences ◽  
Large Numbers ◽  
The Law

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).

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