scholarly journals Correction to: On the concept of B-statistical uniform integrability of weighted sums of random variables and the law of large numbers with mean convergence in the statistical sense

Test ◽  
2021 ◽  
Author(s):  
Manuel Ordóñez Cabrera ◽  
Andrew Rosalsky ◽  
Mehmet Ünver ◽  
Andrei Volodin
Keyword(s):  
Law Of Large Numbers ◽  
Random Variables ◽  
Weighted Sums ◽  
Uniform Integrability ◽  
Sums Of Random Variables ◽  
Mean Convergence ◽  
Large Numbers ◽  
Statistical Sense ◽  
The Law

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).

Sign in / Sign up

Export Citation Format

Share Document