scholarly journals Convergence rates in the law of large numbers for long-range dependent linear processes

2017 ◽  
Vol 2017 (1) ◽  
Author(s):  
Tao Zhang ◽  
Pingyan Chen ◽  
Soo Hak Sung
Keyword(s):  
Long Range ◽  
Law Of Large Numbers ◽  
Convergence Rates ◽  
Linear Processes ◽  
Large Numbers ◽  
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

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).

A pair of complementary theorems on convergence rates in the law of large numbers

1967 ◽  
Vol 63 (1) ◽  
pp. 73-82 ◽  
Author(s):  
C. C. Heyde ◽  
V. K. Rohatgi
Keyword(s):  
Law Of Large Numbers ◽  
Convergence Rates ◽  
Random Variables ◽  
Strong Law ◽  
Large Numbers ◽  
The Law ◽  
Independent And Identically Distributed

Introduction. Let Xi (i= 1, 2, 3,…) be a sequence of independent and identically distributed random variables with law ℒ(X) and write The Kolmogorov-Marcinkiewicz strong law of large numbers (Loève(6), p. 243) has the following statement:If E|X|r < ∞, then with cr = 0 or EX according as r 1 or r ≥ 1.

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