Mean Convergence Theorems and Weak Laws of Large Numbers for Arrays of Measurable Operators under Some Conditions of Uniform Integrability

2019 ◽  
Vol 40 (8) ◽  
pp. 1218-1229
Author(s):  
Nguyen Van Quang ◽  
Do The Son ◽  
Tien-Chung Hu ◽  
Nguyen Van Huan
2006 ◽  
Vol 2006 ◽  
pp. 1-15 ◽  
Author(s):  
Le Van Thanh

For a double array of random variables {Xmn, m ≥ 1, n ≥ 1}, mean convergence theorems and weak laws of large numbers are established. For the mean convergence results, conditions are provided under which ∑i=1km∑j=1lnamnij(Xij−EXij)→Lr0(0<r≤2) where {amnij;m,n,i,j≥1} are constants, and {kn,n≥1} and {ln,n≥1} are sequences of positive integers. The weak law results provide conditions for ∑i=1Tm∑j=1τnamnij(Xij−EXij)→p0 to hold where {Tm,m≥1} and {τn,n≥1} are sequences of positive integer-valued random variables. The sharpness of the results is illustrated by examples.


2017 ◽  
Vol 58 (3-4) ◽  
pp. 455-463
Author(s):  
YAN-JIAO MENG

The $L_{r}$ convergence and a class of weak laws of large numbers are obtained for sequences of $\widetilde{\unicode[STIX]{x1D70C}}$-mixing random variables under the uniform Cesàro-type condition. This is weaker than the $p$th-order Cesàro uniform integrability.


1985 ◽  
Vol 8 (4) ◽  
pp. 805-812 ◽  
Author(s):  
Xiang Chen Wang ◽  
M. Bhaskara Rao

Under uniform integrability condition, some Weak Laws of large numbers are established for weighted sums of random variables generalizing results of Rohatgi, Pruitt and Khintchine. Some Strong Laws of Large Numbers are proved for weighted sums of pairwise independent random variables generalizing results of Jamison, Orey and Pruitt and Etemadi.


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