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
Keyword(s):  
Uniform Integrability ◽  
Convergence Theorems ◽  
Laws Of Large Numbers ◽  
Mean Convergence ◽  
Large Numbers ◽  
Measurable Operators ◽  
Weak Laws

scholarly journals Mean convergence theorems and weak laws of large numbers for double arrays of random variables

2006 ◽  
Vol 2006 ◽  
pp. 1-15 ◽  
Author(s):  
Le Van Thanh
Keyword(s):  
Positive Integer ◽  
Random Variables ◽  
Convergence Theorems ◽  
Laws Of Large Numbers ◽  
Mean Convergence ◽  
Convergence Results ◽  
Large Numbers ◽  
The Mean ◽  
Weak Laws ◽  
Positive Integers

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.

THE CONVERGENCE AND WEAK LAWS OF LARGE NUMBERS FOR -MIXING RANDOM VARIABLES

2017 ◽  
Vol 58 (3-4) ◽  
pp. 455-463
Author(s):  
YAN-JIAO MENG
Keyword(s):  
Random Variables ◽  
Uniform Integrability ◽  
Type Condition ◽  
Laws Of Large Numbers ◽  
Large Numbers ◽  
Weak Laws ◽  
Mixing Random Variables

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.

scholarly journals A note on convergence of weighted sums of random variables

1985 ◽  
Vol 8 (4) ◽  
pp. 805-812 ◽  
Author(s):  
Xiang Chen Wang ◽  
M. Bhaskara Rao
Keyword(s):  
Integrability Condition ◽  
Random Variables ◽  
Independent Random Variables ◽  
Weighted Sums ◽  
Uniform Integrability ◽  
Laws Of Large Numbers ◽  
Sums Of Random Variables ◽  
Large Numbers ◽  
Pairwise Independent Random Variables ◽  
Weak Laws

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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