• Journal of the Korean Mathematical Society
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• v.36 no.1
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• pp.37-49
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• 1999

We derive the almost sure convergence for weighted sums of random variables which are either pairwise positive quadrant dependent or pairwise positive quadrant dependent or pairwise negative quadrant dependent and then apply this result to obtain the almost sure convergence of weighted averages. e also extend some results on the strong law of large numbers for pairwise independent identically distributed random variables established in Petrov to the weighted sums of pairwise negative quadrant dependent random variables.

• Communications of the Korean Mathematical Society
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• v.14 no.4
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• pp.797-804
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• 1999

In this paper we derive the almost sure convergence of weighted sums of 2-dimensional arrays of random variables which are either pairwise positive quadrant dependent or associated. Our re-sults imply and extension of Etemadi's(1983) strong laws of large numbers for weighted sums of nonnegative random variables to the 2-dimensional case.

• Journal of applied mathematics & informatics
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• v.30 no.1_2
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• pp.327-336
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• 2012

Let {$X_{ni}$, $i{\geq}1$, $n{\geq}1$} be an array of rowwise and pairwise negatively quadrant dependent random variables with mean zero, {$a_{ni}$, $i{\geq}1$, $n{\geq}1$} an array of weights and {$b_n$, $n{\geq}1$} an increasing sequence of positive integers. In this paper we consider some results concerning complete convergence of ${\sum}_{i=1}^{bn}a_{ni}X_{ni}$.

• Bulletin of the Korean Mathematical Society
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• v.35 no.4
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• pp.709-718
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• 1998

In this paper we obtain strong laws of large numbers for 2-dimensional arrays of random variables which are either pairwise positive quadrant dependent or associated. Our results imply extensions of Etemadi`s strong laws of large numbers for nonnegative random variables to the 2-dimensional case.

• Communications of the Korean Mathematical Society
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• v.19 no.4
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• pp.765-773
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• 2004

In this paper we define the notion of asymptotically quadrant independent random field and derive the strong laws of large numbers for this random field.

• Communications of the Korean Mathematical Society
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• v.9 no.4
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• pp.951-959
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• 1994

A sequence ${X_j : j \geq 1}$ of random variables is said to be pairwise positive quadrant dependent (pairwise PQD) if for any real $r-i,r_j$ and $i \neq j$ $$ P{X_i > r_i,X_j > r_j} \geq P{X_i > r_i}P{X_j > r_j} $$ (see [8]) and a sequence ${X_j : j \geq 1}$ of random variables is said to be associated if for any finite collection ${X_{i(1)},...,X_{j(n)}}$ and any real coordinatewise nondecreasing functions f,g on $R^n$ $$ Cov(f(X_{i(1)},...,X_{j(n)}),g(X_{j(1)},...,X_{j(n)})) \geq 0, $$ whenever the covariance is defined (see [6]). Instead of association Cox and Grimmett's [4] original central limit theorem requires only that positively linear combination of random variables are PQD (cf. Theorem $A^*$).