Abstract
Let \({\{X_n, n\geqq 1\}}\) be a sequence of identically distributed negatively orthant dependent random variables and let \({\{a_{ni}, 1 \leqq i \leqq n, n \geqq 1\}}\) be an array of constants satisfying \({\sum_{i=1}^n |a_{ni} |^\alpha = O(n)}\) for some \({0 < \alpha < 2}\). Set \({b_n = n^{1/\alpha}({\rm log} n)^{1/\gamma}}\) and \({\gamma > \alpha}\). We give necessary and sufficient conditions for complete convergence of the form
As corollaries, strong laws of large numbers for weighted sums are obtained.
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Chen, P., Sung, S.H. Complete convergence and strong laws of large numbers for weighted sums of negatively orthant dependent random variables. Acta Math. Hungar. 148, 83–95 (2016). https://doi.org/10.1007/s10474-015-0559-9
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DOI: https://doi.org/10.1007/s10474-015-0559-9
Key words and phrases
- complete convergence
- weighted sum
- strong law of large numbers
- negatively orthant dependent random variable
- Rosenthal inequality