In this paper the authors study the convergence properties for arrays of rowwise pairwise negatively quadrant dependent random variables. The results extend and improve the corresponding theorems of T. C. Hu, R. L. Taylor: On the strong law for arrays and for the bootstrap mean and variance, Int. J. Math. Math. Sci 20 (1997), 375–382.

In this paper, some new results on complete convergence and complete moment convergence for sequences of pairwise negatively quadrant dependent random variables are presented. These results improve the corresponding theorems of S. X. Gan, P. Y. Chen (2008) and H. Y. Liang, C. Su (1999).

Let $\{{X}_{n,j},1\le j\le m\left(n\right),n\ge 1\}$ be an array of rowwise pairwise negative quadrant dependent mean 0 random variables and let $0<{b}_{n}\to \infty $. Conditions are given for ${\sum}_{j=1}^{m\left(n\right)}{X}_{n,j}/{b}_{n}\to 0$ completely and for ${max}_{1\le k\le m\left(n\right)}\left|{\sum}_{j=1}^{k}{X}_{n,j}\right|/{b}_{n}\to 0$ completely. As an application of these results, we obtain a complete convergence theorem for the row sums ${\sum}_{j=1}^{m\left(n\right)}{X}_{n,j}^{*}$ of the dependent bootstrap samples $\{\{{X}_{n,j}^{*},1\le j\le m\left(n\right)\},n\ge 1\}$ arising from a sequence of i.i.d. random variables $\{{X}_{n},n\ge 1\}$.

The structure of linearly negative quadrant dependent random variables is extended by introducing the structure of $m$-linearly negative quadrant dependent random variables ($m=1,2,\cdots $). For a sequence of $m$-linearly negative quadrant dependent random variables $\{{X}_{n},n\ge 1\}$ and $1<p<2$ (resp. $1\le p<2$), conditions are provided under which ${n}^{-1/p}{\sum}_{k=1}^{n}({X}_{k}-E{X}_{k})\to 0$ in ${L}^{1}$ (resp. in ${L}^{p}$). Moreover, for $1\le p<2$, conditions are provided under which ${n}^{-1/p}{\sum}_{k=1}^{n}({X}_{k}-E{X}_{k})$ converges completely to $0$. The current work extends some results of Pyke and Root (1968) and it extends and improves some...

The authors provide a correction to “Some mean convergence and complete convergence theorems for sequences of $m$-linearly negative quadrant dependent random variables”.

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