### A generalization of the global limit theorems of R. P. Agnew.

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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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