Some mean convergence and complete convergence theorems for sequences of $m$-linearly negative quadrant dependent random variables

summary: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,\dots $). 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 $n^{-1/p} \sum _{k=1}^{n} (X_k - EX_k) \to 0$ in $L^1$ (resp. in $L^p$). Moreover, for $1\le p < 2$, conditions are provided under which $n^{-1/p} \sum _{k=1}^{n} (X_k - EX_k)$ converges completely to $0$. The current work extends some results of Pyke and Root (1968) and it extends and improves some results of Wu, Wang, and Wu (2006). An open problem is posed