Geodesic
Some mean convergence and complete convergence theorems for sequences of $m$-linearly negative quadrant dependent random variables
Applications of Mathematics, Tome 58 (2013) no. 5, pp. 511-529.

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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 $1$ (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.
DOI : 10.1007/s10492-013-0030-6
Classification : 60F15, 60F25
Keywords: $m$-linearly negative quadrant dependence; mean convergence; complete convergence 
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Wu, Yongfeng; Rosalsky, Andrew; Volodin, Andrei. Some mean convergence and complete convergence theorems for sequences of $m$-linearly negative quadrant dependent random variables. Applications of Mathematics, Tome 58 (2013) no. 5, pp. 511-529. doi : 10.1007/s10492-013-0030-6. http://geodesic.mathdoc.fr/articles/10.1007/s10492-013-0030-6/

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