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.
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 
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
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     title = {Some mean convergence and complete convergence theorems for sequences of $m$-linearly negative quadrant dependent random variables},
     journal = {Applications of Mathematics},
     pages = {511--529},
     year = {2013},
     volume = {58},
     number = {5},
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     zbl = {06282094},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10492-013-0030-6/}
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