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

Yongfeng Wu; Andrew Rosalsky; Andrei Volodin

Applications of Mathematics (2013)

  • Volume: 58, Issue: 5, page 511-529
  • ISSN: 0862-7940

Abstract

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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 , ⋯ ). For a sequence of m -linearly negative quadrant dependent random variables { X n , n ≥ 1 } and 1 < p < 2 (resp. 1 ≤ p < 2 ), conditions are provided under which n - 1 / p ∑ k = 1 n ( X k - E X k ) → 0 in L 1 (resp. in L p ). Moreover, for 1 ≤ p < 2 , conditions are provided under which n - 1 / p ∑ 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 results of Wu, Wang, and Wu (2006). An open problem is posed.

How to cite

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Wu, Yongfeng, Rosalsky, Andrew, and Volodin, Andrei. "Some mean convergence and complete convergence theorems for sequences of $m$-linearly negative quadrant dependent random variables." Applications of Mathematics 58.5 (2013): 511-529. <http://eudml.org/doc/260633>.

@article{Wu2013,
abstract = {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 $\lbrace X_n, n\ge 1\rbrace $ and $1<p<2$ (resp. $1\le p <2$), conditions are provided under which $n^\{-1/p\} \sum _\{k=1\}^\{n\} (X_k - EX_k) \rightarrow 0$ 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.},
author = {Wu, Yongfeng, Rosalsky, Andrew, Volodin, Andrei},
journal = {Applications of Mathematics},
keywords = {$m$-linearly negative quadrant dependence; mean convergence; complete convergence; -linearly negative quadrant dependence; -convergence; complete convergence},
language = {eng},
number = {5},
pages = {511-529},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Some mean convergence and complete convergence theorems for sequences of $m$-linearly negative quadrant dependent random variables},
url = {http://eudml.org/doc/260633},
volume = {58},
year = {2013},
}

TY - JOUR
AU - Wu, Yongfeng
AU - Rosalsky, Andrew
AU - Volodin, Andrei
TI - Some mean convergence and complete convergence theorems for sequences of $m$-linearly negative quadrant dependent random variables
JO - Applications of Mathematics
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 5
SP - 511
EP - 529
AB - 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 $\lbrace X_n, n\ge 1\rbrace $ and $1<p<2$ (resp. $1\le p <2$), conditions are provided under which $n^{-1/p} \sum _{k=1}^{n} (X_k - EX_k) \rightarrow 0$ 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.
LA - eng
KW - $m$-linearly negative quadrant dependence; mean convergence; complete convergence; -linearly negative quadrant dependence; -convergence; complete convergence
UR - http://eudml.org/doc/260633
ER -

References

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