Complete convergence theorems for normed row sums from an array of rowwise pairwise negative quadrant dependent random variables with application to the dependent bootstrap

Andrew Rosalsky; Yongfeng Wu

Applications of Mathematics (2015)

  • Volume: 60, Issue: 3, page 251-263
  • ISSN: 0862-7940

Abstract

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Let { X n , j , 1 ≤ j ≤ m ( n ) , n ≥ 1 } be an array of rowwise pairwise negative quadrant dependent mean 0 random variables and let 0 < b n → ∞ . Conditions are given for ∑ j = 1 m ( n ) X n , j / b n → 0 completely and for max 1 ≤ k ≤ m ( n ) | ∑ j = 1 k X n , j | / b n → 0 completely. As an application of these results, we obtain a complete convergence theorem for the row sums ∑ j = 1 m ( n ) X n , j * of the dependent bootstrap samples { { X n , j * , 1 ≤ j ≤ m ( n ) } , n ≥ 1 } arising from a sequence of i.i.d. random variables { X n , n ≥ 1 } .

How to cite

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Rosalsky, Andrew, and Wu, Yongfeng. "Complete convergence theorems for normed row sums from an array of rowwise pairwise negative quadrant dependent random variables with application to the dependent bootstrap." Applications of Mathematics 60.3 (2015): 251-263. <http://eudml.org/doc/270120>.

@article{Rosalsky2015,
abstract = {Let $\lbrace X_\{n,j\}, 1\le j\le m(n), n\ge 1\rbrace $ be an array of rowwise pairwise negative quadrant dependent mean 0 random variables and let $0<b_n\rightarrow \infty $. Conditions are given for $\sum \nolimits _\{j=1\}^\{m(n)\}X_\{n,j\}/b_n\rightarrow 0$ completely and for $\max \nolimits _\{1\le k\le m(n)\}\Bigl |\sum \nolimits _\{j=1\}^kX_\{n,j\}\Big |/b_n\rightarrow 0$ completely. As an application of these results, we obtain a complete convergence theorem for the row sums $\sum \nolimits _\{j=1\}^\{m(n)\}X_\{n,j\}^*$ of the dependent bootstrap samples $\lbrace \lbrace X_\{n,j\}^*, 1\le j\le m(n)\rbrace , n\ge 1\rbrace $ arising from a sequence of i.i.d. random variables $\lbrace X_n, n\ge 1\rbrace $.},
author = {Rosalsky, Andrew, Wu, Yongfeng},
journal = {Applications of Mathematics},
keywords = {array of rowwise pairwise negative quadrant dependent random variables; complete convergence; dependent bootstrap; sequence of i.i.d. random variables; array of rowwise pairwise negative quadrant dependent random variables; complete convergence; dependent bootstrap; sequence of i.i.d. random variables},
language = {eng},
number = {3},
pages = {251-263},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Complete convergence theorems for normed row sums from an array of rowwise pairwise negative quadrant dependent random variables with application to the dependent bootstrap},
url = {http://eudml.org/doc/270120},
volume = {60},
year = {2015},
}

TY - JOUR
AU - Rosalsky, Andrew
AU - Wu, Yongfeng
TI - Complete convergence theorems for normed row sums from an array of rowwise pairwise negative quadrant dependent random variables with application to the dependent bootstrap
JO - Applications of Mathematics
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 3
SP - 251
EP - 263
AB - Let $\lbrace X_{n,j}, 1\le j\le m(n), n\ge 1\rbrace $ be an array of rowwise pairwise negative quadrant dependent mean 0 random variables and let $0<b_n\rightarrow \infty $. Conditions are given for $\sum \nolimits _{j=1}^{m(n)}X_{n,j}/b_n\rightarrow 0$ completely and for $\max \nolimits _{1\le k\le m(n)}\Bigl |\sum \nolimits _{j=1}^kX_{n,j}\Big |/b_n\rightarrow 0$ completely. As an application of these results, we obtain a complete convergence theorem for the row sums $\sum \nolimits _{j=1}^{m(n)}X_{n,j}^*$ of the dependent bootstrap samples $\lbrace \lbrace X_{n,j}^*, 1\le j\le m(n)\rbrace , n\ge 1\rbrace $ arising from a sequence of i.i.d. random variables $\lbrace X_n, n\ge 1\rbrace $.
LA - eng
KW - array of rowwise pairwise negative quadrant dependent random variables; complete convergence; dependent bootstrap; sequence of i.i.d. random variables; array of rowwise pairwise negative quadrant dependent random variables; complete convergence; dependent bootstrap; sequence of i.i.d. random variables
UR - http://eudml.org/doc/270120
ER -

References

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