Strong Convergence for Weighted Sums of Negatively Associated Arrays |
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| Authors: | Hanying LIANG and Jingjing ZHANG |
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| Institution: | Department of Mathematics,Tongji University,Shanghai 200092,China |
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| Abstract: | Let “X
ni
” be an array of rowwise negatively associated random variables and $
T_{nk} = \sum\limits_{i = 1}^k {i^\alpha X_{ni} }
$
T_{nk} = \sum\limits_{i = 1}^k {i^\alpha X_{ni} }
for α ≥ −1, $
S_{nk} = \sum\limits_{\left| i \right| \leqslant k} {\varphi \left( {\tfrac{i}
{{n^\eta }}} \right)\tfrac{1}
{{n^\eta }}X_{ni} }
$
S_{nk} = \sum\limits_{\left| i \right| \leqslant k} {\varphi \left( {\tfrac{i}
{{n^\eta }}} \right)\tfrac{1}
{{n^\eta }}X_{ni} }
for η ∈ (0, 1], where ϕ is some function. The author studies necessary and sufficient conditions of
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$
\sum\limits_{n = 1}^\infty {A_n P\left( {\mathop {max}\limits_{1 \leqslant k \leqslant n} \left| {T_{nk} } \right| > \varepsilon B_n } \right) < \infty and \sum\limits_{n = 1}^\infty {C_n P\left( {\mathop {\max }\limits_{0 \leqslant k \leqslant m_n } \left| {S_{nk} } \right| > \varepsilon D_n } \right) < \infty } }
$
\sum\limits_{n = 1}^\infty {A_n P\left( {\mathop {max}\limits_{1 \leqslant k \leqslant n} \left| {T_{nk} } \right| > \varepsilon B_n } \right) < \infty and \sum\limits_{n = 1}^\infty {C_n P\left( {\mathop {\max }\limits_{0 \leqslant k \leqslant m_n } \left| {S_{nk} } \right| > \varepsilon D_n } \right) < \infty } } |
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| Keywords: | Tail probability Negatively associated random variable Weighted sum |
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