Strong Convergence for Weighted Sums of Negatively Associated Arrays

Let “X _ni ” be an array of rowwise negatively associated random variables and $T_{nk} = sumlimits_{i = 1}^k {i^alpha X_{ni} } $T_{nk} = sumlimits_{i = 1}^k {i^alpha X_{ni} } for α ≥ −1, $S_{nk} = sumlimits_{left| i right| leqslant k} {varphi left( {tfrac{i}{{n^eta }}} right)tfrac{1}{{n^eta }}X_{ni} } $S_{nk} = sumlimits_{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

$sumlimits_{n = 1}^infty {A_n Pleft( {mathop {max}limits_{1 leqslant k leqslant n} left| {T_{nk} } right| > varepsilon B_n } right) < infty and sumlimits_{n = 1}^infty {C_n Pleft( {mathop {max }limits_{0 leqslant k leqslant m_n } left| {S_{nk} } right| > varepsilon D_n } right) < infty } } $sumlimits_{n = 1}^infty {A_n Pleft( {mathop {max}limits_{1 leqslant k leqslant n} left| {T_{nk} } right| > varepsilon B_n } right) < infty and sumlimits_{n = 1}^infty {C_n Pleft( {mathop {max }limits_{0 leqslant k leqslant m_n } left| {S_{nk} } right| > varepsilon D_n } right) < infty } }
Keywords:	Tail probability  Negatively associated random variable  Weighted sum
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