Strong convergence for weighted sums of negatively associated arrays

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} } $ 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} } $ for η ∈ (0, 1], where ϕ is some function. The author studies necessary and sufficient conditions of $\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 } } $ for all ɛ > 0, where A _n, B _n, C _n and D _n are some positive constants, m _n ∈ ℕ with m _n/n ^η → ∞. The results of Lanzinger and Stadtmüller in 2003 are extended from the i.i.d. case to the case of the negatively associated, not necessarily identically distributed random variables. Also, the result of Pruss in 2003 on independent variables reduces to a special case of the present paper; furthermore, the necessity part of his result is complemented.

Keywords

Tail probability / Negatively associated random variable / Weighted sum

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