Complete moment and integral convergence for sums of negatively associated random variables

For a sequence of identically distributed negatively associated random variables “X _n ; n ≥ 1” with partial sums S _n = Σ _i=1 ⁿ X _i , n ≥ 1, refinements are presented of the classical Baum-Katz and Lai complete convergence theorems. More specifically, necessary and sufficient moment conditions are provided for complete moment convergence of the form

$ \sum\limits_{n \geqslant n_0 } {n^{r - 2 - \tfrac{1} {{pq}}} a_n E\left( {\mathop {\max }\limits_{1 \leqslant k \leqslant n} \left| {S_k } \right|^{\tfrac{1} {q}} - \varepsilon b_n^{\tfrac{1} {{pq}}} } \right)^ + < \infty } $ \sum\limits_{n \geqslant n_0 } {n^{r - 2 - \tfrac{1} {{pq}}} a_n E\left( {\mathop {\max }\limits_{1 \leqslant k \leqslant n} \left| {S_k } \right|^{\tfrac{1} {q}} - \varepsilon b_n^{\tfrac{1} {{pq}}} } \right)^ + < \infty }
Keywords:	Baum-Katz＇s law  Lai＇s law  complete moment convergence  complete integral convergence  convergence rate of tail probabilities  sums of identica/ly distributed and negatively associated random variables
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