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

Abstract

For a sequence of identically distributed negatively associated random variables “Xn; n ≥ 1” with partial sums Sn = Σi=1nXi, 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 $$ \sum\limits_{n \geqslant n_0 } {n^{r - 2 - \tfrac{1} {{pq}}} a_n E\left( {\mathop {\max }\limits_{1 \leqslant k \leqslant n} \left| {S_k } \right|^{\tfrac{1} {q}} - \varepsilon b_n^{\tfrac{1} {{pq}}} } \right)^ + < \infty } $$ to hold where r > 1, q > 0 and either n0 = 1, 0 < p < 2, an = 1, bn = n or n0 = 3, p = 2, an = (log n)−1/2q, bn = n log n. These results extend results of Chow and of Li and Spataru from the independent and identically distributed case to the identically distributed negatively associated setting. The complete moment convergence is also shown to be equivalent to a form of complete integral convergence.