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

Abstract

For a sequence of identically distributed negatively associated random variables $\{X_n; n\geq 1\}$ with partial sums $S_n=\sum_{i=1}^nX_i, n\geq 1$, 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 $$ \sum_{n \ge n_0} n^{r -2 -\frac{1}{pq}} a_n E(\max_{1 \le k \le n}|S_k|^{\frac{1}{q}} - \epsilon b_n^{\frac{1}{pq}})^+ < \infty $$ to hold where $r>1, q>0$ and either $n_0=1, 0<p<2, a_n=1, b_n=n$ or $n_0=3, p=2, a_n=(\log n)^{-\frac{1}{2q}}, b_n=n\log n$. These results extend results of Chow (1988) and Li and Sp\u{a}taru (2005) 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.