On complete convergence in Marcinkiewicz-Zygmund type SLLN for negatively associated random variables

Abstract

We present a generalization of Baum-Katz theorem for negatively associated random variables satisfying some cover condition.

Complete Moment Convergence for Arrays of Rowwise Negatively Associated Random Variables and Its Application in Non-Parametric Regression Model

“…In Theorem 3.3, we not only consider the case αp = 1, but also verify that (3.8) and (3.10) can be equivalent to a much stronger result (3.9). Consequently, our result markedly improve the corresponding one of Kuczmaszewska [14] for NA random variables and thus the corresponding ones for independent random variables.…”

“…Proof: The statement (3.8) ⇒ (3.9) ⇒ (3.10) follows immediately from Theorems 3.1 and 3.2, and similar to the proof in Kuczmaszewska [14], we have that (3.10) implies…”

“…The results obtained in this paper correct the result obtained in Ko [13] and also improve and extend Theorems A and B for NA random variables. The meaningful case αp = 1 that was not considered in either Kuczmaszewska [14] or Ko [13] is also considered in this paper. It is worth mentioning that the proof of our main results is different from that of Ko [13].…”

“…Kuczmaszewska [14] only obtained that (3.8) and (3.10) are equivalent for the case αp > 1. In Theorem 3.3, we not only consider the case αp = 1, but also verify that (3.8) and (3.10) can be equivalent to a much stronger result (3.9).…”

“…A familiar extension of independence is negative association, which was introduced by Joag-Dev and Proschan [11] as follows. Kuczmaszewska [14] extended the Hsu-Robbins-Erdös's result from independent sequence to NA sequence and weakened the identical distribution assumption as follows.…”