Throughout this paper, let denote the set of nonnegative integer, let be a sequence of random variables defined on probability space (), and put . The symbol will denote a generic constant () which is not necessarily the same one in each appearance.

In [1], Jajte studied a large class of summability method as follows: a sequence is summable to by the method () if

The main result of Jajte is as follows.

Theorem 1.1.

Let be a positive, increasing function and a positive function such that satisfies the following conditions.

(1)For some , is strictly increasing on with range .

(2)There exist C and a positive integer such that , .

(3)There exist constants and such that , .

Then, for i.i.d. random variables ,

where is the inverse of function , is the indicator of event .

Motivated by Jajte [1], the present paper is devoted to the study of the limiting behavior of sums when are dependent RVs In particular, we willl consider the case when are NA RVs and obtain some general results on the complete convergence of dependent RVs First, we shall give some definitions.

Definition 1.2.

A finite family of random variables is said to be negatively associated (abbreviated NA) if, for every pair of disjoint subsets and of , we have

whenever and are coordinatewise increasing and the covariance exists.

Definition 1.3.

A finite family of random variables is said to be positively associated (abbreviated PA) if

whenever and are coordinatewise increasing and the covariance exists.

An infinite family of random variables is NA (resp., PA) if every finite subfamily is NA (resp., PA).

Let be the -algebra generated by RVs, .

Definition 1.4.

A sequence of random variables is said to be -dependence if and are independent for all and such that .

Definition 1.5.

A sequence of random variables is said to be -mixing (or uniformly strong mixing), if

These concepts of dependence were introduced by Esary et al. [2] and Joag-Dev and Proschan [3]. Their basic properties may be found in [2, 3] and the references therein.

Definition 1.6.

Let be a sequence of random variables which is said to be: uniformly bounded by a random variable (we write ) if there exists a constant , for almost every , such that

Remark 1.7.

The uniformly bounded random variables in (1.6) can be insured by moment conditions. For example, if

then there exists a uniformly bounded random variable such that .

The structure of this paper is as follows. Some needed technical results will be presented in Section 2. The strong law of large numbers for NA RVs will be established in Section 3. The Spitzer and Hus-Robbins-type law of large numbers will be presented in Sections 4 and 5, respectively.