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.