Strong Law of Large Numbers for a 2-Dimensional Array of Pairwise Negatively Dependent Random Variables ()
1. Introduction and Main Results
Let
be a sequence of random variables. We say that
satisfies the strong law of large numbers
(SLLN) if there exist sequences of real numbers 
and 
such that 
as
. where 
and the abbreviation a.s. stands for almost surely.
To study the strong law of large numbers, there is a simple question come in mind. When does the sequence 
satisfy the SLLN? Many conditions of the sequence 
have been found for this question. The SLLN are investigated extensively in the literature especially to the case of a sequence of independent random variables (see for examples in [1-3]). After concepts of dependence was introduced, it is interesting to study the SLLN with condition of dependence.
A sequence 
of random variables is said to be pairwise positively dependent (pairwise PD) if for any 
and
,
and it is said to be pairwise negatively dependent (pairwise ND) if for any 
and
,
Theorem 1.1-1.5 are examples of SLLN for a sequence of pairwise PD and pairwise ND random variables.
Theorem 1.1. (Birkel, [4]) Let 
be a sequence of pairwise PD random variables with finite variances. Assume 1) 
2) 
Then 
as 
Theorem 1.2. (Azarnoosh, [5]) Let 
be a sequence of pairwise ND random variables with finite variances. Assume 1) 
2) 
Then 
as 
Theorem 1.3. (Nili Sani, Azarnoosh and Bozorgnia, [6]) Let 
be a positive and increasing sequence such that 
as 
Let 
be a sequence of pairwise ND random variables with finite variances such that 1) 
2) 
Then 
as 
In this work, we study the SLLN for a 2-dimensional array of pairwise ND random variables. We say that
satisfies the SLLN if there exist double sequences of real numbers 
and 
such that 
as 
where 
In 1998, Kim, Beak and Seo investigated SLLN for a 2-dimensional array of pairwise PD random variables and it was generalized to a case of weighted sum of 2-dimensional array of pairwise PD random variables by Kim, Baek and Han in one year later. The followings are their results.
A double sequence 
is said to be pairwise positively dependent (pairwise PD) if for any 
and 
Theorem 1.4. (Kim, Beak and Seo, [7]) Let
be a 2-dimensional array of pairwise PD random variables with finite variances. Assume 1) 
2) 
Then 
as 
Theorem 1.5. (Kim, Baek and Han, [8]) Let 
be a 2-dimensional array of positive numbers and 
such that 
and 
as 
Let 
be a 2-dimensional array of pairwise PD random variables with finite variances such that 1) 
2) 
Then 
as 
where 
Observe that, for a double indexed sequence of real number 
the convergence as 
implies the convergence as
. However, a double sequence 
where 
shows us that the converse is not true in general.
Our goal is to obtain the SLLN for 2-dimensional array of random variables in case of pairwise ND.
A double sequence 
is said to be pairwise negtively dependent (pairwise ND) if for any 
and 
The followings are SLLNs for a 2-dimensional array of pairwise ND random variables which are all our results.
Theorem 1.6. Let 
and 
be increasing sequences of positive numbers such that 
which 
as 
and 
as 
Let 
be a 2-dimensional array of pairwise ND random variables with finite variances. If there exist real numbers 
such that
then for any double sequence 
such that
for every 
as 
The next theorem is the SLLN for the difference of random variables which independent and identically distributed conditions are regarded.
Theorem 1.7. Let
and 
be 2dimensional arrays of random variables on a probability space (Ω, F, P). If
then
as 
Corollary 1.8 and Corollary 1.9 follow directly from Theorem 1.6 by choosing 
and 
where 
and 
with p = q = 4, respectively.
Corollary 1.8. Let 
and 
be increasing sequences of positive numbers such that 
which 
as 
and 
as 
Let 
be a 2-dimensional array of pairwise ND random variables with finite variances. If there exist 
such that
then for any 
as 
Corollary 1.9. Let 
be a 2-dimensional array of pairwise ND random variables with finite variances. If
then
as 
2. Auxiliary Results
In this section, we present some materials which will be used in obtaining the SLLN’s in the next section.
Proposition 2.1. (Móricz, [9]) Let 
be a double sequence of positive numbers such that for all 
and 
as 
Let 
be a double sequence of real numbers. Assume that 1) 
2) 
for every 
and 
for every 
Then 
as 
The following proposition is a Borel-Cantelli lemma for a sequence of double indexed events Proposition 2.2. Let 
be a double sequence of events on a probability space 
Then
where 
Proof. Let 
be such that 
First note that
where 
denote the greatest integer smaller than or equal 
and hence
Therefore 
and
This completes the proof. □
3. Proof of Main Results
Proof of Theorem 1.6
Let 
and define 
and 
Clearly, f and g are increasing whose facts
and 
which imply that 
and
.
Let 
be given. By using the fact that 
for 
([10], p. 313), we have
From this fact and Chebyshev’s inequality, we have
For each 
let
and 
and
. Since 
and
, we have 
and
. From this facts and (3.1), we have
Since 
and
, we have 
and
.
From this facts and (3.2) together with our assumption 2), we have
By Proposition 2.2 with
we have 
and this hold for every 
By using the same idea with Theorem 4.2.2 ([11], p. 77), we can prove that
as 
Proof of Theorem 1.7
Let 
By Proposition 2.2, we have
For every 
we will show that
(3.3)
for every
,
(3.4)
and for every 
(3.5)
From (3.3), (3.4) and (3.5), we can apply Proposition 2.1 with 
that
as 
We here note that 
as 
implies 
as
. Hence
as 
To prove (3.3), (3.4) and (3.5), let 
Then there exists 
such that for 
(3.6)
Thus for each 
and
are different only finitely many terms. This implies that (3.3) holds.
For fixed 
we can find a large 
such that (3.6) holds for all 
which means that there are only finitely many different terms of 
and 
So for fixed 
.
Similarly, for fixed 
Now (3.4) and (3.5) are now proved and this ends the proof.
Remark 3.1. In case of m fixed and 
by considering the limit as 
we also obtain the corresponding results for a case of 1-dimensional pairwise ND random variables.
4. Example
Example 4.1 A box contains pq balls of p different colors and q different sizes in each color. Pick 2 balls randomly.
Let 
and 
be a random variable indicating the presence of a ball of the ith color and the jth size such that
For 
let 
be a random variable defined by
Proof. By a direct calculation, we have
’s are pairwise ND random variables, i.e. for 
that 
and 
Note that
and 
Hence,
By applying Theorem 1.6, for any double sequence 
such that 
for every m
, we have 
as 
5. Acknowledgements
The authors would like to thank referees for valuable comments and suggestions which have helped improving our work. The first author gives an appreciation and thanks to the Institute for the Promotion of Teaching Science and Technology for financial support.