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 i^{th} color and the j^{th} 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.