ABSTRACT

In this paper, we obtain the strong law of large numbers for a 2-dimensional array of pairwise negatively dependent random variables which are not required to be identically distributed. We found the sufficient conditions of strong law of large numbers for the difference of random variables which independent and identically distributed conditions are regarded. In this study, we consider the limit as which is stronger than the limit as when m, n are natural numbers.

1. Introduction and Main Results

as

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.

REFERENCES