Strong Law of Large Numbers for a 2-Dimensional Array of Pairwise Negatively Dependent Random Variables ()
1. Introduction and Main Results
![](6-1240165.files/image003.gif)
Let
![](https://www.scirp.org/html/6-1240165\9ed984be-9d41-4673-9094-d76a0377c8da.jpg)
be a sequence of random variables. We say that
![](https://www.scirp.org/html/6-1240165\eb7fb1bf-caef-4974-adec-3593849601ac.jpg)
satisfies the strong law of large numbers
(SLLN) if there exist sequences of real numbers ![](https://www.scirp.org/html/6-1240165\df06968e-298a-4c2e-b69c-dc70d4cd9b67.jpg)
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
,
![](https://www.scirp.org/html/6-1240165\90d07047-0063-45eb-b3e3-0298842da19c.jpg)
and it is said to be pairwise negatively dependent (pairwise ND) if for any
and
,
![](https://www.scirp.org/html/6-1240165\c9f95dc7-7959-40d8-baf2-d809a79097ae.jpg)
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) ![](https://www.scirp.org/html/6-1240165\a4fefdf6-ad36-487d-8d40-90aa2bd080cb.jpg)
2) ![](https://www.scirp.org/html/6-1240165\88c6433e-8079-4f02-9393-a47f2361acca.jpg)
Then
as ![](https://www.scirp.org/html/6-1240165\0370bec9-9f84-4293-b3e3-42304dbac6a4.jpg)
Theorem 1.2. (Azarnoosh, [5]) Let
be a sequence of pairwise ND random variables with finite variances. Assume 1) ![](https://www.scirp.org/html/6-1240165\ca27240e-8578-4ee9-afbb-c1a9fdb9eb77.jpg)
2) ![](https://www.scirp.org/html/6-1240165\a2a44047-561c-47b8-8286-9cb2211b55bd.jpg)
Then
as ![](https://www.scirp.org/html/6-1240165\e7f229f8-febe-47cb-8fa2-abd18c14962d.jpg)
Theorem 1.3. (Nili Sani, Azarnoosh and Bozorgnia, [6]) Let
be a positive and increasing sequence such that
as ![](https://www.scirp.org/html/6-1240165\7cc225c3-26bb-4095-a311-e32680e619a6.jpg)
Let
be a sequence of pairwise ND random variables with finite variances such that 1) ![](https://www.scirp.org/html/6-1240165\e5be2609-4fb5-42f3-8900-4071b7b7d495.jpg)
2) ![](https://www.scirp.org/html/6-1240165\b415e236-053f-4275-926a-e778b82f63ca.jpg)
Then
as ![](https://www.scirp.org/html/6-1240165\703ed9f9-aa12-4d11-91ed-7571be6160fb.jpg)
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 ![](https://www.scirp.org/html/6-1240165\051877f1-01e3-4d1c-9227-cdd24ebd1993.jpg)
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 ![](https://www.scirp.org/html/6-1240165\65793736-b2fb-49d7-b28b-3a51ab0a574a.jpg)
![](https://www.scirp.org/html/6-1240165\7a26511d-123f-49a9-a47c-887feacea8e4.jpg)
Theorem 1.4. (Kim, Beak and Seo, [7]) Let
be a 2-dimensional array of pairwise PD random variables with finite variances. Assume 1) ![](https://www.scirp.org/html/6-1240165\a938f08e-7bf3-4580-8c5b-bbe247665a8d.jpg)
2) ![](https://www.scirp.org/html/6-1240165\4f84fa38-58b0-4d1b-8098-2fafea4b21f7.jpg)
Then
as ![](https://www.scirp.org/html/6-1240165\c1ec5108-7c5f-4a23-a3c5-eb65e588c8a1.jpg)
Theorem 1.5. (Kim, Baek and Han, [8]) Let
be a 2-dimensional array of positive numbers and
such that
and
as ![](https://www.scirp.org/html/6-1240165\4469783f-bd60-46ab-a8ab-41d38b5e0b37.jpg)
Let
be a 2-dimensional array of pairwise PD random variables with finite variances such that 1) ![](https://www.scirp.org/html/6-1240165\b78aaec8-605f-46d2-8545-ca79d1058f3e.jpg)
2) ![](https://www.scirp.org/html/6-1240165\af564feb-b3e3-4945-b055-749b0e2531a8.jpg)
Then
as
where ![](https://www.scirp.org/html/6-1240165\67e16fc6-cb47-4052-989a-6d571f7ea964.jpg)
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 ![](https://www.scirp.org/html/6-1240165\b25bd0f3-5b8d-4f2f-928c-0fde471eb1ce.jpg)
![](https://www.scirp.org/html/6-1240165\c9b90cdb-da40-4c58-a3dd-cfb7adb5e75c.jpg)
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 ![](https://www.scirp.org/html/6-1240165\080f28f4-321e-4a3b-9bc5-80581447e0cc.jpg)
Let
be a 2-dimensional array of pairwise ND random variables with finite variances. If there exist real numbers
such that
![](https://www.scirp.org/html/6-1240165\09e97532-f193-46f0-b049-8e9dbcc0bc24.jpg)
then for any double sequence
such that
for every ![](https://www.scirp.org/html/6-1240165\b0972d22-8326-4c77-bddf-dfc71360f02a.jpg)
as ![](https://www.scirp.org/html/6-1240165\2952fa1c-d40e-47de-85c4-0427fd98dc4d.jpg)
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
![](https://www.scirp.org/html/6-1240165\01ab77b5-b2ea-4ddc-b5ca-3ebac2e6ad75.jpg)
then
![](https://www.scirp.org/html/6-1240165\de8419ce-8f5d-4ffe-b418-d2458ff35373.jpg)
as ![](https://www.scirp.org/html/6-1240165\202f6d5f-27ba-4b16-84da-846a707d4200.jpg)
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 ![](https://www.scirp.org/html/6-1240165\9315128f-74b1-4e33-b9f2-fb026fea3fec.jpg)
Let
be a 2-dimensional array of pairwise ND random variables with finite variances. If there exist
such that
![](https://www.scirp.org/html/6-1240165\91cc8306-3815-4c52-a235-5ea4fa761c3e.jpg)
then for any ![](https://www.scirp.org/html/6-1240165\934fffe1-0cc1-4f39-9873-9a199224decc.jpg)
as ![](https://www.scirp.org/html/6-1240165\e900f0c0-f048-4348-bcc5-8369833c880c.jpg)
Corollary 1.9. Let
be a 2-dimensional array of pairwise ND random variables with finite variances. If
![](https://www.scirp.org/html/6-1240165\dc69a2dd-7a75-4dc8-860c-a4986d1bfc8d.jpg)
then
as ![](https://www.scirp.org/html/6-1240165\43024d5a-873a-4386-b6c2-06ad52ff46d5.jpg)
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 ![](https://www.scirp.org/html/6-1240165\ee0b8794-d477-44a6-b6c8-6b07419bec61.jpg)
![](https://www.scirp.org/html/6-1240165\30463a04-4740-42ce-9c7f-1513c6539edc.jpg)
and
as ![](https://www.scirp.org/html/6-1240165\4ce7ff17-94ab-40ea-b744-ead1cf41dc0b.jpg)
Let
be a double sequence of real numbers. Assume that 1) ![](https://www.scirp.org/html/6-1240165\13e0f634-7742-4798-8e52-9a2023a0cf96.jpg)
2)
for every
and
for every
Then
as ![](https://www.scirp.org/html/6-1240165\3d16da2e-5c0d-4a62-b7f1-3e0c303f713b.jpg)
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
![](https://www.scirp.org/html/6-1240165\e2ec3d52-00e3-4209-b81a-3c474d0e8134.jpg)
where ![](https://www.scirp.org/html/6-1240165\a33d98c2-faa8-4c56-8677-f8a156785123.jpg)
Proof. Let
be such that
First note that
![](https://www.scirp.org/html/6-1240165\d4e0bd73-f345-4413-b4dd-3d96331eb4f0.jpg)
where
denote the greatest integer smaller than or equal
and hence
![](https://www.scirp.org/html/6-1240165\9262773c-1886-4635-8850-a319aae44e38.jpg)
Therefore
and
![](https://www.scirp.org/html/6-1240165\6de4b637-ca06-4db2-a77a-93da1994e8f6.jpg)
This completes the proof. □
3. Proof of Main Results
Proof of Theorem 1.6
Let
and define
and ![](https://www.scirp.org/html/6-1240165\adc288f2-6bbf-4180-ad51-82c67586841f.jpg)
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
![](https://www.scirp.org/html/6-1240165\10e9c0de-aca0-4f77-bd0e-5314f8bae5d8.jpg)
From this fact and Chebyshev’s inequality, we have
![](https://www.scirp.org/html/6-1240165\de5476a3-2707-4eaf-bfa7-349a438df30c.jpg)
For each
let
![](https://www.scirp.org/html/6-1240165\884b9242-e31a-4308-98d8-3b49ad00fcee.jpg)
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
![](https://www.scirp.org/html/6-1240165\8c8d2636-deb7-499f-8ea6-1c28d32f34c9.jpg)
By Proposition 2.2 with
![](https://www.scirp.org/html/6-1240165\eddc71ad-4051-4530-bffc-ed972f3576dd.jpg)
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
![](https://www.scirp.org/html/6-1240165\0943627d-4a2c-46c0-979e-fd677916954c.jpg)
For every
we will show that
(3.3)
for every
,
(3.4)
and for every ![](https://www.scirp.org/html/6-1240165\2e73c7c1-74d8-4e3d-a38f-a4dccc132270.jpg)
(3.5)
From (3.3), (3.4) and (3.5), we can apply Proposition 2.1 with
that
![](https://www.scirp.org/html/6-1240165\339dc0ba-4a99-4a20-835f-555863113b47.jpg)
as
We here note that
as
implies
as
. Hence
![](https://www.scirp.org/html/6-1240165\56dfcef7-28fd-455c-827c-fc2b3e160d92.jpg)
as ![](https://www.scirp.org/html/6-1240165\6b20ae0a-c2ba-4295-9b5f-72ff6f235d42.jpg)
To prove (3.3), (3.4) and (3.5), let
Then there exists
such that for ![](https://www.scirp.org/html/6-1240165\d80fe91a-b3fe-4c8b-bcaf-7c2221a6bc78.jpg)
(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 ![](https://www.scirp.org/html/6-1240165\b1b76fb2-0c78-49f8-b993-5b99650e8512.jpg)
.
Similarly, for fixed ![](https://www.scirp.org/html/6-1240165\0158334c-28a0-41e3-875f-1e427ca395b4.jpg)
![](https://www.scirp.org/html/6-1240165\536be77d-9b4d-4cb5-a49b-d71d76485dac.jpg)
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
![](https://www.scirp.org/html/6-1240165\fd56480e-d0f4-408a-b978-c3dd38a72d0b.jpg)
For
let
be a random variable defined by
![](https://www.scirp.org/html/6-1240165\323e5860-4514-4d3b-8124-f7e0a60eaaf4.jpg)
Proof. By a direct calculation, we have
’s are pairwise ND random variables, i.e. for
that
and ![](https://www.scirp.org/html/6-1240165\60ad10dd-2a3f-489b-8186-dabcb03a25f7.jpg)
![](https://www.scirp.org/html/6-1240165\6994876c-0d10-4dbe-90cc-59154e0c6340.jpg)
Note that
![](https://www.scirp.org/html/6-1240165\baa0a28a-179d-4469-aaf5-38b41443ef40.jpg)
and ![](https://www.scirp.org/html/6-1240165\26605a1b-e558-4bd4-80be-fa991ccb521a.jpg)
Hence,
![](https://www.scirp.org/html/6-1240165\38496141-1089-4b9d-84f9-137dbe024d35.jpg)
By applying Theorem 1.6, for any double sequence
such that
for every m
, we have
as ![](https://www.scirp.org/html/6-1240165\b6588a16-57bb-420a-bbbf-9f7515a203d4.jpg)
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.