Open Journal of Statistics, 2013, 3, 42-46 http://dx.doi.org/10.4236/ojs.2013.31006 Published Online February 2013 (http://www.scirp.org/journal/ojs) Strong Law of Large Numbers for a 2-Dimensional Array of Pairwise Negatively Dependent Random Variables Karn Surakamhaeng1, Nattakarn Chaidee1,2, Kritsana Neammanee1 1 Department of Mathematics and Computer Science, Chulalongkorn University, Bangkok, Thailand 2 Centre of Excellence in Mathematics, CHE, Bangkok, Thailand Email: nattakarn.c@chula.ac.th Received October 30, 2012; revised November 30, 2012; accepted December 14, 2012 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 m  n   which is stronger than the limit as m, n   when m, n are natural numbers. Keywords: Strong Law of Large Numbers; Negatively Dependent; 2-Dimensional Array of Random Variables 1. Introduction and Main Results Let  X i iN be a sequence of random variables. We say that  X i iN satisfies the strong law of large numbers (SLLN) if there exist sequences of real numbers  an nN S  an a.s. and  bn nN such that n  0 as n   . bn Theorem 1.1. (Birkel, [4]) Let  X i iN be a sequence of pairwise PD random variables with finite variances. Assume iN 2) i 1 almost surely. To study the strong law of large numbers, there is a simple question come in mind. When does the sequence  X i iN satisfy the SLLN? Many conditions of the sequence  X i iN 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  X i iN of random variables is said to be pairwise positively dependent (pairwise PD) if for any a, b  R and i  j , P  X i  a, X j  b   P  X i  a  P  X j  b  and it is said to be pairwise negatively dependent (pairwise ND) if for any a, b  R and i  j , P  X i  a, X j  b   P  X i  a  P  X j  b  . Theorem 1.1-1.5 are examples of SLLN for a sequence of pairwise PD and pairwise ND random variables. Copyright © 2013 SciRes.  n Cov  X i , Si  i 1 i2  n where Sn   X i and the abbreviation a.s. stands for  1) sup E X i  E  X i   ,  . Sn  E  Sn  a.s.   0 as n  . n Theorem 1.2. (Azarnoosh, [5]) Let  X i iN be a sequence of pairwise ND random variables with finite variances. Assume Then 1) sup E  X i   , iN 2) n Var  X i  i 1 i2   . Sn  E  Sn  a.s.   0 as n  . n Theorem 1.3. (Nili Sani, Azarnoosh and Bozorgnia, [6]) Let  an nN be a positive and increasing sequence such that an   as n  . Let  X i iN be a sequence of pairwise ND random variables with finite variances such that Then   n E Xi  E  Xi  1) sup   an nN  i 1  2)  Var  X i  i 1 ai2     ,    . OJS K. SURAKAMHAENG Sn  E  Sn  Then i , j N quences of real numbers Sm , n  am , n bm, n m a  m , n m , nN a.s.  0 and b  m , n m , nN as m, n   where n Sm, n   X i , j . i 1 j 1 43 AL. n a.s.   0 as n  . an In this work, we study the SLLN for a 2-dimensional array of pairwise ND random variables. We say that  X i, j  satisfies the SLLN if there exist double se- such that ET m Wm, n    ai , j X i , j . i 1 j 1 Observe that, for a double indexed sequence of real number  am , n m , nN , the convergence as m  n   implies the convergence as m, n   . However, a dou- a  ble sequence i , j N positively dependent (pairwise PD) if for any a, b  R and  i, j    k , l  , P  X i , j  a, X k ,l  b   P  X i , j  a  P  X k ,l  b  Theorem 1.4. (Kim, Beak and Seo, [7]) Let be a 2-dimensional array of pairwise PD  X i, j  i , j N random variables with finite variances. Assume  P  X i , j  a, X k ,l  b   P  X i , j  a  P  X k ,l  b  . The followings are SLLNs for a 2-dimensional array of pairwise ND random variables which are all our results. Theorem 1.6. Let  am mN and  bn nN be increasing sequences of positive numbers such that am , bn  e which am   as m   and bn   as n  . be a 2-dimensional array of pairwise Let  X i , j  i , jN ND random variables with finite variances. If there exist real numbers p, q such that  Sm, n  E  Sm, n  mn i 1 j 1 i , j i , j N p cm, n i , j N be a 2-dimensional array of pairwise   ai , j ak ,l Cov  X i , j , X k ,l     . 2)     bi2, j i , j ,i  j 1  k , l , k l  i  j   Wm, n  E Wm, n  bm, n Copyright © 2013 SciRes.  , c  m , n m , nN such that a.s.   0 as m  n   where a.s.   0 as m  n  . The next theorem is the SLLN for the difference of random variables which independent and identically distributed conditions are regarded. Theorem 1.7. Let  X i , j  and Yi , j  be 2i , jN i , jN dimensional arrays of random variables on a probability space (Ω, F, P). If 1) sup E X i , j  E  X i , j   , Then q q Sm, n  E  Sm,n  PD random variables with finite variances such that i , j N p ai2  b j2 cm, n  am2  bn2 for every m, n  N , a.s.   0 as m  n  .  Var  X i , j  then for any double sequence be a 2-dimensional array of positive numbers and n m am , n bm, n   ai , j such that  0 and bm, n   as bm , n i 1 j 1 m, n  . X     Cov  X i , j , X k ,l       . 2  i , j , i  j 1  k , l , k l  i j i  j    Theorem 1.5. (Kim, Baek and Han, [8]) Let  ai , j  Let m  n negtively dependent (pairwise ND) if for any a, b  R and  i, j    k , l  ,   Then mn mn 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  X i , j  is said to be pairwise m , n m , n N 1) sup E X i , j  E  X i , j   , 2)  1 i , j N 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  X i , j  is said to be pairwise i , j N where am, n     P  X i , j  Yi , j   , i 1 j 1 then 1 m n a.s. 0   X i , j  Yi , j   m  n i 1 j 1 as m  n  . OJS 44 K. SURAKAMHAENG Corollary 1.8 and Corollary 1.9 follow directly from k Theorem 1.6 by choosing cm, n   am  bn  and cm, n  am  bn where am  3m and bn  3n with p = q = 4, respectively. Corollary 1.8. Let  am mN and  bn nN be increasing sequences of positive numbers such that am , bn  e which am   as m   and bn   as n  . Let  X i , j  be a 2-dimensional array of pairwise ET AL. max m, n  . The following proposition is a Borel-Cantelli lemma for a sequence of double indexed events Proposition 2.2. Let  Ei , j  be a double sequence i , j N of events on a probability space     Var  X i , j  p 2 i 1 j 1 ai  b q 2 j  ,  am  bn  k E i, j i 1 j 1  k  k      P  Ei , j     0 as m  n  . a.s. i , j i , j N  Var  X i , j  i  j  i 1 j 1 be a 2-dimensional ar- 2 j 1 i , j ,i  j  k  , equal 2 k  i 1, j  i , j  0, i , j 1  i , j  0 i 1, j 1  i 1, j  i , j 1  i , j  0, and i , j   as max i, j  . be a double sequence of real numbers. Let  ai , j  i , j N Assume that   a i, j  , 1)  i , j k 1   for every i  N and every j  N . Then  1 m,n Copyright © 2013 SciRes. m n  ai , j  0 i 1 j 1 ak , j  k 1 i ,k i 1 j 1 i , j ,i  j  k P  Ei , j  as k, j   for k i 1 j 1 k  double sequence of positive numbers such that for all i, j  N , a   P  Ei , j   klim   Therefore lim a.s.   0 as m  n  . i , j N  i 1 j 1  k  k    k 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  i , j  be a  i ,k k  lim  P  Ei , j   L. 2. Auxiliary Results i 1 j 1 k k and hence k  81 m  n  P  Ei , j    P  Ei , j  where  k  denote the greatest integer smaller than or L  lim then Sm, n  E  Sm,n   First note that X      i.o.     Ei , j . k 1  i , j , i  j  k   ray of pairwise ND random variables with finite variances. If 2) where i 1 Corollary 1.9. Let  i 1 j 1 Proof. Let L  R be such that L   P  Ei , j . then for any k  p  q, Sm, n  E  Sm, n    P  Ei , j     P Ei , j i.o.  0 i , j N ND random variables with finite variances. If there exist p, q  N such that  , F , P  . Then  i , j ,i  j  k P  Ei , j   L and   P  Ei , j i.o.  lim P   Ei , j   lim  P  Ei , j  k   i , j ,i  j  k  k  i , j ,i j  k      lim   P  Ei , j   P  Ei , j    0. k  i , j ,i j  k 1  i 1 j 1  This completes the proof. □ 3. Proof of Main Results Proof of Theorem 1.6 Let m, n  N and define f  m    ln am  and g  n    ln bn  . Clearly, f and g are increasing whose facts f  m   ln am  f  m   1 and g  n   ln bn  g  n   1 f m f m 1 which imply that e    am  e   and g n g  n  1  bn  e . e Let   0 be given. By using the fact that Cov  X i , j , X k ,l   0 for  i, j    k , l  ([10], p. 313), we have  m n  m n Var   X i , j    Var  X i , j .  i 1 j 1  i 1 j 1 From this fact and Chebyshev’s inequality, we have OJS K. SURAKAMHAENG     C  Var  Sm, n  m 1 n 1  2 m,n c  m 1 n 1     C  Var  X i , j    i 1 j 1    C       m n 2 m , n i 1 j 1 c 1    C  Var  X i , j    i 1 j 1 e For each i, j  N , let     t  N : e   Bj f s 1 f t 1 .  b     m  i n  j i 1 j 1     C  Var  X i , j  i 1 j 1    C  Var  X i , j  i 1 j 1    m  f i  1 e  e  and for every j  N , 1 1 (3.5) e 1 m n a.s. 0   X i , j  Yi , j   m  n i 1 j 1   pm 2 n g  j  as max m, n  . We here note that am,n  a as max m, n   implies am,n  a as m  n   . Hence 1 qn 2 1 m n a.s. 0   X i , j  Yi , j   m  n i 1 j 1 (3.2) 1 e f  i   e 1 and ai as m  n  . To prove (3.3), (3.4) and (3.5), let   c0 . Then there exists k  N such that for i, j  N , i  j  k  X i , j    Yi , j   . Thus for each   c0 ,  X i , j    1 . bj  Sm, n  E  Sm, n     Var  X i, j      C  p  . q cm, n i 1 j 1   2 2 ai  b j By Proposition 2.2 with  Sm, n  E  Sm, n     , Em , n   cm, n   Copyright © 2013 SciRes. (3.4) j 1  pm  qn    2   P m 1 n 1 1 From (3.3), (3.4) and (3.5), we can apply Proposition 2.1 with i , j  i  j that From this facts and (3.2) together with our assumption 2), we have  (3.3) i 1 Since i  Ai and j  B j , we have 1   i  j  X i , j    Yi, j       e e       1  1    C  Var  X i , j   pf i   qg j  .    i 1 j 1  2  2   e  e g  j  1  i  j  X i , j    Yi , j     .  m f i  n  g  j   for every i  N , j  pf  m   qg  n        Yi , j  i.o.  1. i 1 j 1 1 e i, j  i  j  X i , j    Yi , j     ,  ai  Sm, n  E  Sm, n    P cm, n m 1 n 1     X For every   c0 , we will show that   C  Var  X i , j      P c0  1  P  0   1  P (3.1) and i  min Ai and j  min B j . Since i  Ai and j  B j , we have i  i and j  j . From this facts and (3.1), we have  By Proposition 2.2, we have  pf  m   qg  n   Ai  s  N : e □ k 1 i , j , i  j  k 1 m i n  j a.s.   0 as m  n  .   X i , j  Yi , j . Let  0   1 p q a m  i n  j m  bn  cm, n Proof of Theorem 1.7 c  C  Var  X i , j    i 1 j 1 Sm, n  E  Sm, n   Var  X i , j  1 2 m,n m i n  j 45 AL. we have P  Em, n i.o.  0 and this hold for every   0. By using the same idea with Theorem 4.2.2 ([11], p. 77), we can prove that  Sm, n  E  Sm, n    P cm, n m 1 n 1    ET Y    i, j i , j N (3.6) and are different only finitely many terms. i , j N This implies that (3.3) holds. For fixed i  N , we can find a large j0  N such that (3.6) holds for all j  j0 which means that there are only finitely many different terms of  X i , j    and Yi , j    i , j N i , j N  . So for fixed i  N ,  i  j  X i , j    Yi, j      . 1 j 1 OJS 46 K. SURAKAMHAENG Similarly, for fixed j  N ,  1  i  j  X i , j    Yi , j     . i 1 Now (3.4) and (3.5) are now proved and this ends the proof. □ Remark 3.1. In case of m fixed and n  , by considering the limit as m  n  , 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 X i , j , i  1, 2, , p and j  1, 2, , q be a random variable indicating the presence of a ball of the ith color and the jth size such that X i , j  X ,if 1  i  p and 1  j  q, X i, j   i, j 0, otherwise. Proof. By a direct calculation, we have X i , j ’s are pairwise ND random variables, i.e. for i, j , k , l  R that  i, j    k , l  and a, b  R, pq  1 2 E  X i, j     pq  pq    2     Var  X i , j  i 1 j 1 i  j  2  m n Var  X i , j     lim   2 m , n     i 1 j 1  i  j    2 4    1     . 2  pq  pq 2   i 1 j 1  i  j    By applying Theorem 1.6, for any double sequence c  m , n m , nN such that cm,n  81 m  n  Copyright © 2013 SciRes. 2 cm, n a.s.   0 as m  n  . 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 [1] S. Csörgő, K. Tandori and V. Totik, “On the Strong Law of Large Numbers for Pairwise Independent Random Variables,” Acta Mathematica Hungarica, Vol. 42, No. 34, 1983, pp. 319-330. doi:10.1007/BF01956779 [2] N. Etemadi, “An Elementary Proof of the Strong Law of Large Numbers,” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, Vol. 55, No. 1, 1981, pp. 119-122. [3] R. G. Laha and V. K. Rohatgi, “Probability Theory,” John Wiley & Sons, Hoboken, 1979. [4] T. Birkel, “A Note on the Strong Law of Large Numbers for Positively Dependent Random Variables,” Statistics & Probability Letters, Vol. 7, No. 1, 1989, pp. 17-20. doi:10.1016/0167-7152(88)90080-6 [5] H. A. Azarnoosh, “On the Law of Large Numbers for Negatively Dependent Random Variables,” Pakistan Journal of Statistics, Vol. 19, No. 1, 2003, pp. 15-23. [6] H. R. Nili Sani, H. A. Azarnoosh and A. Bozorgnia, “The Strong Law of Large Numbers for Pairwise Negatively Dependent Random Variables,” Iranian Journal of Science & Technology, Vol. 28, No. A2, 2004, pp. 211-217. [7] T. S. Kim, H. Y. Beak and H. Y. Seo, “On Strong Laws of Large Numbers for 2-Dimensional Positively Dependent Random Variables,” Bulletin of the Korean Mathematical Society, Vol. 35, No. 4, 1998, pp. 709-718. [8] T. S. Kim, H. Y. Beak and K. H. Han, “On the Almost Sure Convergence of Weighted Sums of 2-Dimensional Arrays of Positive Dependent Random Variables,” Communications of the Korean Mathematical Society, Vol. 14, No. 4, 1999, pp. 797-804. [9] F. Móricz, “The Kronecker Lemmas for Multiple Series and Some Applications,” Acta Mathematica Academiae Scientiarum Hungaricae, Vol. 37, No. 1-3, 1981, pp. 3950. doi:10.1007/BF01904871 2 4 and Var  X i , j   .  pq  pq 2 Hence, Sm, n  E  Sm, n  □ P  X i , j  a, X k , l  b   P  X i , j  a  P  X k , l  b  . Note that AL. n  N , we have 1,if the i th color and the j th size of ball is picked,  0, otherwise. For i, j  N , let X i , j be a random variable defined by ET [10] N. Ebrahimi and M. Ghosh, “Multivariate Negative Dependence,” Communications in Statistics—Theory and Methods, Vol. A10, No. 4, 1981, pp. 307-337. [11] K. L. Chung, “A Course in Probability Theory,” Academic Press, London, 2001. for every m OJS