Strong Law of Large Numbers for a 2 Dimensional Array of Pairwise Negatively Dependent Random Variables

Strong Law of Large Numbers for a 2-Dimensional Array

of Pairwise Negatively Dependent Random Variables

Karn Surakamhaeng1_{, Nattakarn Chaidee}1,2_{, Kritsana Neammanee}1

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 when m, n

are natural numbers.

,

m n 

Keywords: Strong Law of Large Numbers; Negatively Dependent; 2-Dimensional Array of Random Variables

1. Introduction and Main Results

Let

 

Xi i N be a sequence of random variables. We say

 

X

that _{i i N} satisfies the strong law of large numbers (SLLN) if there exist sequences of real numbers

 

_n

n N

a _

and

 

_n such that

n N

b _ n n a.s. 0

n

S a b



 as .

where and the abbreviation a.s. stands for

n 

1

n





n i

S  X_i

almost surely.

To study the strong law of large numbers, there is a simple question come in mind. When does the sequence

i N satisfy the SLLN? Many conditions of the se-

quence _{i N} have been found for this question. The

SLLN are investigated extensively in the literature espe- cially 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.

 

Xi 

 

Xi 

A sequence _{i N} of random variables is said to be

pairwise positively dependent (pairwise PD) if for any

and i j,

 

Xi 



,

a bR



i , j





i





j



P X a X b P X a P X b

and it is said to be pairwise negatively dependent (pair- wise ND) if for any a b, R and i j,



i , j





i





j



.

P X a X b P X a P X b

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 _{i N} be a se- quence of pairwise PD random variables with finite variances. Assume

 

Xi

1) sup



i

 

i



,

i N

E X E X    

2)



₂



1

Cov ,

.

n

i i i

X S i



 



Then Sn E S

 

n a.s. ₀

n



 as n .

Theorem 1.2. (Azarnoosh, [5]) Let _{i i N} be a se- quence of pairwise ND random variables with finite variances. Assume

 

X

1) sup

 

i ,

i N

E X 

 

2)

 

₂

1 Var

.

n

i i

X i



 



Then Sn E S

 

n a.s. ₀

n



 as 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

1)



 



1

sup n i i ,

n N i n

E X E X

a  

 _ 

 _{  }

 







2)

 

₂

1

Var

. i i i

X

a 



Then n

 

n a.s. ₀

n

S E S

a



 as n .

In this work, we study the SLLN for a 2-dimensional array of pairwise ND random variables. We say that

 

X_{i j i j N}, _, satisfies the SLLN if there exist double se- quences of real numbers

 

am n, _{m n N,} and

 

bm n, _{m n N,}

such that , , a.s.

,

0 m n m n

m n

S a

b



 as where

,

m n 

, ,

1 1

.

m n m n i j

i j

S X

 





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

 

_,

,

i j _{i j N}

X

 is said to be pairwise

positively dependent (pairwise PD) if for any a b, R

and

   

i j,  k l, ,



i j, , k l,

 

i j,





k l,



P X a X b P X a P X b

Theorem 1.4. (Kim, Beak and Seo, [7]) Let

 

X_{i j i j N}, _, be a 2-dimensional array of pairwise PD random variables with finite variances. Assume

1)



,

 

,



,

sup i j i j , i j N

E X E X 

  

2)









, ,

2

, , 1 , ,

Cov ,

.

i j k l i j i j k l k l i j

X X

i j     

 

   

 _ 

 





Then Sm n, E S

 

m n, a.s. ₀ m n





 as m n  .

Theorem 1.5. (Kim, Baek and Han, [8]) Let

 

ai j, _{i j N,} be a 2-dimensional array of positive numbers and

such that ,

1 1

n m m n i j

i j

b

 





a_, ,

,

0 m n m n

a

b  and bm n,   as

, .

m n 

Let be a 2-dimensional array of pairwise

PD random variables with finite variances such that

 

Xi j, _{i j N,}

1)



,

 

,



,

sup i j i j , i j N

E X E X

   

2) , , ₂



, ,



, , 1 , , ,

Cov ,

.

i j k l i j k l i j i j k l k l i j i j

a a X X

b     



   

 

 







Then as m n   where

, X

double indexed sequence of real ,

m n 



,

1 1

.

n m i j i j i j

W a







Observe that, for a number

 

am n, _{m n N,} ,





, , a.s.

,

0

m n m n m n

W E W

b





m n   the convergence as

im

nce

plies the convergence as m n,  . However, a dou-

ble seque

 

a _,

,

m n _{m n N} where

  



,

m n

1m n

a

m n

 



Our goal is t he SLLN -

A double sequence

m n

 

shows us that the conv o obtai ray of random variab

erse is not true in general.

n t for 2-dimensional ar

les in case of pairwise ND.

 

Xi j, _{i j N,} is said to be pairwise negtively dependent (pairwise ND) if for any a b, R and

   

i j,  k l, ,



i j, , k l,

 

i j,





k l,



.

P X a X b P X a P X 

The

b

followings are SL of pairwise ND ran sults.

LNs for a 2-dimensional array dom variables which are all our re-

Theorem 1.6. Let

 

am m N and

 

bn n N be increas-

ing sequences of positive numbers such that ,

w m n

a b e

hich am  as m  and bn  as n .

Let

 

_,

,

i j i j N

X _ be a 2-dimensional array of

ND ran nite va . If th st

pairwise

with fi riances ere exi

dom variables real numbers p q, such that

 

,

1 1 ₂

Var

,

i j p q i j

X

a b  

 

  



2

i j

nce such that

then for any double seque

 

_,

,

m n _{m n}

c N

2 2

,

p q m n m n

c a b for every ,m nN,

 

, , a.s.

,

m n

c 0

m n m n

S E S

 as   n .

The next theorem is the SLLN for the difference of random variables which independent and identically dis- tributed conditions are regarded.

m

Theorem 1.7. Let

 

X_{i j i j N}, _, and

 

Yi j, _{i j N,} be 2- dimensional arrays of random variables on a probability sp

then

ace (Ω, F, P). If



, ,



1

,

i j i j i

P X Y

  

  



1

j





a.s.

, ,

1 1

1

0

m n

i j i j i j

X Y

m n  

 





Corollary 1.8 and Corollary 1.9 follow directly from

Theorem 1.6 by choosing nd

b where with p = q =

ing sequences of positive nu an

a 2-dim l array

such





,

k m n m n

c  a b

3

a m and bn3n

a ,

m n

4, r m n .

c a 

espectively m



Corollary 1.8. Let

 

am and

 

b N be increas-

mbers such that a bm, ne

which am  as d as n .

m N n n

m  bn 

Let

 

_,

,

i j i j N

X _ be ensiona of pairwise

ND random variables with finite variances. If there exist

,

p qN that

 

, Var Xi j

 

1 1 _{2 2}

,

p q i j

a b

 

  



i j

then for any k p q,

 





, , a.s. ₀

m n m n k m n

S E S

a b





 as m n  .

Corollary 1.9. Let

 

X_{i j i j N}, _, be a 2-dimensional ar-

ray of pairwise ND random va s with finite vari-

ances. If

riable

 





, 2

1 1

,

i j i j

  



then

Var Xi j

 

 





, , a.s.

2 0

81

m n m n

S E S

m n





 as m n .

2. Auxiliary Results

In this section, we present some materials which will be used in obtaining the SLLN’s in the next section.

icz, [9]) Let

 

Proposition 2.1. (Mór

 

i j, _{i j N,} be a

and

double sequence of positive numbers such that for all

, ,

i jN

1, , 0, , 1 , 0

i j i j i j i j

_    _  

1, 1 1, , 1 , 0,

i j i j i j i j

_{ } _  _  

,

i j

  

Let

 

a

as

ence of r numbers.

Assume that

 

max ,i j  .

be a double sequ

, _,

i j i j N eal

1) ,

1 1 ,

, i j





2)

i i j

a 

 

 

j



,

i k

a 



1 ,

k i k

  for every iN and ,

1 ,

k j k k j

a 





 



for

every jN. Then _,

1 1

m n i j i j

a

 



,

1 ₀

m n

  as

 

max m n,  .

The

for a sequence of d

osition

following proposition is a Borel-Cantelli lemma ouble indexed events

t

Prop 2.2. Le

 

Ei j, _{i j N,} be a sequence probability space

double

of events on a



, ,F P



.Then

 

i j,



i j, i.o



P E P E

 

1 1

i j

. 0

  





where



Ei j, i.o.



Ei j,





 

 

1 , ,

.

k i j i j k 

 

 

Proof. Let

 

,

1 1

.

i j i j

L P E

   





LR be such that

First note that

 

,

 

,

 

1 1

k k _{k k}

i j i j i j i j

P E P E P E

      

 

 

 





,

1 1 , ,

i j i j i j k 

where



k

 

  denote the greatest integer smaller than or

equal k and hence

 

 



1 1 , ,

1 1

lim

k

i j i j i j k k k

k i j

P



   

  





, ,

,

lim lim

. k k

i j i j

k

i j

L P E P E

E L

       



 

 

 



Therefore

 

_,

, ,

lim i j k

i j i j k

P E L





_{ }  and



,







 

, ,

, , , ,

, ,

1 1 , , 1

i.o. lim lim

lim 0.

i j i j i j

k k _{i j i j k} i j i j k

i j i j k

i j i j i j k

P E P E P E

P E E

 _{ }  _{ }

 

 _{    }

 

  

 



_{ }



P

 _{ }

 









This completes the proof. □

3. Proof of Main Results

Proof of Theorem 1.6

Let m n, N and define f m

 

 _lna_m_ and

 

ln n .

g n  _ b _

Clearly, f and g are increasing whose facts

 

ln m

 

1

f m  a  f m  and g n

 

lnb_n g n

 

1

 1

f m

e  and

which imply that f m 

e  



g g n m

a

 n 1

n

e b e  .

Let 0 be given. By using the fact that



, ,



ov Xi j,Xk l 0

C  for

   

i j,  k l, ([10], p. 313), we

From this fact and Chebyshev’s ineq have

 

Var m n , Var , .

m n

i j i j

X  X









1 1 1 1

i j i j

 

 

 

_{ }

 

 

 

_{    } , ,

1 1 ,

1 1 _, 1 1 _, 1 1

, 2

1 1 ,

, 1 1 , 1 1 Var 1 Var 1 Var .

m n m n m n m n

m n m n m n m ni j

i j

i j m i n j m n

i j p q i j m i n j m n

i j _{pf m qg n} j m i n j

S E S P c c c C X c C X a b C X e                                     _           





 



 

 

1 (3.1) i   



For each let

, , 2 2 Var ₁ Var m n m n i j S

C C X

   













, ,

i jN

 





 





1 1 : : f s i i f t j j

A s N e a

B t N e b

 

  

  

and iminAi and jminBj . Since iAi and j

jB (3.1

, we have ), we have

ii and j j. From this facts and

 

 

_{    }

 

   

 

   

 

_{ } , , 1 , 1 1 ,

1 1 ₂

,

1 1 _{2 2}

, 1 1 2 1 1 Var 1 1 Var 1 Var

m n m n

i j _{pf m qg n} m i n j

i j pm qn i j m f i n g j

i j pm qn i j m f i n g j

i j _{pf i} i j

S E S

P c X e C X e C X e e C X e                     _ _                 _              _   



 



 









         1 , Var

m n m n

i j C      



2

1 _{. (3.2)}

qg j e           

Since iA_i and jBj, we have  

1 1 f i i e a e  

 and

  1 1 g j j e e  assum 2) ve b   .

From this facts and (3.2) together with our ption

, we ha

 

 

,

m n

c

, ,

1 1 , 1 1 2 2

Var

.

m n i j

p q

m n m n i j

S E S X

P  C

         _   _ _{  }    





ition 2.2 with

i j a b    By Propos

 

, , , , ,

m n m n m n

m n

S E S E c   _    _  _    



m n, i.o.



0

P E 

we have and this hold for every 0.

, p. 77)

By us ith Theorem 4.2.2 ([11]

we can prove that

ing the same idea w ,

 

, , a.s.

m n m n

S E S ,

0

m n

c  as m n  . □

Proof of Theorem 1.7

Let ₀





By Proposition 2.2,

1 , ,

.

k i j i j k

X Y

   

 

_{ }

_{i j,}  _{i j,}

we have

 

c 1

P   ₀ P

 

  ₀ 1 P





X_{i j,} Y_{i j,}



i.o.



1.

0,

c

 

For every we will show that

 

 



,

1 1 i j

i j X i j   



,



1 _, i j Y

     (3.3)

fo ry

 

r eve iN,

 

 



, ,



1

1

i j i j j

X Y

i j  

 

 





 (3.4)

and for every jN,

 

 



, ,



1

1 _.

i j i j i

X Y

i j  

 

 





 (3.5)

From (3.3), (3.4) and (3.5), we can apply Proposition 2.1 with i j_, i j that





a.s.

, ,

1 1

1

0

m n

i j i j i j

X Y

m n  

 





as max

 

m n,  . We here note that a_{m n,} a as

 

x m n,   imp

ma lies am n, a as m n  . Hence





a.s.

, ,

1 1 i j i j

i j

m n  

as m n

1

0

m n

X Y 



.

  

To prove (3.3), (3.4) and (3.5), let Then

there exists 0

. c

 

k_N such that for i j, N,

 

 

, , .

i j i j

X Y



i j k     (3.6)

Thus for each c



 



0, Xi j, _{i j N,}

 



  and

 



Yi j, 



_{i j N,} are different only finitely many terms.

plies th t (3.3) holds.

This im a

For fixed iN, we can find a large j0N such

that (3.6) holds for all j j0 which m

are only finitely many different terms of

eans that there

 

, _,

i j _{i j N}



X 





 



Y 



. So for fixed

and _,

,

i j _{i j N} iN,

 

 



, ,



1

i j i j

X Y

i j     .

1

j

 

 

, , a.s.

,

0

m n m n m n

S E S

c





Similarly, for fixed jN,

 

 



, ,



1 _.

i j i j

X Y 



1

i i j

   



_

Now ) are now pr

proo

(3.4) and (3.5 oved and this ends the

f. □

Remark 3.1. In case of m fixed and by con-

sidering the limit as we the

corresponding results for a case of 1-dime nal pair-

wise N

4. Example

Example 4.1 A box contains pq balls of p different col- or

,

n 

also obtain

,

m n  

nsio D random variables.

s and q different sizes in each color. Pick 2 balls ran- domly.

Let Xi j, ,i1, 2, , p and j1, 2, q random

variable indicating th of a ball of the ith color

and the jth size such that

1,if the_ithcolo





,

ce

be a presen

e

,

0,other



r and the size of ball is picked, wise.

th i j

j X  

For i j, N, let Xi j, be a random variable defined by

, ,

1 ,

0, otherwise. i j

i j

,if and 1

X i q

X    



p j

  

 

Proof. By a direct calculation, we have Xi j_, ’

,

k l R

s are pairwise ND random variables, i.e. for i j, ,  that



and

  

i j,  k l, a b, R,



i j, , k l,

 

i j,





k l,



.

P X a X b P X a P X b

Note that

 

,

1 2 2

i j

pq E X

pq pq



 

 

 

 

and

 

 

, 2

2 4 _. i j

pq

 

Var X

pq

Hence,

 





 





 





, ,

2 _, 2

1 1 1 1

2 2

1 1

Var Var

lim

2 4 1

.

m n

i j i j

m n

i j i j

i j

X X

i j i j

pq pq i j

 



   

   

 

 



 

 _  _

 

 

  

  _

 









By applying Theorem 1.6, for any double sequence

that n for ever

nN, we have as m n  .

5. Acknowledgements

have hel

ork. hor gives an ap

thanks to the Institute for the Prom hnology for financial su

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□

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Th comm our w

Scienc

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e and Tec pport.

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