Complete Convergence and Weak Law of Large Numbers for ρ Mixing Sequences of Random Variables

Complete Convergence and Weak Law of Large Numbers

for

ρ



-Mixing Sequences of Random Variables

Qunying Wu1,2

1

College of Science, Guilin University of Technology, Guilin, China

2

Guangxi Key Laboratory of Spatial Information and Geomatics, Guilin, China Email: [email protected]

Received September 5,2012; revised October 7, 2012; accepted October 20, 2012

ABSTRACT

In this paper, the complete convergence and weak law of large numbers are established for -mixing sequences of random variables. Our results extend and improve the Baum and Katz complete convergence theorem and the classical weak law of large numbers, etc. from independent sequences of random variables to -mixing sequences of random variables without necessarily adding any extra conditions.

Keywords: 



,,P





,,P





X ;n1



SN



_i;



-Mixing Sequence of Random Variables; Complete Convergence; Weak Law of Large Number

 

1. Introduction

f t , then the sequence

Let be a probability space. The random variables we deal with are all defined on . Let

n be a sequence of random variables. For each

nonempty set , write S  X iS . Given

 -algebras  , in  , let





,



sup corr



X Y,



;XL2

 

 ,YL2

 





,





  

where corr ,

Var Var

EXY EXEY X Y

X Y



 . Define the 

 

n sup



_S, _T



,

    

n

, of

S T N



S T,





-mix-

ing coefficients by

(1.1)

where (for a given positive integer ) this sup is taken over all pairs of nonempty finite subsets

such that dist n.





Obviously 0 n1 

 

n 1,n0, and except in the trivial case where all of the ran- dom variables

 

0 1

 

i

X are degenerate.

Definition 1.1. A sequence of random variables

n is said to be a



X ;n1



-mixing sequence of ran- dom variables if there exists kN such that 

 

k <1



X ;n1 

 

1

. Without loss of generality we may assume that

n is such that (see [1]). Here we

give two examples of the practical application of



1

 _- mixing.

Example 1.1. According to the proof of Theorem 2 in

[2] and Remark 3 in [1], if

spectral density



f X

 

i ;i1





 

1 1

 

has the property that . There- fore, instantaneous functions



f Xi ;i1



of such a

sequence provides a class of examples for -mixing se- quences.



X_n;n1



has a bounded positive

Example 1.2. If

 

t , i.e., 0 m f t

 

M for f

spectral density

 



X i_i; 1



is a strictly sta- tionary Gaussian sequence which has a bounded positive

every t, then  1  1 m M1. Thus,



Xn;n1



is a -mixing sequence.

-mixing is similar to -mixing, but both are quite different. 

 

k is defined by (1.1) with index sets re- stricted to subsets S of

 

1,n and subsets T of



nk,



, ,n kN. On the other hand, -mixing se- quence assume condition 

 

k 0，but 

k N

-mixing sequence assume condition that there exists  such that 

 

k 1, from this point of view,  -mixing is weaker than -mixing.

A number of writers have studied  -mixing se- quences of random variables and a series of useful results have been established. We refer to [2] for the central limit theorem [1,3], for moment inequalities and the strong law of large numbers [4-9], for almost sure con- vergence, and [10] for maximal inequalities and the in- variance principle. When these are compared with the corresponding results for sequences of independent ran- dom variables, there still remains much to be desired.

complete convergence theorems and the weak law of large numbers. Our results in this paper extend and im- prove the corresponding results of Feller [11] and Baum and Katz [12].

Lemma 1.1. ([10], Theorem 2.1) Suppose K is a posi-

tive integer, , and . Then there exists a positive constant such that the follow- ing statement holds:

0 r



1 q2



, ,



D D K r q



X i_i; 1



If is a sequence of random variables such that 

 

K r and EX_i0 and E X_iq 

1

for all , then for every ,

i n1





2

2

1 1

,

q

i

EX



 _{ }

 

 _{ } 

 

1

=

i i

j i

S X







1

1

max

n n

q q

i i

i n

i i

E S D E X

  _{ }



 







where .

Lemma 1.2. Let



Xn;n be a 

0

x

1

n

-mixing se- quence of random variables. Then for any , there exists a positive constant c such that for all ,













2 1

1

1 (max )

max .

n

k n

k

k k n

P X x

cP X x

  _

 

 

 



1

k P Xk x

Proof. Let Ak 



Xk x



and



1



=1

1 1 max

n

n k k

k n k

P A P X x



 

 

  _{ }    

> 0

. Without loss of

generality, assume that n . By the Cauchy-Schwarz

inequality and Lemma 1.2,

 





 





Thus





 

j=1

j=1

1 1

, n j

A A n

n n

n k

k k

j

I

P A

P A



 

 

 

 

 

     













j 1

j 1

j 1

1 1 1

1 1

1 2 2

2 2

j 1 1

1 ,

+

+

k

n

k k

j

n

k k

j

k k

n n n n

k k

k k k

n n

A A k

A

k k

A A j

A A

k

j

n n

A

E I

E I

E

P A P A A E I

E I I P A

E I I E

c E I I P A



 

  

 

 



 

 

 _{ }

 



 



  

 

 _{ } 

 

 _{ }

 _{ } 

 

   

 _{  }



 















 







 





 





 





  

1 2

1

1 2

1 1

1

+ 1

1

+ 1

1 1

+ + 1

2 n n k

k

n n

n

n k n

k k

n n

n k

k k

n

P A c

P A c

P A





 

 

 





 





 _

 _



 

 _ _

 

 

 _ 









 

















 

1

k

n n

n k

P A P A



  

  



2 1

1 ,

n

n k n

k

P A c

 



 



i.e.,

















2 1

1 1

1 max

max .

n

k k

k n

k k

k n

P X x P X x

cP X x

  _

 

  

 



2. Complete Convergence

 

 

a x b x denote In the following, let

   

1,

a x b x  x  a b



a_n b



> 0

n n

a cb

, and n n n de-

note that there exists a constant c such that



an cbn



for sufficiently large n, logx mean





 



ln max x, e

1

= _i

j n

n

S X





. , and

 

Definition 2.1. A measurable function





is said to be a slowly varying function at if for any

0 0

l x  x



> 0

c

 

 

, lim 1 x

l cx l x

  .

Lemma 2.1 ([13], Lemma 1). Let be a slowly

varying function at

 

l x

. Then

 

 

i)

1

2 2

lim sup 1

2

k k k

k x

l x

l 

 _{ }  .

 

lim ,

x x l x



  limx x l x

 

0,

 

  for any 0

ii)  

0

r 0

.

iii) For any and 

1

c c r,

, there exist positive constants and 2 (depending only on , and the

function l

 

·

 

 

 

1 2

1

2 2 2 2 2 2 .

k

kr k jr j kr k

j

c l l c l

) such that for any positive number k,

  









< 0

r 0

iv) For any and 

1

d d r,

, there exist positive constants and 2 (depending only on , and the

function l

 

·

 

 

 

12 2 2 2 22 2

kr k jr j kr k

j k

d l l d l

) such that for any positive number k,

.

 

 











X_n;n1



be a

Theorem 2.1. Let  -mixing se-

quence of identically distributed random variables. Sup- pose that l x

 

0 is a slowly varying function at , and also assume that for each a0, the function l x





 

0,a 0 p 2

is bounded on the interval . Suppose  

1

p

and   ; and if  1 then suppose also that

1 0

EX  . Then





1





1 1

p

 



S n



 ,

1,

p

2

1 1

max

0 p

j j n n

n l n P



 

  

 



_(2.2)

are equivalent. For  

 

l x



1

we also have the following theorem un- der adding the condition that is a monotone non- decreasing function.

Theorem 2.2. Let



Xn;n be a 

 

0

l x  

-mixing se- quence of identically distributed random variables. Let

is a slowly varying function at and mono- tone non-decreasing function. Suppose 1 2

1

; and if

 then suppose also that EX10. Then







1 1



1 1

E X l X    (2.3) and

 



1 1 1

max _j j n n

n l n P S n

 

  







   ,  0

 

log

l x  x



1

(2.4)

are equivalent.

Taking and respectively in Theorems 2.1 and 2.2 we can immediately obtain the following corollaries.

 

1

l x 

Corollary 2.1. Let



Xn;n be a  -mixing se-

quence of identically distributed random variables. Sup- pose 0 p 2 and p1; and if  1

0

then sup- pose also that EX1 . Then

1

E X p 

and



S n



,

 _  _{ }



1

2 1 1

max

0 p

j j n n

n P



 

  

 



are equivalent.

Corollary 2.2. Let



Xn;n be a  -mixing se-

quence of identically distributed random variables. Sup- pose 0 p 2 and p1; and if  1

0

then sup- pose also that EX1 . Then



1 log X1



 

p

E X

and



S n



,

 _  _{ }



; 1

X n

2

1 1

log max

0 p

j j n n

n nP



 

  

 



are equivalent.

Remark 2.1. When



n i.i.d., Corollary 2.5

becomes the Baum and Katz [12] complete convergence theorem. So Theorems 2.1 and 2.2 extend and improve the Baum and Katz complete convergence theorem from the i.i.d. case to -mixing sequences.

Remark 2.2. Letting take various forms in

Theorems 2.1 and 2.2, we can get a variety of pairs of equivalent statements, one involving a moment condition

 

l x

and the other involving a complete convergence condi- tion.



Proof of Theorem 2.1. 2.1

  

 2.2

 



| _i|



n

i i i _{X n}

Y Y X I

. Let

,

  

 







, 1, 2, ,

i

n

i i i _{X n}

Y Y X I _ i n



    . Firstly, we prove that

1 1

max 0, .

j

i j n

i

n EY n

 



_    (2.5)

By Lemma 2.1 and (2.1), it is easy to show that

1 , for any 0. p

E X     

1

(2.6)

i) For   , we have p11, and EX10.

Let 0  min p 1,p 1





 

  _  _

  in (2.6), by

1 ,1 0

p

E X    p  ,





 





1

1

1

1 1

1 1

1 1 1

1 1

1

1

max

0.

j n

i i

j n

i i

X n

p

p X n

p p

n EY n EY

n EX I

X E X

n I

n

n

E X 



 



 

 

  

 

  _{ }



   

  

  















1,p 1

ii) For    , let 0   1

 

  in (2.6), then

1 1

E X    and 1  0. Hence



1



1 1

1 ₁

1 1

1

max

0.

j

i _{X n}

j n _i

n EY n E X I

n E X



 

  

 

   _

  



 



1,p 1

  , iii) For 





 





1

1

1

1 1

1 1

1 1

1

1 ₁

1

max

.

j n

i i

j n

i i

X n

n

i X i

i

n EY n EY

n EY n E X I

n E X I



 

 

 



 

  _{ }

 

 

   



 









1, 1

p p

Noting    , let 0  p 1

   

1 p 0

in (2.6). By

 

   and 1

p

 





 





 



1 1 1 ₁ 1 1 1 1 1 1 1 i X i p p i p i

i E X I

i E X

E X I





1 1 1 . i

i X i i X i

I                _     _    







         1

i   

By and the Kronecker lemma,

 



1



1 1 ₁ 1 n i X i

n E X I _{ }

  





0, .

i  n 

Hence (2.5) holds. So to prove (2.2) it suffices to prove that

 



X n



,

 _  _{ }    0 2 1 1 n p i i n

n l n P

     



 (2.7)

and   ,

 





2 1 1 1 max j p i i j n n i

n l n P Y EY

      . n   _{ } _{ }     0 a

 

l x

 

0,a





(2.8)

By Lemmas 2.1 (i), (iii), (2.1), and for each , the function is bounded on the interval ,

 





 





 





 

  



 



 



 



 



 



 











1 2 1 1 1 1 1 1 0 2 2

1 1 1 1 1 1 1 1 1 1

2 2 2

2 2 2

2 2 2

2 2 2

j j n p i i n p n p j n

j p j j j

pj j k j k j

k

pj j k k j

pk k k k

p

n l n P X

n l n P X n n l n P

l P

l P

l P

l P

E X l X

                                     1 1 1 1 1 1 1 2 2 2 2 j k k k n X n X X X X                            





 











    . 0 a  

i.e., (2.7) holds.

By the Markov inequality, Lemma 1.2, Lemmas 2.1 (i), (iv), (2.1), and for each , the function l x

 

 

0,a

 

is bounded on the interval ,

 





 





1 1 1 1 2 1 1 1 2 2 2 1 1

1 2 2

1 1

1 2 2

1

2 2 (| | )

1

( 2) 2

1 _{(| | 2 )} 1 ( 2) 1 1 1 max ( )

2 (2 )

2 (2 )

j j j j p i i j n n i n p i i n i p X n n p

n X n

j

j p j

X j

j

p j j

j k

n l n P Y EY n

n l n E Y EY

n l n EX I

n l n EX I

l EX I

l EX                                                         





















    

 

   













( 1) 1 ( 1) 1 1 1 2

(2 | | 2 )

( 2) 2

1 _{(2 | | 2 )}

1 2 2 1 2 2 1 1/ 1 1

2 (2 )

2 2 2

.

k k

k k

k k

X

p j j

X k j k

p

p k k p k

X k

p

I

l EX I

l E X I

E X l X

                              





  

 

Hence, (2.8) holds.

Now we prove that (2.2) (2.1). Obviously (2.2) implies





2 1 1 max , 0. p k k n n

n l n P X n

          



2 1 p (2.9)

Noting     , by Lemma 2.1 (ii), we have

 









 





1 1 1 2 1 1

0 2 2

1 1 2 1 1 max 2 1 max 1 max max . m m m m j j m j j n m n j j n n p j j n n P X

P X n

n

P X n

n

n l n P X n

                               _            











 Thus,





  1 2 1 2 2 1 1 2

max max 2

max 2 0.

m m m j j n n m j j

P X n

P X                 _  _  

Therefore, for sufficiently large n,



2



1

1

max 2 ,

2

j j n

P X n

   

which, in conjunction with Lemma 1.2, gives



2





2



1 1

2 4 max 2 .

k j

j n k

n

P X   n cP X n

  

  

Putting this one into (2.9), we get furthermore

 





1 2

2 n ,

 _ _  _{ }

1 1

0.

p n

n l n P X



 



 



Thus, by Lemmas 2.1 (i), (iii),

 





 





 



 



 



 



 



 



 



 











1

1 2

1 1

1

1 2 2

1

0 1

1

0 1

1 1

0 1

1

1/

1 1

2

2 2 2 2

2 2 2

2 2 2

2 2 2

j j

p

n

p

j n

j

j p j

j

pj j k

j k j

pj j k

k j

pk k k

k

p k

n l n P X

n l n P X

l P X

l P

l P

l P X

E X l X

 

 

 

 

 







   

    

 

 

 

   



  

 

 

 

 























 .

1 2

1

1 0

1 0

1 0

1 0

2

2 ˆ

2

2

2

j

k

k

k

n

n

X

X



  

 









 

 

 

 







 







This completes the proof of Theorem 2.1.

Proof of Theorem 2.2. (2.3) (2.4). Let

 



i



n

i i i _{X n}

Y Y X I _ ,i1, 2,,n

 

0

l x 

, the method of proof

of Theorem 2.2 is similar to method used to prove the above Theorem 2.1. Only the method of prove of (2.5) is not the same. In what follows, we prove that (2.5) holds. Since is a monotone non-decreasing function, we have

 



  

 





 

1

1 1

1 1 1

1 1

1 1

1 1

1 1

1

X

X X I

X l X

X l X

 

 

 





 



1

| | 1 1

1

1 1 . 1

X

I l X

l

 

Hence, by (2.3)，

1

1 .

E X    (2.10) i) For  1, by EX10 and (2.10),

1

1

1

1 (| | ) 1

1

1 1

1 1/

1 _{(| |}

max

| |

| |

j

i _{X n}

j n i

X n

n EY n EX I

X n E X

n

E X I





 



 

 

   _







 

 _{ }

 



1

1/ 1 (| | )

) 0,

X n

I 

  

 

ii) For  1, i.e., 1 1,





 





1

1

1 1 1

1 1

1

1

1 1 1

max

0,

j

i j n

i

X n

n

i X i

i

n EY n EY

n E X I

n E X I



 

 





 

  _ 

 

   





 





from the Kronecker lemma and

 





 





1

1

1

1 ₁ 1

1 1

1 1

1 1 .

i X i i

i X i i

i E X I E X I E X

 

 







 

   



   



  





Hence (2.5) holds. The rest of the proof is similar to the corresponding part of the proof of Theorem 2.1, so we omit it.

3. Weak Law of Large Numbers

Theorem 3.1. Suppose p1 2. Let



Xn;n1



be a

 _{-mixing sequence of identically distributed random} variables satisfying



1



lim p 0.

n

nP X n

   (3.1)

Then



1



1

1 p 0.

p n

p X n

P

S

n EX I n



 



1

p



X ;n1



(3.2)

Remark 3.1. When and n i.i.d.,

then Theorem 3.1 is the weak law of large numbers (WLLN) due to Feller [11]. So, Theorem 3.1 extends the sufficient part of the Feller’s WLLN from the i.i.d. case to a  -mixing setting.

Proof of Theorem 3.1. Let



p



j

j j _{X n}

X X I _

1 j n

for



=1 n

n j

j

S X

and

 



. Then, for each n2,



Xj;1 jn



are 

0

-mixing identically distributed random variables and for every  ,













1

1 1

0,

n n n n

p p p p

n

j j j

n

p p

j j

S S S S

P P

n n n n

P X X

P X n nP X n







 

 _{ } _  _ 

   

 

 

 

  _  _

 





   



via (3.1). So that (3.1) entails

0.

P

n n

p p

S S

n n



Thus, to prove (3.2) it suffices to verify that



1



0.

p

X n P



1 1 p n p

S

n EX I n

 _



 (3.3)

By (3.1) and the Toeplitz lemma,





2 2

1 1

2 2

1 n

p p

k

n p

k

k kP X k

k

 

 

 





0, n .

2 2

Thus, together with 2 1 1

n

p _n p k



k for p1 2, we have





2 1 2 1 1 1

n p p

k

n  k P X k



 



p 0, n ,

0

which, in conjunction with Lemma 1.1, yields for every

 ,





















 

































1

1

2

2 2

1

2

2 2

1 1

2 1 2

1

2 1 2

1 ₁

1

2 1 2

1 1

1 2

2 1 2

1 1

1

1

1

0

p

p p

p p

n n

n p

j j

j

n

p p

j j

j

p

X n

n p

k X k

k

n

p

p p

k

n

p

p p

k

P S ES n n E S ES

n E X EX

n E X EX n E

n EX I

n EX I

n k P X k

n k k P X

P X n

 

 

 

  

  

   

  

  



   

 

 

 _  _

 

  





 

 

  

 

 















2

1 2

1

n n

p

p

X

P X k

k



 



  











2 1

2 1 2 1

1 1

n

p p p

k

P X

n  k P X k





 

 

 







 1 0.

p p

n _

Thus



1



0.

p

X n P

 

1 1 p

n n n

p p

S ES S

n EX I

n n



 

  



1

 

i.e. (3.3) holds.

4. Examples

In this section, we give two examples to show our Theo- rems.

Example 4.1. Let



Xn;n be a  -mixing se-

quence of identically distributed random variables. Sup- pose 0 p 2 and p1; and if 1

0 = logr_{x r}, >0

1

then sup- pose also that . Assume that

and

1

EX  l x

 

X has a distribution with



1



1

1

~ , +1

log

P X r

x x

x _{ } 

  .



Is easy to verify that l x



satisfies the conditions of Theorems 2.1 and 2.2, and





1





1 1

p

E X l X   . Thus, by Theorems 2.1 and 2.2,





2

1 1

log max , 0

p r

j j n n

n nP S n 

 

  

    



.

Example 4.2. Suppose p1 2. Let



Xn;n1



be a



1

-mixing sequence of identically distributed random variables. Assume that X has a distribution with



1



1

1 , p

P X x o

x

 

 _{  }

 

then obviously,



1



lim p 0.

nnP X n 

Thus, by Theorem 3.1,



1



1

1 p 0.

p n

p X n

P

S

n EX I n



 



5. Acknowledgements

The work is supported by the National Natural Science Foundation of China (11061012), project supported by Program to Sponsor Teams for Innovation in the Con- struction of Talent Highlands in Guangxi Institutions of Higher Learning ([2011] 47), the Guangxi China Science Foundation (2012GXNSFAA053010), and the support program of Key Laboratory of Spatial Information and Geomatics (1103108-08).

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