A important method for the probability limit theory of exchangeable random variables

Journal of Chemical and Pharmaceutical Research, 2014, 6(6):2323-2328

Research Article

ISSN : 0975-7384

CODEN(USA) : JCPRC5

A important method for the probability limit theory of exchangeable

random variables

Huang Zhaoxia

An Kang University, China

_____________________________________________________________________________________________

ABSTRACT

Limit theory mainly study independent random variables, but in many practical problems, samples are not independent, or the function of independent sample is not independent, or the verification of independent is more difficult. So the concept of dependent random variables in probability and statistics is mentioned. Exchangeable random variables is a major type of dependent random variable. By using reverse martingale, censored and other methods, in certain relevant conditions, we extend some conclusions to the exchangeable random variables, and obtain several conclusions of the convergence of exchangeable random variables.

Key words: limit theory; exchangeable random variables; reverse martingale

_____________________________________________________________________________________________

INTRODUCTION

If the joint distribution of

1, 2, , n

X X L X is permutation invariant, for each permutation

π

of

1, 2,

L

, n

,the joint distribution ofX X₁, ₂,L,X_nis the same of the joint distribution of _{( ) ( ) ( )}

1, 2, , n

X_π X_π L X_π ,so the finite random variable sequence X X₁, ₂,L,X_nis exchangeable. Obviously, the independent and identically distributed random variable sequence is the simplest exchangeable random variable sequence. The concept of exchangeability was first proposed by De Finett[1] in 1930, people using the De Finetti theorem has made some results (see [2]-[3]). In this paper we extend the results of Katz and Baum theorem in the condition of independent and identically distributed random variable sequence to the results of Katz and Baum theorem in the condition of exchangeable random variables. We obtain the Katz and Baum theorem for specific forms of expression in the case of exchangeable random variables.

PRELIMINARY CONCEPTS

Definition [4]: when on the

[ )

0,∞ the positive functionl x

( )

(x→ ∞) is slowly varying function, as for all

0

c> ,

( )

( )

lim 1

x l cx

l x

→∞ =

.About slowly varying function with the following properties:if it is slowly varying function

when l x

( )

(x→ ∞), so

(1)

( )

( )

lim 1, x

l tx l x

→∞ = ∀ >t 0,

(

)

( )

lim 1,

x

l x u

l x

→∞

+

= ∀ ≥u 0；

(2)

( )

( )

2 2

( )

( )

2 2

lim sup lim inf 1

2 k k l 2

k k l k k

k _x k x

l x l x l + l +

→∞ _{≤ <} = →∞ ≤ < =

；

(3) limx l xδ

( )

,

→∞ = ∞ ∀ >δ 0,limx l x

( )

0 δ

______________________________________________________________________________

Lemma 1 Suppose

{

X_n;n≥1

}

are exchangeable random variables, and satisfy

( ) ( )

(

1 1 , 2 2

)

≤0 Cov f X f X

2,

m

∀ ≥ A A1, 2,LAmis

{

1, 2,L, n

}

twenty-two disjoint non-empty set,if

f i

i

,

=

1, 2,

L

,

m

each argument is a

function of both the non-drop (non-liters,so (1) if

f

_i

≥

0,

i

=

1, 2,

L

,

m

_{,we obtain}

(

)

(

)

1 1

, ,

n n

i j i i j i

i i

E f X j A Ef X j A

= =

 

∈ ≤ ∈

 



∏



∏

(2)

∀

∈

,

=

1, 2,

L

, ,

i

x

R i

m

(

1 1

)

(

)

1

,

n

m m i i

i

P X x X x P X x

=

< L < ≤

∏

<

Lemma 2 Suppose

{

Xn;n≥1

}

are exchangeable random variables,

( ) ( )

(

1 1 , 2 2

)

≤0

Cov f X f X

0,

n n n

EX

=

X

≤

b

a s

. .

(

n

=

1, 2,

L

)

,

_t

>

₀

_and

1

max

_i

1

i n

t

b

≤ ≤

≤

,so

∀ >

u

0

,we obtain

2 2

1 1

2 exp

n n

i i

i i

P X u tu t EX

= =

   

≥ ≤ − + 

 

 



∑



∑

Proof: because

( )

0 !

=

→

∑

 i

k n

tX i

n k

tX

Y e

k

,

tX

i

≤

1

a s

. .

we obtain

Y

n

≤

e

a s

. .

.so

By Lebesgue Theorem control convergence,we obtain

( )

( )

( )

0 0

lim lim

! !

i

k k

n n

i i

tX

n n

k k

tX E tX E e E

k k

→∞ ₌ →∞ ₌

 

 

= _{ }=



∑



∑

( )

_{( )}

2

0 2

1 1

! !

k n

i

i

k k

E tX

E tX

k k

∞

= =

=

∑

≤ +

∑

2 2

2 2

1

t EXi

i

t EX

e

≤ +

≤

By Markov inequality and lemma1(1),we obtain

∀ >

u

0,

t

>

0

,有

1 1

1

n n

i i

i i

n t X t X

tu tu i

i

P

X

u

P e

=

e

e

−

Ee

=

=



_∑



_∑





_

_

>

=

>

≤





_

_









∑

2 2

1 1

i i

n n

tX t EX

tu tu

i i

e− Ee e− e

= =

≤

∏

≤

∏

2 2

1

exp

n i i tu t EX

=

 

= − + 



∑



We use−X_iinstead of

X

_i

(

)

( )

2 2

1 1 1

exp

n n n

i i i

i i i

P X u P X u tu t EX

= = =

     

− > = < − ≤ − +

     



∑

 

∑

 

∑



1 1 1

n n n

i i i

i i i

P X u P X u P X u

= = =

     

≥ = > + < −

     

   



∑



∑

∑

2 2

1

2 exp

n i i tu t EX

=

 

≤ − + 

______________________________________________________________________________

THE MAIN CONCLUSION

Theorem: Suppose

{

Xn;n≥1

}

are exchangeable random variables, and the P moments is exist,when 0< ≤p 1it is

an arbitrary random sequence,whenp>1it is exchangeable random variables of zero mean,when0< ≤p 3,

(

1

)

1

max

n

p p

j i

j n

i E S c E X

≤ ≤ ≤

∑

₌

whenp>3,

(

)

3

2

1

1 1

max

p

n n

p p

j j j

j n

j j

E S c E X EX

≤ ≤ _{= =}

 _{ } 

 

≤  +  

 

 

 

∑

∑

(1)

Proof: when0< ≤p 1,by Jensen inequality, we obtain

(

1

)

1 1

1 1

max max max

p p

j j

p

j i i

j n j n j n

i i

E S E X E X

≤ ≤ ≤ ≤ ₌ ≤ ≤ ₌

   _{ } 

   

= _{ }≤ _   _

 



∑

 

∑



1 1

p

j j

p

i i

i i

E X E X

= =

 

≤   ≤



∑



∑

so

(

)

1

1

max

n

p p

j i

j n

i E S c E X

≤ ≤ ≤

∑

₌ is right.

when

p

>

1

,it is easy to prove the following inequality:

3

1

+

t

p

≤ +

1

pt

+

2

−p

t

p

,

∀ ∈

t

R

,

0

< ≤

p

3

2 2

1

+

t

p

≤ +

1

pt

+

2

p

p t

+

2

p

t

p

,

∀ ∈

t

R

,

p

>

3

suppose

t

=

x y

,by the above formula

∀

x

,

y

∈

R

, we obtain

3

1

1

2

p p

p p

x

p

x

_p

x

x

y

y

y

p

y

y

y

−













+

=



+



≤

_

+

+

_





_

_

1 2

2

p p p

sgn ,

x

px y

−

y

−

≤

+

0

< ≤

p

3

(2)

p

x

+

y

≤

x

p

+

y

p

+

px

sgn

y

+

2

p

p y

2 p−2

,

p

>

3

{

}

1 1

max

max

p p

j j

j n

S

j n

S

≤ ≤

=

≤ ≤

(

)

{

max 0,

1

,

,

}

{

max 0,

(

1

,

,

)

}

p p

n n

S

S

S

S

≤

L

+

−

L

−

(

)

{

}

{

(

)

}

1 1

1 1

2

p

max 0,

S

,

,

S

n p

2

p

max 0,

S

,

,

S

n p

− −

≤

L

+

−

L

−

1

2

p−

≤

1

max

p j j n

S

≤ ≤

( )

1 1

2

max

p p

j j n

S

− ≤ ≤

+

−

(3)

because

0

≤ ≤ −

j

n

1

,we note

(

)

,

max

1

,

1 2

,

,

1 2

,

,

j n j j j j j n

M

=

X

₊

X

₊

+

X

₊

L

X

₊

+

X

₊

+

L

+

X

(

)

, 1 1 2 1 2

0

≤

M

%

_{j n}

=

max 0,

X

_j+

,

X

_j+

+

X

_j+

,

L

,

X

_j+

+

X

_j+

+

,

L

,

+

X

_n

First we prove

(

)

1

1

max

n

p p

j i

j n

i E S c E X

≤ ≤ ≤

∑

₌ ,from

0

< ≤

p

3

,we use the formula of (2), we obtain

2 1

0, 1 1, 1 1, 1 1,

1

max

2

p

p

p p p p p

j n n n n

j n

S

M

X

M

−

X

M

pX M

−

≤ ≤

=

+

+

≤

+

+

%

%

%

2 1

2

p p p p

X

M

pX M

− −

______________________________________________________________________________

1

2 1

,

1 1

2

n n

p

p p

i j j n

i i

X

p

X M

−

− −

= =

≤

∑

+

∑

%

₍₄₎

because

X

_j and

M

%

_{j n}p_,−1 are non-drop each on the corresponding variable, they also satisfy

( )

(

)

(

1

)

1 , 2 , 0

p

j j n

Cov f X f M% − ≤ , and we notice

EX

i

=

0

,so

(

1

)

1

, , 0,

p p

j j n j j n

E X M% − ≤EX EM% − =

0

≤ ≤ −

j

n

1

by the above formula and (4), we obtain

2 1

1

max 2

p n

p p

j i

j n

i

E S − E X

≤ ≤ ₌

 

≤

 

 

∑

In the course of discussion, we use

−

X

_iinstead of

X

_i,we also obtain

( )

2

1

1

max 2

p n

p p

j i

j n

i E S − E X

≤ ≤ ₌

 

− ≤

 

 

∑

(5)

form(3),(4) ,(5),we know

(

)

1

1 max

n

p p

j i

j n

i

E S c E X

≤ ≤ ≤

∑

₌

is right.

Let us prove the case whenp>3, In order to improve(1),we first prove

(

)

3

2 1

1

max

p n p

j j

j n

j E S cE X

≤ ≤ ₌

 

≤  



∑



We note

c

₁

=

3 ,

p

c

₂

=

3

p

p

3.because

p

>

3

,

x

+

y

p

≤

x

p

+

y

p

+

px

sgn

y

+

3

p

p y

3 p−3,we obtain

0, 1 1,

1

max

p

p p

j n n

j n

S M X M

≤ ≤

= = + %

1 2 2

1 1 1, 1 1, 2 1 1,

p _{p p p}

n n n

c X M pX M − c X M −

≤ + % + % + %

1 2 2

1 1 1, 1 1, 2 1 1,

p

p p p

n n n

c X

M

pX M

−

c X M

−

≤

+

+

%

+

%

M

1 1

1 2 2

1 , 2 ,

1 1 1

n n n

p _{p p}

i j j n j j n

i j j

c X p X M c X M

− −

− −

= = =

≤

∑

+

∑

% +

∑

%

obviously,对

0

≤ ≤ −

j

n

1

,we obtain

(

)

(

)

3

2

, max , 1, 1 2, , 1 , ,

p p

j n j j j j j j j j n j

M% − = S S +X ₊ S +X ₊ +X ₊ L S +X ₊ +L +X −S −

3 _{3 3}

1

max max

p _{p p}

k j j

j k n j n

c S S c S

− _{− −}

≤ ≤ ≤ ≤

 

≤  + ≤

 

from

(

p_, 1

)

p_, 1

0,

j j n j j n

E X M

%

−

≤

EX EM

%

−

=

0

≤ ≤ −

j

n

1

and all of the above, we obtain

3 2

1 1

1 1

max max

p

p n n

p

j i i j

j n j n

i i

E S c E X E X S

≤ ≤ _{= =} ≤ ≤

  

   

≤  +  

  _{ }

  

∑



∑



(6)

The result also is right of

( )

1

max

p j j n E S

≤ ≤ − . So from (3)and (6), we use the Inequality of Hölder,we obtain

3 2

1 1

1 1

max max

p

p n n

p

j i i j

j n j n

i i

E S c E X E X S

≤ ≤ _{= =} ≤ ≤

  

   

≤  +  

  _{ }

  

∑



∑



(

)

( )

3

3 ₂

2

1

1 1

max

p

p _{p p}

n n _p

p

i i j

j n

i i

c E X E X E S

−

≤ ≤

= =

 _{ } 

 

   

≤  +    

 

 

   

 

______________________________________________________________________________

(

)

3

3 2

1

1 1

3 2

max

p

n n

p

p _p

i i j

j n

i i

p

c E X c E X E S

p p ≤ ≤

= =

−

 

≤ +   +

 

∑

∑

We move the formula 1 max _j p

j n

E S

≤ ≤ above from right to left, we obtain

(

)

3

2

1

1 1

max

p

n n

p p

j i i

j n

i i

E S c E X E X

≤ ≤ _{= =}

 _{ } 

 

≤  + _  _

 

  



∑

∑



from

p

>

3

,so

0

<

3

p

<

1

,so

3

3 3

2

1 1 1 1

p

p p

n n n n

p p p

i i i i

i i i i

E X E X E X E X

= = = =

 

      

=  =    ≤  

      

∑

∑

∑

∑

We obtain

(

)

3

2

1

1

max

p n p

j j

j n

j E S cE X

≤ ≤ ₌

 

≤  



∑



Finally we prove the formula of

(

)

3 2 1

1 1

max

p

n n

p p

j j j

j n

j j

E S c E X EX

≤ ≤ _{= =}

 _{ } 

 

≤  +  

 

 



∑

∑



We note _{( )}

0

i

i i X

X+ =X I _≥ , _{( )}

0

i

i i X

X− = −X I _< ,and suppose

( )

2

( )

2

,

i Xi E Xi

ξ ₌ + ₋ +

( )

2

( )

2

i Xi E Xi

η ₌ − ₋ −

obviously,E X

( )

_i+ 2≤EX_i2,

( )

2 2

i i

E X− ≤EX ,from(1),we obtain

(

)

3 3

2 2

1

1 1

max

p p

p n n

j i i i

j n

i i

E S cE X cE X+ X−

≤ ≤ _{= =}

     

≤ = −

     

   

 

∑

∑

( )

2 3

( )

2 3

1 1

p p

n n

i i

i i

c E X+ E X−

= =

 _{   } 

 

≤    +   

   

 



∑

∑



( )

2 3

( )

2 3

1 1 1 1

p p

n n n n

i i i i

i i i i

c E

ξ

E X+ E

η

E X−

= = = =

 _{   } 

 

≤   +  +  +  

   

 



∑

∑

∑

∑



3 3 3

2

1 1 1

p p p

n n n

i i i

i i i

c E

ξ

E

η

EX

= = =

 _{     } 

 

≤  _{ } + _{ } +_{ } 

     

 



∑

∑

∑



(7)

Here we use (7) ,and use the proof of induction on

p

,so we prove(1). First we notice that

ξ

_iisnon-decreasing function of

X

_i,

η

_iis non-increasing function of

X

_i,

{

ξ

i

;

i

≥

1

}

and

{

η

i

;

i

≥

1

}

Still is the exchangeable

random variables with zero mean,And satisfies

( ) ( )

(

1 1 , 2 2

)

0,

Cov f

ξ

f

ξ

≤ Cov f

(

1

( ) ( )

η

1 ,f2

η

2

)

≤0

When 2

3< ≤p 3 ,1< p 3 3≤ ,from the above, we obtain

(

1

)

1

max

n

p p

j i

j n

i

E

S

c

E X

≤ ≤

≤

∑

₌

( )

(

( )

)

3 3 ₃

2

1 1 1 1

p p

n n n _p n p

i i i i

i i i i

E

ξ

c

E

ξ

c

E X

+

E X

+

= = = =









≤

≤



+









∑



∑



∑

∑



3 2

p p

n n

i i

c

E X

EX

= =



_

_







≤



+

















∑

∑



______________________________________________________________________________

The formula

3

1

p n

i i E η

=

 

 



∑

 also is right ,from the formula(8)we obtain (1) is right.

We suppose that when

3

k−1

< ≤

p

3

k(

k

≥

2

) is right,so (8) is right,when

3

k

< ≤

p

3

k+1,(8) also is right.Use the induction hypothesis

3 3 4

2

1 1 1

p p p

n n n

i i i

i i i

E

ξ

c

E

ξ

E

ξ

= = =

















≤



+













∑



_

_

∑



∑



_

_

3 9

2 2

1 1 1

p p

p

n n n

i i i

i i i

c

E X

EX

EX

= = =



_

_

_

_







≤



+





+





















∑

∑

∑



when

p

>

4

, from the inequality of Hölder,we obtain

( )

( ) ( )

( )

( )

{

_{4 2} 3 2

}

4 2

1 1

n n _p

p p p

i i i

i i

EX

E

X

− −

X

−

= =

=

∑

∑

( )

( 4) ( 2)

(

)

3( 2)

2 1

n _{p p p}

p

i i

i

EX

− −

E X

−

=

≤

∑

( 4) ( 2) 2( 2)

2

1 1

p p p

n n

p

i i

i i

EX

E X

− − −

= =









≤











∑





∑



From the real number of Hölder inequality, we know

( ) ( ) ( )

4 4 4 2 3 2

4 2

1 1 1

p p p p p p

n n n

p

i i i

i i i

EX

EX

E X

− − −

= = =













≤















∑





∑





∑



(

)

(

)

3 2

1 1

4

2

2

2

2

p

n n

p

i i

i i

p

p

EX

E X

p

=

p

=

−





≤

₋





+

₋



∑



∑

(9)

from(8) and (9), we obtain

3 3

2

1 1 1

p p

n n n

p

i i i

i i i

E

ξ

c

E X

EX

= = =

















≤



+













∑



_

_

∑



∑



_

_

(10)

We also obtain

3 3

2

1 1 1

p p

n n n

p

i i i

i i i

E

η

c E X EX

= = =

 

     

≤  + 

   



∑

 

∑



∑

 

(11)

from(7),(10)and (11),we know (1 )is right.

REFERENCES

[1]B. De Fintti. Funzione Caratteristica Di Unfenomeno Aleatorio. Atti Accad. NaZ. Lincei Rend. cl. Sci. Fis. Mat. Nat. vol.4 pp.405-422, 1930.

[2]J. Blum, H. Chernoff, M. Rosenblatt, H. Teicher. Central Limit Theorems for Interchangeable Processes, Canad. Math. vol.10, 1958.

[3]R. L. Taylor. Laws of Large Numbers for Dependent Random Variables. Colloquia mathematica societatics Janos Bolyai, 36, Limit Theorems in Probability and Statistics, Veszprem(Hungary), pp.1023-1042, 1982.