Available online www.jocpr.com
Journal of Chemical and Pharmaceutical Research, 2014, 6(2):200-204
Research Article
ISSN : 0975-7384
CODEN(USA) : JCPRC5
Some conclusions of the exchangeable random variables and the independent
identical distribution random variables
Huang Zhaoxia
Department of Mathematics and Statistics, Ankang University, An kang, China
_____________________________________________________________________________________________
ABSTRACT
As the fundamental structure theorem of infinite exchangeable random variables sequences, the Definetti’s theorem
does not work to finite exchangeable random variables sequences, it is therefore necessary to find other techniques
to solve the approximate behavior problems of finite exchangeable random variables sequences. By using reverse
martingale approach, some scholars have given some results. In this paper we do some researches about the
similarity and difference of identically distributed random variables and exchangeable random variables sequences,
mainly discuss the limit theory of exchangeable random variables.
Key words: Exchangeable random variable; Exchangeability; Law of large numbers
_____________________________________________________________________________________________
INTRODUCTION
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 independence is more
difficult. So the concept of dependent random variables in probability and statistics is mentioned. Exchangeable
random variables are major type of dependent random variables. If the replacement the joint distribution of
X 1 , X 2 , , X n is unchanged, that is, for each replacement of 1, 2, , n the joint distribution of X 1 , X 2 , , X n is
the same with that of X π (1) , X π ( 2) , , X π ( n ) ; then the random variable finite series X 1 , X 2 , , X n is known as the
exchangeable. Obviously, the independent identical distribution random variables are the simplest exchangeable
random variables. The concept of exchangeable random variables is the first proposed by De Finetti 1930.The most
famous property of exchangeable random variables is its basic structured theorem, called De Finetti theorem; that is,
the infinite series of exchangeable random variables is independent identical distribution, if its tail is σ algebra.
Some scholars have given some results about exchangeable random variables sequences ([1]-[5]).The aim of this
paper is generalize the independent identical distribution variables [6] and[7] to the exchangeable random variables.
As the selection method for truncated random variables is different when deal with random variables, so the prove
method is more simple than that of [6] and [7].
THEDEFINITION AND THE LEMMA
Definition [8]. The positive valued function l
( x)
defined on [ 0, ∞ ) is called slowly changed, if for any c > 0 ,we
have lim l ( cx ) = 1 Suppose {ani ,1 ≤ i ≤ n, n ≥ 1} is real positive series that satisfy Aaa, n = n −1
x →∞
l ( x)
Aaa,n = n −1 ∑ i =1 ani
n
∑
n
a
a and
i =1 ni
a
(1)
Lemma [8]. Suppose { X n , n ≥ 1} are exchangeable random variables, which satisfy Cov ( f1 ( X 1 ) , f 2 ( X 2 ) ) ≤ 0
200
Huang Zhaoxia
J. Chem. Pharm. Res., 2014, 6(2):200-204
______________________________________________________________________________
Let A1 , A2 , Am be the disjoint non-empty subset of {1, 2, , n} with m ≥ 2, suppose
fi , i = 1, 2, , m is a
non-increase
(Non-decrease) function, then
fi ≥ 0, i =
1, 2, , m then E  ∏ fi ( X j , j ∈ Ai )  ≤ ∏ Efi ( X j , j ∈ Ai )
n
(1)If
n


=i 1 =i 1
(2)Particularly, for any xi ∈ R, i
n
=
1, 2, , m, we have P ( X 1 < x1 , X m < xm ) ≤ ∏ P ( X i < xi )
i =1
Subsequently, we will outline several lemmas, which will be used in the proof of the main theorems. If necessary,
we will also give the proof.
Lemma2 Suppose X 1 , X 2 , , X n are exchangeable random variables, that satisfy Cov ( f1 ( X 1 ) , f 2 ( X 2 ) ) ≤ 0
σ k2 EX k2 < ∞=
EX k = 0 ,=
( k 1, 2, , n ) , Suppose there exists a positive constant H such that
m ! 2 m−2
1, 2, , n )
⋅σ k H ( k =
2
n
n
n
then we have P  ∑ X ≥ x  ≤ exp  − x 2 4 ∑ σ 2  , 0 ≤ x ≤ σ 2 H
i
i 
∑



i
i =1
=
i 1
 i 1=



EX km ≤
n
n
 n



P  ∑ X i ≤ − x  ≤ exp  − x 2 4 ∑ σ i2  , 0 ≤ x ≤ ∑ σ i2 H
i 1
i =1
 i 1=



Proof. Based on Theorem 2.5 in[9] and Lemma 1 in [7], this Lemma is easy to prove.
Lemma 3. Suppose { X , X n ; n ≥ 1} are the exchangeable random variables and there exist
(
)
r
h > 0, r > 0 , such that E exp h ( x )  < ∞ (2) { X ni ,1 ≤ i ≤ n, n ≥ 1} are the exchangeable random variables that


satisfy Cov ( f1 ( X n1 ) , f 2 ( X n 2 ) ) ≤ 0
EX ni = 0, {ani ,1 ≤ i ≤ n, n ≥ 1} are real constant array that satisfy
( i ) There exist β , 0 < β ≤ r with
u X
lim un = 0 and {un , n ≥ 1} , such that ani X ni ≤ n i
n →∞
log n
β
a.s.
δ
v X
vn = 0 , X ∑ i =1 a ≤ n i
( ii ) There exists δ > 0 and array {vn , n ≥ 1} , that satisfy lim
n →∞
log n
2
ni
n
2
ni
Proof. Based on Theorem 2.5 in [9]and Theorem 18 in[10], the lemma is easy to prove
Lemma 4. Suppose { X , X n ; n ≥ 1} are the exchangeable random variables and there exist
(
)
r
h > 0, r > 0 , such that E exp h ( x )  < ∞


{ X ni ,1 ≤ i ≤ n, n ≥ 1} are the exchangeable random variables that satisfy
C ov ( f1 ( X n1 ) , f 2 ( X n 2 ) ) ≤ 0
EX ni = 0, 1 ≤ i ≤ n, n ≥ 1, {ani ,1 ≤ i ≤ n, n ≥ 1} are real constant array that satisfy
(
)
r
(1) E exp h ( x )  < ∞

β
(2)
β , 0 < β ≤ r and constant c>0 ,such that ani X ni
c Xi
≤
a.s.
log n
201
a.s.
Huang Zhaoxia
J. Chem. Pharm. Res., 2014, 6(2):200-204
______________________________________________________________________________
(3) There exists
n
then
∑a
ni
i =1
δ > 0 and array {υn , n ≥ 1} , lim vn = 0 , such that X ni2 ∑ i =1 ani2 ≤
n
n →∞
δ
vn X i
a.s.
log n
X i → 0 a.s. n → ∞
THEMAIN RESULTS AND PROOF
Theorem 1. Suppose
{ X , X n ; n ≥ 1}
are the exchangeable random variables that satisfy Cov ( f1 ( X 1 ) , f 2 ( X 2 ) ) ≤ 0
fi , i = 1, 2 are functions satisfy the above rule and non-decrease with X 1 , X 2 , EX 1 = 0, α p > 1, l ( x ) > 0 is
Suppose
monotonous non-decrease function when
x → +∞ , {ani ,1 ≤ i ≤ n, n ≥ 1} are real constant array with,
a
suppose Aαα
Aaa,n = n −1 ∑ i =1 ani ,Further,=
lim sup A , n < ∞, E X
n
β
n →∞
< ∞, EX = 0
and 1 < α , β < ∞,1 < p < 2, 且 1= 1 + 1 ,then
p α β
1
n1 p
n
∑a
ni
i =1
X i → ∞ a.s. ( n → ∞ )
(3)
Proof. Without loss of generality, for any 1 ≤ i ≤ n, n ≥ 1 , suppose ani
> 0 , as { X , X n ; n ≥ 1} are the
exchangeable random variables, and an1 X 1 , an 2 X 2 , , ann X n also satisfy
Cov ( f1 ( an1 X 1 ) , f 2 ( an 2 X 2 ) ) ≤ 0
and
En
1 1 1
=
+ ,Then p < α ∧ β ∧ 2 . from (1) we have
p α β
−1 p
n
∑ ani X i
a ∧β ∧2
≤ Cn −a ∧ β ∧ 2
n
p
∑ ani
a ∧β ∧2
E X
a ∧β ∧2
≤ Cn −α ∧ β ∧ 2
p +1
Aαα∧∧ββ∧∧2,2 n → 0, n → ∞
i 1 =i 1
then n
−1 p
n
∑a
i =1
ni
p
Xi →0 n → ∞
From the symmetrized inequality proved in Lemma 14 in [10], we know that, in order to prove
n
p
n −1 p ∑ ani X i → 0, n → ∞ , we just need to prove
i =1
1
ani X i S → 0
n1 p
where X i
S
a.s. n → ∞
is the symmetrized form of X n ,From Lemma 3 in [11], we have the symmetrized series of
Cov ( f1 ( X 1 ) , f 2 ( X 2 ) ) ≤ 0
(
)
also satisfy the inequality, i.e. Cov f1 ( X 1S ) , f 2 ( X 2 S ) ≤ 0 ,Without loss of generality, we assume that { X n , n ≥ 1} are
the symmetrized exchangeable random variables that satisfy Cov ( f1 ( X 1 ) , f 2 ( X 2 ) ) ≤ 0 for all 1 ≤ i ≤ n,
n ≥ 1 ,Letting
′ X i I ( X i ≤ n1 β ) + n1 β I ( X i > n1 β ) − n1 β I ( X i < −n1 β )
X=
i
′′ X i I ( X i > n1 β ) − n1 β I ( X i > n1 β ) + n1 β I ( X i < −n1 β )
X=
i
=
X i′′ X i I ( X i > n1 β )
202
Huang Zhaoxia
J. Chem. Pharm. Res., 2014, 6(2):200-204
______________________________________________________________________________
=
ani′ ani I ( ani ≤ n1 a )
ani′′ = ani − ani′ = ani I ( ani > n1 a )
then
n
n
n
n
∑ ani X i = ∑ ani′ X i′ + ∑ ani′′ X i′ + ∑ ani X i′′
(4)
=i 1 =i 1 =i 1 =i 1
As
β (αα
−1)
1 1 1
=
+ , β= α 1 + β 1 − 1   , X i′′ ≤ X i′′
n −(1−1 p )
p α β
α −1 
 p 
and E X
(
β
∞
< ∞ which is equivalent to ∑ n =1 P X
(
From Borel-Cantelli Lemma, we have P X n
β
β
)
(
> n < ∞ ,then ∑ n =1 P X n
∞
β
)
>n <∞,
)
> n, i.o. =
0 。hence
β
1 n
∑ X i′′ → 0 a.s. ( n → ∞ )
n i =1
From
''
''

Holder
inequality, X i ≤ X i 及 X i′′ ≤ X i′′
β (αα
−1)
⋅ n −(1−1 p ) and
β
1 n
∑ X i′′ → 0 a.s. ( n → ∞ )
n i =1
then
n
−1 p
n
∑a
X ′′ ≤ n
−1 p
n
∑a
X ′′ ≤ n
−1
n
∑a
ni i
ni
i
=i 1 =i 1 =i 1
β
1 n

≤ Aα ,n  ∑ X i′′ 
 n i =1



−1)
(αα
ni
X i′′
β (a −1) a
→ 0 a.s. ( n → ∞ )
Therefore, we have
n
n −1 p ∑ ani X i′′ → 0 a.s. ( n → ∞ )
(5)
i =1
As
1+
1 1 1
=
+ , α ∨ β < 1 ,then
p α β
(2 −α )
α
+
+
(2 − β )
+
β
n
Therefore, we have
 2
if α ∧ β < 1

=
α ∧ β
1
if α ∧ β ≥ 1

2
∑ E ( ani′ X i′) ≤ CnAaa∧∧2,2 n ⋅ n( 2−a )
+
+
a +( 2− β ) β
X
i =1
Moreover, for any 1 ≤ i ≤ n, n ≥ 1 ,we have n
−1 p
max {n 2 a , n 2 β , n} = O ( n 2 p log −2 n ) , From
β ∧2
β ∧2
(
=
O max {n 2 a , n 2 β , n}
ani′ X i′ ≤ n1 a n1 β n −1 p =
1 ,and
Lemma 2, for sucient small " and sucient large n, we have

n
−e 2


P  n1 p ∑ ani′ X i′ > e  ≤ exp  −2 p
 4n O max {n 2 a , n 2 β , n}
i =1



(
203
)

2
 ≤ exp −e 2 ( log n )


(
)
)
Huang Zhaoxia
J. Chem. Pharm. Res., 2014, 6(2):200-204
______________________________________________________________________________

By the same procedures, we can also prove that P  n
−1 p


n
So
n
2
i =1
ni
i
2
),

n
∑ P  n ∑ a′ X ′ > ε  < ∞ 。then
−1 p
ni
=i 1 =i 1
n

∑ a′ X ′ < −e  ≤ exp ( −e ( log n )
−1 p
i
n
∑ a′ X ′ → 0 a.s. ( n → ∞ )
ni
i =1
(6)
i
n
n
(
∑ ani′′ X i′ ≤ n−1 p n1 β ∑ ani I a ni > n1 a
−1 p
For 1= 1 + 1 , we have n
p α =
β
i
n
≤ n −1 p +1 β n(1−a ) a ∑ ani
a
i =1
1 =i 1
=
Aaa,n
)
(7)
Then from (4),(5),(6),(7), we have
lim sup n −1 p
n →∞
n
∑a
i =1
ni
X i ≤ Aaa a.s. ( n → ∞ )
By replacing X i with tX i we have
lim sup n −1 p
n →∞
Let t
1
n1 p
n
∑ ani X i ≤
i =1
→ ∞ ,we have
n
∑a
i =1
ni
Aaa
a.s. ( n → ∞ )
t
X i → ∞ a.s. ( n → ∞ )
the inequality (3) is true.
CONCLUSION
As the fundamental structure theorem of infinite exchangeable random variables sequences, the Definetti’s theorem
states that infinite exchangeable random variables sequences is independent and identically distributed with the
condition of the tail σ-algebra. So some results about independent identically distributed random variables is similar
to exchangeable random variables. As the fundamental structure theorem of infinite exchangeable random variables
sequences, the Definetti’s theorem does not work to finite exchangeable random variables sequences, it is therefore
necessary to find other techniques to solve the approximate behavior problems of finite exchangeable random variables
sequences. By using reverse martingale approach, some scholars have given some results. In this paper we do some
researches about the similarity and difference of identically distributed random variables and exchangeable random
variables sequences, mainly discuss the limit theory of exchangeable random variables.
REFERENCES
[1] Fintti, B. De. Funzione Caratteristica Di Unfenomeno Aleatorio. Atti Accad. NaZ. Lincei Rend. cl. Sci. Fis.Mat.
Nat.vol 4,pp. 86-133, 1930
[2] J. F. C. Kingman. Uses of Exchangeability, Ann. Prob, p. 183-197, June 1978.
[3] W. G. Mcginley, R. Sibson. Dissociated Random Variables. Math Proc Camb. Phil Soc, vol 77:pp.185-188, 1975
[4] R. F. Patterson, R. L. Taylor. Strong Laws of Large Numbers for Triangular Arrays of Exchangeable Random
Variables. Stochastic Analysis and Applications, vol 3, pp. 171~187, 1985.
[5] A. Gut. Precise A sympototics for Record Times and the Associated Counding Process. Stoch Proc Appl, vol
101,pp. 233~239 ,2002.
[6] Bai, Z., & D. Cheng, P. E. Marcinkiewicz Strong Laws for Linear Statistics. Statist Probab Lett, vol 46,
pp.105-112, 2000
[7] Sung, S. H. Strong Laws for weighted suns of i.i.d. random variables. Statist Probab Lett, vol 52, pp. 413-419,
2001
[8] Wu Qunying. Probability limit theory of mixing sequences. Science Press, pp. 132-133, 2006
[9] Taylor, R. L. & Patterson, R. F. A strong laws of large numbers for arrays of row wise negatively dependent
random variables. Stochastic Anal. .vol20, pp. 643-65.,2002
[10] Petrov, B. Sums of Independent Random Variable. Hefei: USTC Press, pp.83-84, 2002
204
Huang Zhaoxia
J. Chem. Pharm. Res., 2014, 6(2):200-204
______________________________________________________________________________
[11] Chi, X., & Su, C. A weak law of large number of identical distribution NA series. Applied Probab and
Statist,vol 13,pp.199-203,1997
205