Advanced Science and Technology Letters
Vol.121 (AST 2016), pp.27-31
http://dx.doi.org/10.14257/astl.2016.121.06
The Conclusion of Exchangeable Random Variable in
Probability Limit Theory
Zhaoxia Huang1,2, Fucai Qian1, 3, Jun Liu1
1
School of Automation and Information Engineering, Xi'an University of Technology,
Xi'an, 710048, China
2
Department of mathematics and statistics, Ankang University, Ankang, 725000, China
3
Key Laboratory of Spacecraft In-orbit Fault Diagnosis and Repair, Xi'an Satellite Control
Center, Xi'an, 710043, China
[email protected]
Abstract. As the fundamental structure theorem of infinite exchangeable
random variables sequences, the De finetti’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. In this paper, By using reverse martingale, censored and
other methods, in certain relevant conditions, we extend the conclusion of the
exchangeable random variables, and obtain the conclusion of the convergence
of exchangeable random variables. We extend the Baum and Katz theorem in
the condition of independent, and obtain the specific forms of expression of
Baum and Katz theorem in the case of exchangeable random variables.
Keywords: exchangeable random variables; reverse martingale; triangular
arrays of row-wise
1
Introduction
Probability limit theory is one of the branches of probability and also is important
basic theory of the science of probability and statistics. 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.
In early 1930s, DeFinetti put forward the concept of random variable
exchangeability [1]. The so-called exchangeability of finite sequence X k nk 1 refers
to that if the joint distribution of random variables X1 , X 2 , , X n is unchanged in
displacement, that means joint distributions to any displacement of 1, 2, , n on  ,

X  1 , X  2 , , X  n and on X1 , X 2 , , X n are identical. The infinite sequence X k k 1

 
 
of exchangeable random variable is exchangeable, in case that any finite subset
ISSN: 2287-1233 ASTL
Copyright 2016 SERSC
Advanced Science and Technology Letters
Vol.121 (AST 2016)
thereof is exchangeable. Theories have proved that infinite sequence of exchangeable
random variables is independent identically distributed under condition of its tail σ --algebraic condition, so it is no wonder that it has asymptotic properties similar to
independent identically distributed sequence.
As the fundamental structure theorem of infinite exchangeable random variables
sequences, the De finetti’s theorem states that infinite exchangeable random variables
sequences is independent and identically distributed with the condition of the tail σalgebra. 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 De finetti’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 [2-10]. 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.
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. we
extend the Baum and Katz theorem in the condition of independent, and obtain the
specific forms of expression of Baum and Katz theorem in the case of exchangeable
random variables.
2
Preliminary Concept and Lemma
Definition1: If the joint distribution of X1 , X 2 ,
each permutation

of
1, 2,
, X n is permutation invariant, for
, n , the joint distribution of X1 , X 2 , , X n is the
same of the joint distribution of X  1 , X  2 , , X  n , so the finite random variable

 
 
sequence X1 , X 2 , , X n is exchangeable.
Definition 2: when on the  0,   the positive function l  x  ( x  ) is slowly
l  cx 
varying function, as for all c  0 ,
. About slowly varying function with
lim
1
x 
l  x
the following properties: if it is slowly varying function when l  x  ( x  ) , so
lim
x 
l  tx 
l  x  u
 1, t  0, lim
 1, u  0
x

l  x
l  x
lim sup
k  2k  x  2k l
28
l  x
l 2
k

 lim k inf k l
k  2  x  2
(1)
l  x
l  2k 
1
(2)
Copyright © 2016 SERSC
Advanced Science and Technology Letters
Vol.121 (AST 2016)
lim x l  x   ,   0, lim x  l  x   0
x 
 X n ; n  1
Lemma 1 Suppose
(3)
x 
are exchangeable random variables, and satisfy
Cov  f1  X 1  , f 2  X 2    0
m  2, A1 , A2 ,
fi , i  1, 2,
is 1, 2,
Am
, n
twenty-two disjoint non-empty set, if
, m each argument is a function of both the non-drop (non-liters, so
(1) if fi  0, i  1, 2,
, m , we obtain
 n
 n
E   fi  X j , j  Ai     Efi  X j , j  Ai 
 i 1
 i 1
(2)
 xi  R, i  1, 2,
, m,
n
X m  xm    P  X i  xi 
P  X 1  x1 ,
 X k k 1
n
Lemma 2 Suppose
i 1
are exchangeable random variables, and satisfy
Cov  f1  X 1  , f 2  X 2    0
EX n  0, X n  bn a.s.  n  1, 2,
, t  0
and t max bi  1 , so u  0 ,
1i  n
we obtain
n
 n



P   X i  u   2exp tu  t 2  EX i 2 
i

1
i

1




Proof:
because Y
n
n

 tX i 
k
k!
k 0
 etX i
tX i  1 a.s. we obtain Yn  e a.s. . so
,
By Lebesgue Theorem control convergence, we obtain
k
n

 tX i 
E etX i  E  lim 
 n k 0 k !

 
k
n E tX

 i
  lim 

 n k 0
k!

n

E  tX i 
k
k!
k 0
By Markov inequality and lemma1 1, for
 1  E  tX i 
2

1
 k !  1  t 2 EX i 2  et EX
k 2
2
2
i
u  0, t  0 , we obtain
 t  Xi

t  Xi
 n

P   X i  u   P  e i 1  etu   e tu Ee i 1


 i 1



n
n
n
i 1
i 1
 etu  EetX i  etu  et
We use  X i instead of
Copyright © 2016 SERSC
2
EX i 2
n
n


 exp  tu  t 2  EX i 2 
i 1


Xi
29
Advanced Science and Technology Letters
Vol.121 (AST 2016)
n
 n

 n



P     X i   u   P    X i   u   exp  tu  t 2  EX i 2 
i 1
 i 1

 i 1



n
 n

 n

 n



P   X i  u   P   X i  u   P   X i  u   2exp  tu  t 2  EX i 2 
i 1


 i 1

 i 1

 i 1

The Main Result
3
 X n ; n  1
Theorem: Suppose
are exchangeable random variables,
EX1  0 , so lim Sn n1 r  lim X n n1 r  0
and when r  1 ,
n
n
E X1  
n

r
proof: because
so
lim Sn n1 r  0
a.s
n 

suppose
 P  X
n 1
, then
n  0  0
and
 P  X
n 1
r
n
 n1 r   

lim X n n1 r  0
n 
a.s

, so
 P  X
n 1
n
  n1 r   ,   0
a.s
1r
n
, by Kolmogorov 0-1,

 P  X
n 1
so
a.s ,
n
limsup Sn n1 r  limsup X n n1 r  
r
n

0r 2
Lemma 1 , and 2, for
  0 , so
 r  n   P  X 1  r  n  

r
n 1


E X1   , we obtain
r

 P  X
n 1
1r
so limsup X n n
n 

n
 Mn1 r   , M  0 ,
a.s , by Borel-Cantelli
limsup X n n1 r  limsup Sn  Sn1 n1 r
n
n 
 limsup Sn n1 r  limsup Sn1 n1 r = 2 lim sup Sn n1 r
n
1r
so
30
lim sup Sn n
n

n
n 
a.s.
Copyright © 2016 SERSC
Advanced Science and Technology Letters
Vol.121 (AST 2016)
Acknowledgements. This work was Supported by National Natural Science
Foundation of China (61273127, 61304204), the Key Laboratory for Fault Diagnosis
and Maintenance of Spacecraft in Orbit (SDML_OF2015004). The Key Program of
National Natural Science Foundation of China (61533014) and Innovative Research
Team of Shaanxi Province (2013KCT-04).
References
1. De Fintti, B.: Funzione Caratteristica Di Unfenomeno Aleatorio. Atti Accad. NaZ. Lincei
Rend. cl. Sci. Fis. Mat. Nat. (1930) vol. 4 pp. 405-422.
2. Blum, J., Chernoff, H., Rosenblatt, M., Teicher, H.: Central Limit Theorems for
Interchangeable Processes, Canad. Math. (1958) vol. 10.
3. Taylor, R. L.: Laws of Large Numbers for Dependent Random Variables. Colloquia
mathematica societatics Janos Bolyai, 36, Limit Theorems in Probability and Statistics,
(1982) Veszprem (Hungary), pp. 1023-1042.
4. Peng, H., Dang, X., Wang, X.: The distribution of partially exchangeable random variables.
Statistics and Probability Letters. (2010) Vol. 27(1), pp. 932-938.
5. Levental, S.: A maximal inequality for real numbers with applications to exchangeable
random variables. Theory Probab Appl. (2001) Vol. 45, pp. 525-532.
6. Delhi Paiva, N.: Random sums of exchangeable variables and actuarial applications.
Mathematics and Economics. (2008) Vol. 42, pp. 147-153.
7. Blum, L. E., Katz, M.: Convergence rates of large numbers. Trars Amer Math Soc., (1965)
Vol. 120, pp. 108-123.
8. Wu, Q.: Probability limit theory of mixing sequences. Science Press, (2006) pp. 132-133.
Copyright © 2016 SERSC
31