Download On the Hàjek-Rènyi-Type Inequality for Conditionally Associated

DOI: http://dx.doi.org/10.5351/CKSS.2011.18.6.799
Communications of the Korean Statistical Society
2011, Vol. 18, No. 6, 799–808
On the Hàjek-Rènyi-Type Inequality for Conditionally
Associated Random Variables
Jeong-Yeol Choia , Hye-Young Seo a , Jong-Il Baek 1, a
a
School of Mathematics and Informational Statistics, Wonkwang University
Abstract
Let {Ω, F , P} be a probability space and {Xn |n ≥ 1} be a sequence of random variables defined on it. A
finite sequence of random variables {Xi |1 ≤ i ≤ n} is a conditional associated given F if for any coordinatewise nondecreasing functions f and g defined on Rn , CovF ( f (X1 , . . . , Xn ), g(X1 , . . . , Xn )) ≥ 0 a.s. whenever
the conditional covariance exists. We obtain the Hàjek-Rènyi-type inequality for conditional associated random
variables. In addition, we establish the strong law of large numbers, the three series theorem, integrability of
supremum, and a strong growth rate for F -associated random variables.
Keywords: Associated random variables, conditional covariance, conditional associated random
variables, Hàjek-Rènyi-type inequality.
1. Introduction
Let {Ω, F , P} be a probability space and all random variables in this paper are defined on it unless
specified otherwise. A finite sequence of random variables {Xi |1 ≤ i ≤ n}is said to be associated if for
any coordinate-wise nondecreasing functions f and g defined on Rn ,
Cov ( f (X1 , . . . , Xn ), g(X1 , . . . , Xn )) ≥ 0.
Assuming that the covariance exists. An infinite sequence of random variables {Xn |n ≥ 1} is said to
be associated if every finite subsequence is associated. Esary et al. (1967) introduced a concept of
association. Lately significant efforts have been dedicated to prove reliability theory: Limit theorems
and statistics applications for such random variables Birkel (1998) provided some inequalities that
Shao and Yu (1996) later generalized. Ioannides and Roussas (1999) established some exponential
type inequalities, Oliveira (2005) presented some extensions, and Yang and Chen (2007) made further
extensions. Newman and Wright (1981) obtained an invariance principle, and Lin (1997) improved
it. Wang and Zhang (2006) developed a nonclassical law of the iterated logarithm, among authors.
Let X and Y be random variables with EX 2 < ∞ and EY 2 < ∞. Let F be a sub-σ-algebra of
A. Prakasa Rao (2009) defined the notion of the conditional covariance of X and Y given F (F covariance) as
((
)(
))
CovF (X, Y) = E F X − E F X Y − E F Y ,
where E F Z denotes the conditional expectation of a random variable Z given F .
In contrast to the ordinary concept of variance, conditional variance of X given F is defined
as VarF X ≡ CovF (X, X). On the basis of the above definition of conditional covariance, Prakasa
1
Corresponding author: Professor, School of Mathematics and Informational Statistics and Institute of Basic Natural
Science, Wonkwang University, Ik-San, Chunbuk 570-749, South Korea. E-mail: [email protected]
Jeong-Yeol Choi, Hye-Young Seo, Jong-Il Baek
800
Rao proposed a new kind of dependence called conditional association, which is an extension to the
corresponding non-conditional case.
Definition 1. A finite sequence of random variables {Xi |1 ≤ i ≤ n} is said to be conditional associated
given F (F -associated) if for any two coordinate-wise nondecreasing functions f and g defined on Rn ,
CovF ( f (X1 , X2 , . . . , Xn ), g(X1 , X2 , . . . , Xn )) ≥ 0, a.s.
whenever the conditional covariance exists. An infinite sequence of random variables {Xn |n ≥ 1} is
said to be F -associated if every finite subsequence is F -associated. Yuan and Yang (2011) presented
the relation between (positive) association and conditional association where the association does
not imply the conditional association, and vice versa.
Hàjek-Rènyi (1955) proved the following important inequality. If {Xn , n ≥ 1} is sequence of independent random variables with mean zero and finite second moments, and {bn , n ≥ 1} is a sequence of
positive nondecreasing real numbers, then for any ε > 0,
∑k
 n



m
 ∑ EX 2j

j=1 X j 

1 ∑
−2
2

 .


> ε ≤ ε 
P  max +
EX
j

2
2
m≤k≤n bk b
b
m
j
j=m+1
j=1
Many authors have studied the above inequality (Gan, 1997; Liu et al., 1999; Cai, 2000; Prakasa Rao,
2002; Hu et al., (2005); Qiu and Gan, 2005; Rao, 2002; Sung, 2008; etc). Recently, Yuan and Yang
extended the Hàjek-Rènyi inequality to conditional associated random variables.
This paper develops a Hàjek-Rènyi-type inequality for F -associated random variables and uses
a proof method different from Yuan and Yang. Using this result, we obtain the strong law of large
numbers, three series theorem, and integrability of supremum for F -associated random variables.
Finally, throughout this paper, c will represent positive constants where their value may change from
one place to another.
2. Conditional Hàjek-Rènyi-Type Inequality
To prove the Hàjek-Rènyi-type inequality for F -associated random variables, we need the following
Lemma 1.
Lemma 1. Let {Xi |1 ≤ i ≤ n} be a sequence of F -associated random variables with E F Xk = 0
and E F Xk2 < ∞ for each k with 1 ≤ k ≤ n. Then for an arbitrary F -measurable random variables
ε > 0 a.s.,
(
)
P max |S k | ≥ ε | F ≤ cE F S n2 ,
1≤k≤n
where S n =
∑n
i=1
Xi .
Proof: By applying proof of Newman and Wright (1982), we obtain that for ε > 0 a.s. is an arbitrary
F -measurable random variable,
P (max(0, S 1 , S 2 , . . . , S n ) ≥ ε | F ) ≤ cE F (max(0, S 1 , . . . , S n ))2
≤ cE F S n2 .
On the Hàjek-Rènyi-Type Inequality for Conditionally Associated Random Variables
801
Note that −X1 , . . . , −Xn are also F -associated random variables, replacing random variables X1 , . . . , Xn
by −X1 , . . . , −Xn , we obtain that for ε > 0 a.s. is an arbitrary F -measurable random variable,
P (max(0, −S 1 , −S 2 , . . . , −S n ) ≥ ε | F ) ≤ cE F S n2 .
Hence,
(
)
(
(
ε )
ε )
P max |S k | ≥ ε | F ≤ P max(0, S 1 , . . . , S n ) ≥ F + P max(0, −S 1 , . . . , −S n ) ≥ F
1≤k≤n
2
2
F 2
≤ cE S n .
The proof is complete
First, we will give a Hàjek-Rènyi-type inequality for F -associated random variables.
Theorem 1. Let {Xn |n ≥ 1} be a sequence of F -associated random variables with E F Xi = 0 and
E F Xi2 < ∞ a.s. for each i ≥ 1 and let {bn |n ≥ 1} be a sequence of positive nondecreasing real numbers.
Then for an arbitrary F -measurable random variables ε > 0 a.s. and positive integer m ≤ n,


k

)

1 ∑(
F

P  max
Xi − E Xi ≥ ε F 
m≤k≤n bk
i=1
 m

m
n
n
∑
∑
∑
∑ VarF Xi
VarF X j
CovF (X j , Xk ) 
CovF (Xi , Xn )

 a.s.
≤ c 
+
+
+
b j bk
b2m
b2m
b2j
i=1
j=m+1
1≤i,k≤m
m+1≤ j,k≤n
Proof: Noting that {Xn |n ≥ 1} is an F -associated random variables implies {(Xn − E F Xn )|n ≥ 1} is
an also F -associated random variables and that ε > 0 a.s. is an F -measurable random variable, we
obtain




k (

)
1 ∑


F


P  max 
Xi − E Xi  ≥ ε F 
m≤k≤n bk
i=1




m (
k (
m (


) ∑
) ∑
)
1 ∑

F
F
F



Xi − E Xi +
= P  max 
Xi − E Xi −
Xi − E Xi  ≥ ε F 
m≤k≤n bk
i=1
i=1
i=1








m
k
m (

) ε 
) ∑
) ε 
1 ∑(
1 ∑(




F
F
F
≤ P  max 
Xi −E Xi  ≥ F  +P  max 
Xi −E Xi −
Xi −E Xi  ≥ F
m≤k≤n bk
m≤k≤n bk
2
2
i=1
i=1
i=1
=: I1 + I2 .
As to I1 , using Lemma 1, we obtain


I1 = P  max



m (
) ε 
1 ∑

F


Xi − E Xi  ≥ F 
m≤k≤n bk
2
i=1
 m

) ε 
 1 ∑ (
F
≤ P  Xi − E Xi ≥ F 
bm i=1
2


k
 1
) ε 
∑ (
F
≤ P 
Xi − E Xi ≥ F 
max bm 1≤k≤m i=1
2
Jeong-Yeol Choi, Hye-Young Seo, Jong-Il Baek
802
c
≤ 2 EF
bm
 m
2
∑

F
Xi − E Xi 

i=1
 m
∑


c

F 
Xi 
= 2 Var 
bm
i=1


m
m
∑
 1 ∑

1
= c  2
VarF Xi + 2
CovF (Xi , Xk ) .
bm i=1
bm 1≤i,k
Next, noting that
∑ (
∑ (
)
)
(
)
k Xm+ j − E F Xm+ j k Xi − E F Xi − ∑m Xi − E F Xi j=1
i=1
i=1
max
= max
,
1≤k≤n−m
m≤k≤n
bk
bm+k
and using Lemma 1, we have
∑ (
)
(
)


k Xi − E F Xi − ∑m Xi − E F Xi 

i=1
i=1
ε 

I2 = P  max
≥ F 
m≤k≤n

bk
2
)


∑k (
F

j=1 Xm+ j − E Xm+ j ε 

= P  max
≥ F 
1≤k≤n−m

bm+k
2
≤
=
=
cE F
cE F
(∑
n−m
j=1
(
Xm+ j − E F Xm+ j
))2
b2m+ j
(
))2
n
F
j=m+1 X j − E X j
(∑
c VarF
(∑
n
b2j
j=m+1
2
bj
Xj
)
(
)
 n
n
∑
 ∑ VarF X j
CovF X j , Xn 
 .
= c 
+

b j bk
b2j
j=m+1
m+1≤ j,k
The proof is complete.
As corollary to Theorem 1, we obtain the following results.
Corollary 1. In Theorem 1, if bk = k, k = 1, 2, . . . , m, then we obtain the following inequality.
∑ (
)



ki=1 Xm − E F Xi 
≥ ε F 
P  max 
m≤k≤n k
(
)
(
)
 m
n CovF X , X
m
∑ CovF X j , Xk 
∑
∑
∑ VarF Xi
j
k
VarF X j

 a.s.
≤ c 
+
+
+

2
2
2
jk
m
m
j
i=1
j=m+1
m+1≤ j,k
1≤ j,k
On the Hàjek-Rènyi-Type Inequality for Conditionally Associated Random Variables
803
3. Almost Sure Convergence for F -Associated Random Variables
In this section, we will give the strong law of large numbers, the three series theorem, integrability of
supremum, and a strong growth rate for F -associated random variables by using the results which we
have obtained in Section 2.
Theorem 2. Let {Xn |n ≥ 1} be a sequence of F -associated random variables such that
∞
∑
VarF X j +
j=1
∞
∑
CovF (X j , Xk ) < ∞ a.s.
1≤ j,k
Then, conditionally on F ,
n (
∑
)
X j − E F X j → 0 a.s. as n → ∞.
j=1
Proof: Without loss of generality, we may assume that E F X j = 0 for all j ≥ 1, and let ε > 0 a.s. be
an F -measurable random variable. Then by Lemma 1 and Theorem 1,


k (
m (
∑


) ∑
)

P  max X j − EF X j −
X j − E F X j ≥ ε F 
k,m≥n j=1
j=1


k (
m (
∑

) ∑
) ε 
≤ P max X j − EF X j −
X j − E F X j ≥ F 
2
k≥n j=1
j=1


m
n (

∑(
) ∑
) ε 

F
F
+ P max Xj − E Xj −
X j − E X j ≥ F 
2
m≥n j=1
j=1


k (
n (
∑
) ∑
) ε 

F
F
≤ c lim P  max Xj − E Xj −
X j − E X j ≥ F 
2
N→∞
n≤k≤N j=1
j=1
2 

k

n (

∑
∑
(
)
)


≤ c lim E F  max X j − EF X j −
X j − E F X j 

n≤k≤N N→∞
j=1
j=1

∞
∞
∑
∑
(
)
≤ c  VarF X j +
CovF X j , Xk  < ∞ a.s.
j=n
n≤ j,k
Hence, the sequence of F -associated random variables {
∑
implies that nj=1 (X j − E F X j ) → 0 a.s. as n → ∞.
The proof is complete.
∑n
j=1 (X j
− E F X j )|n ≥ 1} is a Cauchy which
To prove Theorem 3, we need the following conditional version of Borel-cantelli lemma that is
proved by Majerak et al. (2005).
Lemma 2. Majerak et al. (2005) Let {Ω, F , P} be a probability space and let F be a sub-σ-algebra
of A. Then the following results hold.
Jeong-Yeol Choi, Hye-Young Seo, Jong-Il Baek
804
∑∞
∑
P(An ) < ∞. Then ∞
n=1 P(An |F ) < ∞ a.s.
∑∞
(ii) Let {An |n ≥ 1} be a sequence of events and let A = {ω| n=1 P(An |F ) < ∞} with P(A) < 1. Then,
only finitely many events from the sequence {An ∩ A, n ≥ 1} hold with probability one, namely
∩ ∪∞
∩
P( ∞
A)) = 0.
n=1
k=n (Ak
(i) Let {An |n ≥ 1} be a sequence of events such that
n=1
Theorem 3. (The three series theorem). Let {Xn |n ≥ 1} be a sequence of F -associated random
variables such that
∑
∑
F
c
c
(1) ∞j=1 VarF X cj + ∞
1≤ j,k Cov (X j , Xk ) < ∞ a.s.
(2)
∑∞
(3)
∑∞
Then
j=1
E F X cj < ∞ a.s.
j=1
P(|X j | ≥ c|F ) < ∞ for some constant c > 0.
∑n
j=1
X j → 0 a.s. as n → ∞.
Proof: Let X cj = X j I(|X j | ≤ c). Then, by Theorem 2 and (1), it follows that for condition on F ,
n (
∑
)
X cj − E F X cj → 0 a.s. as n → ∞,
j=1
∑n
and from (2), j=1 X cj → 0 a.s. as.
Which together with (3), implies that n → ∞.
∞
∞
∑
) ∑
(
)
(
P |X j | ≥ c | F < ∞.
P X j , X cj | F =
j=1
j=1
Hence, by Lemma 2, we obtain that
n
∑
X j → 0 a.s. as n → ∞.
j=1
The proof is complete.
Theorem 4. Let {Xn |n ≥ 1} be a sequence of F -associated random variables such that
∞
∑
VarF X j
j=1
b2j
+
∞
∑
CovF (X j , Xk )
< ∞ a.s.
b j bk
1≤ j,k
and let {bn |n ≥ 1} be a sequence of positive nondecreasing real numbers. Then,
(a) For 0 < r < 2 and conditionally on F ,
)  r
 ∑n (
 i=1 Xi − E F Xi 


E F sup 
 < ∞ a.s.
bn
n≥1 
(b) If 0 < bn → ∞, then
∑n
i=1 (Xi
− E F Xi )/bn → 0 a.s. as n → ∞.
On the Hàjek-Rènyi-Type Inequality for Conditionally Associated Random Variables
805
Proof: Proof of (a). Noting that
)  r
 ∑n (
 i=1 Xi − E F Xi 


E F sup 
 < ∞ a.s.

b
n
n≥1
∑ (
)

∫ ∞ 
n X j − E F X j 
j=1
1
≥ tr
⇔
P sup

b
n
n≥1
1


F  dt < ∞ a.s.

by Lemma 1 and Theorem 1, we obtain that
∑ (
)


∫ ∞ 
n X j − E F X j 

j=1
1 
P sup
≥ t r F  dt


bn
n≥1
1
∞

∫ ∞
∑ CovF (X j , Xk ) 
∑ VarF X j
− 2r 
 dt

≤c
t 
+

2
b
b
b
j
k
1
j
j=1
1< j,k
(
)
∞

∞ CovF X , X  ∫ ∞
∑
∑ VarF X j
j
k 
2


t− r < ∞ a.s.
= c 
+

2
b j bk
bj
1
j=1
1≤ j,k
For ε > 0 a.s. is an F -measurable random variable, by Lemma 1 and Theorem 1,
∑k

j=1 X j − E F X j ≥ ε F 
m≤k≤n
bk

(
))
(
(
))2 
(
F
 E F ∑m X j − E F X j 2 E F ∑n

j=1
j=m+1 X j − E X j


≤ c 
+


b2m
b2j
(
)
(
)
 ∑m
n
F
∑ CovF X j , Xk 
∑ CovF X j , Xk
∑
 j=1 VarF X j
Var
X
j
 .
= c 
+
+
+

2
2
b
b
b2m
b
b
j
k
m
j
j=m+1
m+1≤ j,k
1≤ j,k
Proof of (b).


P  max
But for ε > 0 a.s. is an F -measurable random variable,
∑ (
)


k X j − E F X j ∪

j=1


max
P 
≥ ε F 
 m≤k≤n

b
k
n=m
∑ (
)


k X j − E F X j 

j=1


= lim P  max
≥ ε F 
n→∞ m≤k≤n

bk

(
(
))
(
(
))2 
F
 E F ∑m X j − E F X j 2 E F ∑n

j=1
j=m+1 X j − E X j


≤ c 
+

2
2

bm
bj
(
)
 ∑m

n
∑ CovF (Xi , Xk ) 
∑ CovF X j , Xk
∑
 j=1 VarF X j
VarF X j

 .
= c 
+
+
+

2
2
b
b
b2m
b
b
j
k
m
j
j=m+1
m+1≤ j,k
1≤ j,k
(3.1)
Jeong-Yeol Choi, Hye-Young Seo, Jong-Il Baek
806
By the kronecker Lemma and
∑∞
j=1
VarF X j b2j +
∑
1≤ j,k
CovF (X j , Xk )/(b j bk ) < ∞ a.s., we obtain
that
m
∑
VarF X j
b2m
j=1
+
m
∑
CovF (X j , Xk )
→0
b2m
1≤ j,k
as m → ∞.
(3.2)
Hence, by (3.1) and (3.2), we obtain that
)


∑k (


X j − EF X j
j=1


lim P sup
≥ ε F  = 0,
n→∞  k≥n
bk
)
∑n (
F
j=1 X j − E X j
i.e,
→ 0 a.s. as n → ∞.
bn
The proof is complete.
Remark 1. Taking bn = 1, we can obtain the result
n (
∑
)
X j − E F X j → 0 a.s. as n → ∞.
j=1
Finally, we obtain the almost sure convergence of weighted sums of sequence of F -associated
random variables.
Theorem 5. Assume that {Xn |n ≥ 1} be a sequence of F -associated random variables satisfying
∞
∑
∑
VarF X j +
j=1
CovF (X j , Xk ) < ∞ a.s.
1≤ j,k
∑
and let {ani |1 ≤ i ≤ n, n ≥ 1} be a sequence of real numbers such that ani = 0, i > n, supn≥1 ni=1 |ani | <
∞ and let {bn |n ≥ 1} be a sequence of positive nondecreasing real numbers such that 0 < bn → ∞.
Then
∑n
i=1 ani Xi
→ 0 a.s. as n → ∞.
bn
Proof: Note that {ani Xi |1 ≤ i ≤ n, n ≥ 1} is a sequence of F -associated random variables and let
Sk =
k
∑
Xi
,
b
i=1 k
cni =
bi
(ani − ani+1 ),
bn
for 1 ≤ i ≤ n − 1,
and cnn = ann . Then
n
∑
ani Xi
i=1
bn
=
n
∑
cni S i ,
i=1
and lim |cni | = 0,
n→∞
n
∑
|cni | ≤ 2 sup
i=1
for every fixed 1.
n
∑
|ani |
(3.3)
n≥1 i=1
(3.4)
On the Hàjek-Rènyi-Type Inequality for Conditionally Associated Random Variables
∞,
807
From (3.3) and (3.4), we obtain for every sequence real numbers dn with dn → 0 as n →
∑n
j=1 cn j d j → 0 as n → ∞.
Hence, by Theorem 4(b), (3.3) and (3.4), we obtain the result of Theorem 5.
The proof is complete.
Corollary 2. Let {Xn |n ≥ 1} be a sequence of F -associated random variables E F Xk = 0 and
∑
∑∞
F
F
) < ∞ a.s., and {ani |1 ≤ i ≤ n, n ≥ 1} be a sequence of real
1≤ j,k Cov (X j , Xk∑
j=1 Var X j +
numbers with ani = 0, i > n, supn≥1 ni=1 |ani | < ∞. Then for 0 < t < 1,
n
∑
ani Xi
1
i=1
nt
→ 0 a.s. as n → ∞.
Proof: Note that {ani Xi |1 ≤ i ≤ n, n ≥ 1} is a sequence of F -associated random variables. Then
taking bn = n1/t , from Theorem 5, we obtain the result of Corollary 2.
Acknowledgements
The authors would like to thank the referees for their valuable comments and suggestions. The first
author was supported by a Wonkwang University Research Grant in 2011.
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Received July 2011; Accepted October 2011