Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 823767, 13 pages
doi:10.1155/2010/823767
Research Article
On Inverse Moments for a Class of Nonnegative
Random Variables
Soo Hak Sung
Department of Applied Mathematics, Pai Chai University, Taejon 302-735, Republic of Korea
Correspondence should be addressed to Soo Hak Sung, sungsh@pcu.ac.kr
Received 1 April 2010; Accepted 20 May 2010
Academic Editor: Andrei Volodin
Copyright q 2010 Soo Hak Sung. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
Using exponential inequalities, Wu et al. 2009 and Wang et al. 2010 obtained asymptotic
approximations of inverse moments for nonnegative independent random variables and
nonnegative negatively orthant dependent random variables, respectively. In this paper, we
improve and extend their results to nonnegative random variables satisfying a Rosenthal-type
inequality.
1. Introduction
Let {Zn , n ≥ 1} be a sequence of nonnegative random variables with finite second moments.
Let us denote
Xn n
i1
σn
Zi
,
σn2 n
VarZi .
1.1
i1
We will establish that, under suitable conditions, the inverse moment can be approximated
by the inverse of the moment. More precisely, we will prove that
Ea Xn −α ∼ a EXn −α ,
1.2
where a > 0, α > 0, and cn ∼ dn means that cn dn−1 → 1 as n → ∞. The left-hand side of 1.2
is the inverse moment and the right-hand side is the inverse of the moment. Generally, it is
not easy to compute the inverse moment, but it is much easier to compute the inverse of the
moment.
2
Journal of Inequalities and Applications
The inverse moments can be applied in many practical applications. For example, they
appear in Stein estimation and Bayesian poststratification see Wooff 1 and Pittenger 2,
evaluating risks of estimators and powers of test statistics see Marciniak and Wesołowski
3 and Fujioka 4, expected relaxation times of complex systems see Jurlewicz and Weron
5, and insurance and financial mathematics see Ramsay 6.
For nonnegative asymptotically normal random variables Xn , 1.2 was established
in Theorem 2.1 of Garcia and Palacios 7. Unfortunately, that theorem is not true under
the suggested assumptions, as pointed out by Kaluszka and Okolewski 8. Kaluszka and
Okolewski 8 also proved 1.2 for 0 < α < 3 0 < α < 4 in the i.i.d. case when {Zn , n ≥ 1}
is a sequence of nonnegative independent random variables satisfying EXn → ∞ and L3
Lyapunov’s condition of order 3, that is, ni1 E|Zi − EZi |c /σnc → 0 with c 3. Hu et al. 9
generalized the result of Kaluszka and Okolewski 8 by considering Lc for some 2 < c ≤ 3
instead of L3 .
Recently, Wu et al. 10 obtained the following result by using the truncation method
and Bernstein’s inequality.
Theorem 1.1. Let {Zn , n ≥ 1} be a sequence of nonnegative independent random variables such that
EZn2 < ∞ and EXn → ∞, where Xn is defined by 1.1. Furthermore, assume that
EZi
O1,
1≤i≤n σn
1.3
max
σn−2
n
EZi2 I Zi > ησn −→ 0 for some η > 0.
1.4
i1
Then 1.2 holds for all real numbers a > 0 and α > 0.
For a sequence {Zn , n ≥ 1} of nonnegative independent random variables with only
rth moments for some 1 ≤ r < 2, Wu et al. 10 also obtained the following asymptotic
approximation of the inverse moment:
−α −α
∼ a EXn
E a Xn
1.5
for all real numbers a > 0 and α > 0. Here Xn is defined as
Xn
n
i1
σn
Zi
,
σn2 n
VarZi IZi ≤ Mn ,
1.6
i1
where {Mn , n ≥ 1} is a sequence of positive constants satisfying
Mn → ∞,
Mn O n2−δ/4−r
for some 0 < δ <
r
.
2
1.7
Specifically, Wu et al. 10 proved the following result.
Theorem 1.2. Let {Zn , n ≥ 1} be a sequence of nonnegative independent random variables. Suppose
that, for some 1 ≤ r < 2,
i {Zn } is uniformly integrable,
Journal of Inequalities and Applications
3
ii supn≥1 EZnr < ∞,
iii n−1 ni1 EZi ≥ C for some positive constant C > 0,
iv n−1/2 σn ≥ D for some positive constant D > 0,
where σn is the same as in 1.6 for some positive constants {Mn } satisfying 1.7. Then 1.5 holds
for all real numbers a > 0 and α > 0.
Wang et al. 11 obtained some exponential inequalities for negatively orthant
dependent NOD random variables. By using the exponential inequalities, they extended
Theorem 1.1 for independent random variables to NOD random variables without condition
1.3.
The purpose of this work is to obtain asymptotic approximations of inverse moments
for nonnegative random variables satisfying a Rosenthal-type inequality. For a sequence
{Zn , n ≥ 1} of independent random variables with EZn 0 and E|Zn |q < ∞ for some q > 2,
Rosenthal 12 proved that there exists a positive constant Cq depending only on q such that
⎧
q
q/2 ⎫
n
n
n
⎨
⎬
q
2
.
E Zi ≤ Cq
E|Zi | E|Zi |
i1 ⎩ i1
⎭
i1
1.8
Note that the Rosenthal inequality holds for NOD random variables see Asadian et al. 13.
In this paper, we improve and extend Theorem 1.2 for independent random variables
to random variables satisfying a Rosenthal type inequality. We also extend Wang et al. 11
result for NOD random variables to the more general case.
Throughout this paper, the symbol C denotes a positive constant which is not
necessarily the same one in each appearance, and IA denotes the indicator function of the
event A.
2. Main Results
Throughout this section, we assume that {Zn , n ≥ 1} is a sequence of nonnegative random
variables satisfying a Rosenthal type inequality see 2.1.
The following theorem gives sufficient conditions under which the inverse moment is
asymptotically approximated by the inverse of the moment.
Theorem 2.1. Let {Zn , n ≥ 1} be a sequence of nonnegative random variables. Let μn EXn and
μ
n σn−1 ni1 EZi IZi ≤ Mn , where Xn and σn are defined by 1.6, and {Mn , n ≥ 1} is a sequence
of positive real numbers. Suppose that the following conditions hold:
i for any q > 2, there exists a positive constant Cq depending only on q such that
⎧
q
q/2 ⎫
n
n
n
⎨
⎬
q
E
,
EZni
− EZni
Var Zni
Zni − EZni
≤ Cq
i1
⎩ i1
⎭
i1
where Zni
Zi IZi ≤ Mn Mn IZi > Mn ,
ii μn → ∞ as n → ∞,
2.1
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Journal of Inequalities and Applications
iii μ
n /μn → 1 as n → ∞,
iv Mn /
σn μ
sn O1 for some 0 < s < 1.
Then 1.5 holds for all real numbers a > 0 and α > 0.
Proof. Let us decompose Xn as
Xn Un Vn ,
2.2
and Vn σn−1 ni1 Zi − Mn IZi > Mn . Denote un EUn . Since
where Un σn−1 ni1 Zni
≤ Zi , we have that μ
n ≤ un ≤ μn . It follows by ii and iii that
Zi IZi ≤ Mn ≤ Zni
un
−→ 1,
μn
un
−→ 1,
μ
n
un −→ ∞.
Now, applying Jensen’s inequality to the convex function fx
−α
Ea Xn −α ≥ a EX n . Therefore
2.3
a x−α yields
α −α
lim inf a EX n E a Xn
≥ 1.
2.4
n→∞
Hence it is enough to show that
α −α
lim sup a EXn E a Xn
≤ 1.
2.5
n→∞
Since 0 < s < 1, we can take t such that 0 < s < t < 1 and 2t > s1. Namely, s < s1/2 < t < 1.
Let us write
−α
E a Xn
Q1 Q2 ,
2.6
where Q1 Ea Xn −α IUn ≤ un − un t and Q2 Ea Xn −α IUn > un − un t . Since
Xn ≥ Un , we get that
t t −α
−α ≤ a un − un
I Xn > un − un
,
Q2 ≤ E a Xn
2.7
which implies by ii and 2.3 that
t −α
α
α a un − un
lim sup a EXn Q2 ≤ lim sup a EXn
n→∞
n→∞
lim sup
n→∞
a/μn 1
a/μn un /μn − un t /μn
α
2.8
1.
Journal of Inequalities and Applications
5
It remains to show that
α
lim a EXn Q1 0.
2.9
n→∞
Observe by Markov’s inequality and i that, for any q > 2,
t Q1 ≤ a−α P Un ≤ un − un
t ≤ a−α P Un − un ≥ un
q
n
−tq −α −q
≤ a σn un
E
Z − EZni i1 ni
⎧
q/2 ⎫
n
n
⎨
⎬
−q
−tq
q
.
≤ Cq a−α σn un
EZni
− EZni
Var Zni
⎩ i1
⎭
i1
2.10
By the definition of Zni
, we have that
Z − EZ ≤ max Z , EZ ≤ Mn ,
ni
ni
ni
ni
Var Zni
VarZi IZi ≤ Mn VarMn IZi > Mn 2CovZi IZi ≤ Mn , Mn IZi > Mn VarZi IZi ≤ Mn VarMn IZi > Mn 2.11
− 2EZi IZi ≤ Mn Mn P Zi > Mn ≤ VarZi IZi ≤ Mn Mn2 P Zi > Mn ≤ VarZi IZi ≤ Mn Mn EZi IZi > Mn .
Substituting 2.11 into 2.10, we have that
Q1
≤
−tq
−q Cq a−α σn un
n
q−2 VarZi IZi ≤ Mn Mn EZi IZi > Mn Mn
i1
n
VarZi IZi ≤ Mn Mn EZi IZi > Mn q/2 ⎫
⎬
⎭
i1
≤
−tq
−q Cq a−α σn un
q−2
n
q−1 Mn σn2 Mn
EZi IZi > Mn i1
2q/2−1
n
q/2
VarZi IZi ≤ Mn i1
: I1 I2 I3 I4 .
q/2 ⎫
⎬
Mn EZi IZi > Mn ⎭
i1
2q/2−1
n
2.12
6
Journal of Inequalities and Applications
For I1 , we have by iv that
−tq
I1 Cq a−α un
q−2
Mn
σn μ
sn
sq−2
μ
n
−tq sq−2
≤ Ca−α un
μ
n
.
2.13
For I2 , we first note that
μ
n
σn−1
1
−
μn
n
i1
EZi IZi > Mn ,
μn
2.14
which entails by iii that
μn
−1
σn−1
n
EZi IZi > Mn −→ 0.
2.15
i1
It follows by iv that
n
−1
−q1 −tq
q−1
Mn μn μn σn−1 EZi IZi
un
i1
I2 Cq a−α σn
> Mn −q1 −tq
q−1
Mn μn
un
≤ Ca−α σn
−tq
Ca−α un
Mn
σn μ
sn
q−1
2.16
sq−1 μ
n
μn
−tq sq−1 ≤ Ca−α un
μ
n
μn .
For I3 , we have by the definition of σn that
−tq
I3 Cq 2q/2−1 a−α un
.
2.17
For I4 , we have by 2.15 and iv that
I4 q/2
−q/2 −tq
q/2 Cq 2q/2−1 a−α σn
Mn μn
un
−1
μn σn−1
n
q/2
EZi IZi > Mn i1
q/2
−q/2 −tq
q/2 Mn μn
un
≤ Ca−α σn
−tq q/2
Ca−α un
μn
Mn
σn μ
sn
−tq q/2 sq/2
≤ Ca−α un
μ
n .
μn
q/2
2.18
sq/2
μ
n
Journal of Inequalities and Applications
7
Substituting 2.13 and 2.16–2.18 into 2.12, we get that
α
lim sup a EXn Q1
n→∞
α −α −tq sq−2
−tq sq−1 Ca un
≤ lim sup a EXn
μ
n
Ca−α un
μ
n
μn
n→∞
−tq
−tq q/2 sq/2 Cq 2q/2−1 a−α un
Ca−α un
μ
n
μn
lim sup
n→∞
a μn
μ
n
2.19
α −tq sq−2α
−tq sq−1α Ca−α un
μ
n
Ca−α un
μ
n
μn
−tq α
−tq q/2 sq/2α .
Cq 2q/2−1 a−α un
μ
n Ca−α un
μ
n
μn
Since 0 < s < s 1/2 < t < 1, we can take q > 2 large enough such that tq > max{sq − 1 α 1, sq/2 q/2 α}. Then we have by 2.3 that
un
un
un
un
−tq
−tq
−tq
sq−2α
μ
n
−sq−2−α sq−2α −tqsq−2α
un
μ
n
−→ 0,
un
sq−1α μn
μ
n
−sq−1−α sq−1α −1 −tqsq−1α1
un
μ
n
μ n un
−→ 0,
un
−α α −tqα
μ
αn un μ
n un
−→ 0,
−tq μn
q/2
sq/2α
μ
n
2.20
−sq/2−α sq/2α q/2 −q/2 −tqsq/2q/2α
un
μ
n
−→ 0.
μn
un
un
Hence all the terms in the second brace of 2.19 converge to 0 as n → ∞. Moreover, we have
by ii and iii that
a μn
μ
n
α
a/μn 1
μ
n /μn
α
−→ 1.
2.21
Therefore lim supn → ∞ a EXn α Q1 0 and so 2.9 is proved.
Remark 2.2. In 2.1, {Zni
, 1 ≤ i ≤ n} are monotone transformations of {Zi , 1 ≤ i ≤ n}. If
{Zn , n ≥ 1} is a sequence of independent random variables, then 2.1 is clearly satisfied from
the Rosenthal inequality 1.8. There are many sequences of dependent random variables
satisfying 2.1 for all q > 2. Examples include sequences of NOD random variables
see Asadian et al. 13, φ-mixing identically distributed random variables satisfying
∞ 1/2 n
2 < ∞ see Shao 14, ρ-mixing identically distributed random variables
n1 φ
2/q n
2 < ∞ see Shao 15, negatively associated random variables see
satisfying ∞
n1 ρ
Shao 16, and ρ∗ -mixing random variables see Utev and Peligrad 17.
We can extend Theorem 1.1 for independent random variables to the more general
random variables by using Theorem 2.1. To do this, the following lemma is needed.
8
Journal of Inequalities and Applications
Lemma 2.3. Let {Yn , n ≥ 1} be a sequence of nonnegative random variables with EYn → ∞. Let
{bn , n ≥ 1} be a sequence of positive real numbers satisfying bn → b, where b > 0. Assume that
Ea Yn −α ∼ a EYn −α
∀a > 0, α > 0.
2.22
Then Ebn Yn −α ∼ b EYn −α .
Proof. Take > 0 such that 0 < < b. Since bn → b, there exists a positive integer N such that
0 < b − < bn < b if n > N. We have by 2.22 that, for n > N,
Ebn Yn −α ≤ Eb − Yn −α ∼ b − EYn −α .
2.23
It follows that
lim supb EYn α Ebn Yn −α
n→∞
≤ lim supb EYn α Eb − Yn −α
n→∞
b EYn α
α
−α
α b − EYn Eb − Yn n → ∞ b − EYn α
b/EYn 1
lim sup
b − EYn α Eb − Yn −α 1.
b − /EYn 1
n→∞
lim sup
2.24
Similar to the above case, we get that, for n > N,
Ebn Yn −α ≥ Eb Yn −α ∼ b EYn −α ,
α
2.25
−α
lim infb EYn Ebn Yn n→∞
≥ lim infb EYn α Eb Yn −α
n→∞
lim inf
n→∞
b/EYn 1
b /EYn 1
α
2.26
b EYn α Eb Yn −α 1.
Hence the result is proved by 2.24 and 2.26.
By using Theorem 2.1, we can obtain the following theorem which improves and
extends Theorem 1.1 for independent random variables to the more general random variables
satisfying the Rosenthal-type inequality 2.1.
Theorem 2.4. Let {Zn , n ≥ 1} be a sequence of nonnegative random variables with EZn2 < ∞. Let
Xn and σn be defined by 1.1. Assume that the Rosenthal-type inequality 2.1 with Mn ησn holds
for all q > 2, where η > 0 is the same as in (ii). Furthermore, assume that
i EXn → ∞ as n → ∞,
ii σn−2 ni1 EZi2 IZi > ησn → 0 for some η > 0.
Then 1.2 holds for all real numbers a > 0 and α > 0.
Journal of Inequalities and Applications
n σn−1
Proof. Let μn EXn and μ
Note that
σn2 n
n
i1
9
EZi IZi ≤ Mn , where Xn and σn are defined by 1.6.
VarZi IZi ≤ Mn Zi IZi > Mn i1
n
VarZi IZi ≤ Mn VarZi IZi > Mn 2CovZi IZi ≤ Mn , Zi IZi > Mn i1
σn2 n
n
VarZi IZi > Mn − 2 EZi IZi ≤ Mn EZi IZi > Mn ,
i1
i1
2.27
which implies that
1
σn2
σn2
n
i1
VarZi IZi > Mn σn2
−2
n
i1
EZi IZi ≤ Mn EZi IZi > Mn σn2
.
2.28
But, we have by ii that
σn−2
n
VarZi IZi > Mn ≤ σn−2
i1
n
EZi2 IZi > Mn −→ 0,
i1
n
n
σn−2 EZi IZi ≤ Mn EZi IZi > Mn ≤ σn−2 Mn EZi IZi > Mn i1
i1
≤ σn−2
2.29
n
EZi2 IZi > Mn −→ 0.
i1
Substituting 2.29 into 2.28, we have that
σn−1 σn −→ 1 as n −→ ∞.
2.30
Now we will apply Theorem 2.1 to the random variable Xn . By 2.30 and i, we get that
μn σn−1
n
i1
EZi σn−1 σn EXn −→ ∞.
2.31
10
Journal of Inequalities and Applications
We also get that
−1
μ
n μn
n
σn−1
1−
Mn i1 EZi IZi ≤
σn−1 ni1 EZi
Mn n
σn−1
i1 EZi IZi ≤
σn−1 ni1 EZi
σn−1
n
i1
2.32
EZi IZi > Mn −→ 1,
EXn
since EXn → ∞ by i and σn−1 ni1 EZi IZi > Mn ≤ σn−1 Mn−1 ni1 EZi2 IZi > Mn n
n → ∞ and so we have by
η−1 σn−2 i1 EZi2 IZi > Mn → 0 by ii. From 2.31 and 2.32, μ
2.30 that, for any s > 0,
ησn
Mn
−→ 0.
s σn μ
n σn μ
sn
2.33
Hence all conditions of Theorem 2.1 are satisfied. By Theorem 2.1,
E a
σn−1
n
Zi
−α
∼
a
σn−1
i1
n
EZi
−α
∀a > 0, α > 0.
2.34
i1
Note that the norming constants in 2.34 are different from those in Xn .
To complete the proof, we will use Lemma 2.3. Since σn−1 σn → 1, we have by
Lemma 2.3 that
aσn σn−1
E
σn−1
n
−α
∼
Zi
a
σn−1
i1
n
EZi
−α
.
2.35
.
2.36
.
2.37
i1
Namely,
E a
σn−1
n
Zi
−α
∼
a
σn σn−1
σn−1
i1
n
EZi
−α
i1
By i and 2.30,
a
σn σn−1
σn−1
n
−α
EZi
∼
a
i1
σn−1
n
EZi
−α
i1
Combining 2.36 with 2.37 gives the desired result.
Remark 2.5. Wang et al. 11 extended Wu et al. 10 result see Theorem 1.1 to NOD
random variables without condition 1.3. As observed in Remark 2.2, 2.1 holds for not
Journal of Inequalities and Applications
11
only independent random variables but also NOD random variables. Hence Theorem 2.4
improves and extends the results of Wu et al. 10 and Wang et al. 11 to the more general
random variables.
Theorem 2.6. Let {Zn , n ≥ 1} be a sequence of nonnegative random variables. Let μn EXn and
μ
n σn−1 ni1 EZi IZi ≤ Mn , where Xn and σn are defined by 1.6, and {Mn , n ≥ 1} is a sequence
of positive real numbers satisfying
Mn −→ ∞,
Mn O nt for some 0 < t < 1.
2.38
Assume that the Rosenthal-type inequality 2.1 holds for all q > 2. Furthermore, assume that
i {Zn } is uniformly integrable,
ii n−1 ni1 EZi ≥ C for some positive constant C > 0,
iii n−1/2 σn ≥ D for some positive constant D > 0.
Then 1.5 holds for all real numbers a > 0 and α > 0.
Proof. We first note by i and ii that
Cn ≤
n
EZi ≤ n sup EZi On,
i1
i≥1
n
n
EZi IZi > Mn ≤ sup EZi IZi > Mn o1.
2.39
−1
i≥1
i1
We next estimate σn . By 2.39,
σn2 ≤
n
n
EZi2 IZi ≤ Mn ≤ Mn EZi Mn On.
i1
2.40
i1
Combining 2.40 with iii gives
D2 n ≤ σn2 ≤ C1 nMn
for some constant C1 > 0.
2.41
Now we will apply Theorem 2.1 to the random variable Xn . By ii, 2.41, and 2.38, we get
that
μn
n
i1
EZi
σn
≥
Cn
Cn
Cn1/2
≥
−→ ∞.
σn
C1 nMn
C1 O nt/2
2.42
12
Journal of Inequalities and Applications
We also get by ii and 2.39 that
μ
n σn−1
μn
n−1
1−
n
i1 EZi IZi ≤
σn−1 ni1 EZi
Mn i1 EZi IZi ≤
n−1 ni1 EZi
Mn n
n−1
n
i1 EZi IZi >
n−1 ni1 EZi
2.43
Mn −→ 1.
Since 0 < t < 1, we can take s such that max{2t − 1, 0} < s < 1. Then we have by ii, iii,
2.38, and 2.43 that
Mn
Mn
s −s s
s σn μ
n σn μn μn μ
n
≤
≤
Mn
−s s
−1 s
σn Cn
n
σn
μn μ
byii
2.44
O nt
−s s −→ 0,
Cs D1−s ns1−s/2 μn μ
n
since s 1 − s/2 > t and μn −1 μ
n → 1. Hence all conditions of Theorem 2.1 are satisfied.
The result follows from Theorem 2.1.
Remark 2.7. The conditions of Theorem 2.6 are much weaker than those of Theorem 1.2 in the
following three directions.
i If {Zn , n ≥ 1} is a sequence of independent random variables, then 2.1 is satisfied
from the Rosenthal inequality. Hence 2.1 is weaker than independence condition.
ii If {Mn , n ≥ 1} satisfies 1.7, then it also satisfies 2.38 by the fact that 2 − δ/4 −
r < 2 − δ/2 < 1. Hence 2.38 is weaker than 1.7.
iii The condition supn≥1 EZnr < ∞ in Theorem 1.2 is not needed in Theorem 2.6.
Therefore Theorem 2.6 improves and extends Wu et al. 10 result see Theorem 1.2
to the more general random variables.
Acknowledgments
The author is grateful to the editor Andrei I. Volodin and the referees for the helpful
comments. This research was supported by Basic Science Research Program through the
National Research Foundation of Korea NRF funded by the Ministry of Education, Science
and Technology 2010-0013131.
Journal of Inequalities and Applications
13
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