Volume 2010, Article ID 823767,13pages 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,[email protected]
Received 1 April 2010; Accepted 20 May 2010
Academic Editor: Andrei Volodin
Copyrightq2010 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
i1Zi
σn , σ
2
n n
i1
VarZi. 1.1
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
EaXn−α∼aEXn−α, 1.2
wherea >0, α >0,andcn ∼dnmeans thatcndn−1 → 1 asn → ∞.The left-hand side of1.2
The inverse moments can be applied in many practical applications. For example, they appear in Stein estimation and Bayesian poststratificationsee Wooff 1and Pittenger2, evaluating risks of estimators and powers of test statisticssee Marciniak and Wesołowski
3and Fujioka4, expected relaxation times of complex systemssee Jurlewicz and Weron
5, and insurance and financial mathematicssee Ramsay6.
For nonnegative asymptotically normal random variablesXn, 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 Okolewski8. Kaluszka and Okolewski8also proved1.2for 0< α <30 < α < 4 in the i.i.d. casewhen{Zn, n ≥1}
is a sequence of nonnegative independent random variables satisfyingEXn → ∞andL3 Lyapunov’s condition of order 3, that is,ni1E|Zi−EZi|c/σnc → 0 withc3.Hu et al.9
generalized the result of Kaluszka and Okolewski8by consideringLcfor some 2 < c ≤3
instead ofL3.
Recently, Wu et al.10obtained 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
EZ2
n<∞andEXn → ∞,whereXnis defined by1.1. Furthermore, assume that
max
1≤i≤n
EZi
σn O1,
1.3
σ−2
n n
i1
EZ2
iI
Zi> ησn
−→0 for someη >0. 1.4
Then1.2holds for all real numbersa >0andα >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:
EaXn
−α∼
aEXn
−α 1.5
for all real numbersa >0 andα >0. HereXn is defined as
Xn
n i1Zi
σn , σ
2
n n
i1
VarZiIZi≤Mn, 1.6
where{Mn, n≥1}is a sequence of positive constants satisfying
Mn → ∞, MnO
n2−δ/4−r for some 0< δ < r
2. 1.7
Specifically, Wu et al.10proved the following result.
Theorem 1.2. Let{Zn, n≥1}be a sequence of nonnegative independent random variables. Suppose
that, for some1≤r <2,
iisupn≥1EZr n<∞,
iiin−1n
i1EZi≥Cfor some positive constantC >0,
ivn−1/2σ
n≥Dfor some positive constantD >0,
whereσnis the same as in1.6for some positive constants{Mn}satisfying1.7. Then1.5holds
for all real numbersa >0andα >0.
Wang et al. 11 obtained some exponential inequalities for negatively orthant dependentNODrandom variables. By using the exponential inequalities, they extended Theorem 1.1for 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 withEZn 0 andE|Zn|q <∞for someq >2,
Rosenthal12proved that there exists a positive constantCqdepending only onqsuch that
E
n
i1
Zi
q ≤Cq
⎧ ⎨ ⎩
n
i1
E|Zi|q
n
i1
E|Zi|2
q/2⎫⎬
⎭. 1.8
Note that the Rosenthal inequality holds for NOD random variablessee Asadian et al.13. In this paper, we improve and extendTheorem 1.2for 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
eventA.
2. Main Results
Throughout this section, we assume that{Zn, n ≥ 1}is a sequence of nonnegative random
variables satisfying a Rosenthal type inequalitysee2.1.
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. Letμn EXnand
μnσn−1
n
i1EZiIZi≤Mn,whereXn andσnare defined by1.6, and{Mn, n≥1}is a sequence
of positive real numbers. Suppose that the following conditions hold:
ifor anyq >2,there exists a positive constantCqdepending only onqsuch that
E
n
i1
Zni −EZni
q
≤Cq
⎧ ⎨ ⎩
n
i1
E Zni−EZni q n
i1
VarZni q/2⎫⎬
⎭, 2.1
whereZni ZiIZi≤Mn MnIZi> Mn,
iiiμn/μn → 1asn → ∞,
ivMn/σnμsn O1for some0< s <1.
Then1.5holds for all real numbersa >0andα >0.
Proof. Let us decomposeXnas
XnUnVn, 2.2
whereUn σn−1
n
i1Zniand Vn σn−1
n
i1Zi−MnIZi > Mn.Denoteun EUn.Since
ZiIZi≤Mn≤Zni ≤Zi,we have thatμn≤un ≤μn.It follows byiiandiiithat
un
μn
−→1, u
n
μn −→
1, un−→ ∞. 2.3
Now, applying Jensen’s inequality to the convex function fx ax−α yields
EaXn−α≥aEXn− α
.Therefore
lim inf
n→ ∞
aEXn
α
EaXn
−α≥
1. 2.4
Hence it is enough to show that
lim sup
n→ ∞
aEXn
α
EaXn
−α≤
1. 2.5
Since 0< s <1,we can taketsuch that 0< s < t <1 and 2t > s1.Namely,s <s1/2< t <1.
Let us write
EaXn
−α
Q1Q2, 2.6
whereQ1 EaXn−αIUn ≤ un −untandQ2 EaXn−αIUn > un−unt.Since
Xn ≥Un,we get that
Q2≤EaXn
−α
IXn> un−
un
t≤
aun−
un
t−α
, 2.7
which implies byiiand2.3that
lim sup
n→ ∞
aEXn
α
Q2≤lim sup
n→ ∞
aEXn
α
aun−
un
t−α
lim sup
n→ ∞
a/μn1
a/μnun/μn−unt/μn
α
1.
It remains to show that
lim
n→ ∞
aEXn
α
Q10. 2.9
Observe by Markov’s inequality andithat, for anyq >2,
Q1≤a−αP
Un≤un−
un
t
≤a−αP Un−un ≥
un
t
≤a−ασn−q
un
−tq
E
n
i1
Zni −EZni
q
≤Cqa−ασn−q
un
−tq
⎧ ⎨ ⎩
n
i1
E Zni −EZni q
n
i1
VarZni
q/2⎫⎬ ⎭.
2.10
By the definition ofZni,we have that
Z
ni−EZni ≤max
Zni , EZni
≤Mn,
VarZniVarZiIZi≤Mn VarMnIZi> Mn
2CovZiIZi≤Mn, MnIZi> Mn
VarZiIZi≤Mn VarMnIZi> Mn
−2EZiIZi≤MnMnPZi> Mn
≤VarZiIZi≤Mn Mn2PZi> Mn
≤VarZiIZi≤Mn MnEZiIZi> Mn.
2.11
Substituting2.11into2.10, we have that
Q1≤Cqa−ασn−q
un
−tq
Mqn−2
n
i1
VarZiIZi≤Mn MnEZiIZi> Mn
n
i1
VarZiIZi≤Mn MnEZiIZi> Mn
q/2⎫⎬
⎭
≤Cqa−ασn−q
un
−tq
Mqn−2σn2M q−1
n n
i1
EZiIZi> Mn
2q/2−1
n
i1
VarZiIZi≤Mn
q/2
2q/2−1
Mn
n
i1
EZiIZi> Mn
q/2⎫⎬
⎭
:I1I2I3I4.
ForI1,we have byivthat
I1Cqa−α
un
−tq Mn
σnμsn
q−2
μsnq−2≤Ca−α
un
−tq
μsnq−2. 2.13
ForI2,we first note that
μn
μn
1− σ −1
n
n
i1EZiIZi> Mn
μn ,
2.14
which entails byiiithat
μn
−1
σn−1 n
i1
EZiIZi> Mn−→0. 2.15
It follows byivthat
I2Cqa−ασn−q1
un
−tq
Mqn−1μn
μn
−1
σ−1
n n
i1
EZiIZi> Mn
≤Ca−ασ−q1 n
un
−tq
Mqn−1μn
Ca−αun
−tq Mn
σnμsn
q−1
μsnq−1μn
≤Ca−αun
−tq
μsnq−1μn.
2.16
ForI3,we have by the definition ofσnthat
I3Cq2q/2−1a−α
un
−tq
. 2.17
ForI4,we have by2.15andivthat
I4Cq2q/2−1a−ασn−q/2
un
−tq
Mq/n 2
μn
q/2
μn
−1
σn−1 n
i1
EZiIZi> Mn
q/2
≤Ca−ασ−q/2 n
un
−tq
Mq/n 2
μn
q/2
Ca−αun
−tq
μn
q/2 Mn
σnμsn
q/2
μsq/n 2
≤Ca−αun
−tq
μn
q/2
μsq/n 2.
Substituting2.13and2.16–2.18into2.12, we get that
lim sup
n→ ∞
aEXn
α
Q1
≤lim sup
n→ ∞
aEXn
α
Ca−αun
−tq
μsnq−2Ca−α
un
−tq
μsnq−1μn
Cq2q/2−1a−α
un
−tq
Ca−αun
−tq
μn
q/2
μsq/n 2
lim sup
n→ ∞
aμn
μn
α
Ca−αun
−tq
μsnq−2αCa−α
un
−tq
μsnq−1αμn
Cq2q/2−1a−α
un
−tq
μαnCa−α
un
−tq
μn
q/2
μsq/n 2α
.
2.19
Since 0< s <s1/2< t <1,we can takeq >2 large enough such thattq >max{sq−1
α1, sq/2q/2α}.Then we have by2.3that
un
−tq
μsnq−2α
un
−sq−2−α
μsnq−2α
un
−tqsq−2α−→
0,
un
−tq
μsnq−1αμn
un
−sq−1−α
μsnq−1αμn
un
−1
un
−tqsq−1α1−→
0,
un
−tq
μαn
un
−α
μαn
un
−tqα−→
0,
un
−tq
μn
q/2
μsq/n 2α
un
−sq/2−α
μsq/n 2α
μn
q/2
un
−q/2
un
−tqsq/2q/2α−→
0.
2.20
Hence all the terms in the second brace of2.19converge to 0 asn → ∞.Moreover, we have byiiandiiithat
aμn
μn
α
a/μn 1
μn/μn
α
−→1. 2.21
Therefore lim supn→ ∞aEXnαQ10 and so2.9is 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, then2.1is 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
∞
n1φ1/22n < ∞ see Shao 14, ρ-mixing identically distributed random variables
satisfying∞n1ρ2/q2n < ∞see Shao 15, negatively associated random variablessee
Shao16, andρ∗-mixing random variablessee Utev and Peligrad17.
Lemma 2.3. Let{Yn, n ≥ 1}be a sequence of nonnegative random variables withEYn → ∞.Let
{bn, n≥1}be a sequence of positive real numbers satisfyingbn → b,whereb >0.Assume that
EaYn−α∼aEYn−α ∀a>0, α >0. 2.22
ThenEbnYn−α∼bEYn−α.
Proof. Take >0 such that 0< < b.Sincebn → b,there exists a positive integerNsuch that
0< b− < bn< bifn > N.We have by2.22that, forn > N,
EbnYn−α≤Eb−Yn−α∼b−EYn−α. 2.23
It follows that
lim sup
n→ ∞ bEYn
α
EbnYn−α
≤lim sup
n→ ∞
bEYnαEb−Yn−α
lim sup
n→ ∞
bEYnα
b−EYnα
b−EYnαEb−Yn−α
lim sup
n→ ∞
b/EYn1
b−/EYn1
α
b−EYnαEb−Yn−α1.
2.24
Similar to the above case, we get that, forn > N,
EbnYn−α≥EbYn−α∼bEYn−α, 2.25
lim inf
n→ ∞ bEYn
α
EbnYn−α
≥lim inf
n→ ∞ bEYn
α
EbYn−α
lim inf
n→ ∞
b/EYn1
b/EYn1
α
bEYnαEbYn−α1.
2.26
Hence the result is proved by2.24and2.26.
By using Theorem 2.1, we can obtain the following theorem which improves and extendsTheorem 1.1for independent random variables to the more general random variables satisfying the Rosenthal-type inequality2.1.
Theorem 2.4. Let{Zn, n ≥ 1}be a sequence of nonnegative random variables withEZ2n < ∞.Let
Xnandσnbe defined by1.1. Assume that the Rosenthal-type inequality2.1withMnησnholds
for allq >2,whereη >0is the same as in (ii). Furthermore, assume that
iEXn → ∞asn → ∞,
iiσ−2
n
n
i1EZ2iIZi> ησn → 0 for someη >0.
Proof. LetμnEXn andμnσn−1
n
i1EZiIZi≤Mn,whereXn andσnare defined by1.6.
Note that
σn2 n
i1
VarZiIZi≤Mn ZiIZi > Mn
n
i1
VarZiIZi≤Mn VarZiIZi> Mn 2CovZiIZi ≤Mn, ZiIZi> Mn
σn2 n
i1
VarZiIZi> Mn−2 n
i1
EZiIZi≤MnEZiIZi> Mn,
2.27
which implies that
1 σ
2
n
σ2
n
n
i1VarZiIZi> Mn
σ2
n
−2
n
i1EZiIZi≤MnEZiIZi> Mn
σ2
n
. 2.28
But, we have byiithat
σn−2 n
i1
VarZiIZi> Mn≤σn−2 n
i1
EZ2iIZi> Mn−→0,
σn−2 n
i1
EZiIZi≤MnEZiIZi> Mn≤σn−2Mn n
i1
EZiIZi> Mn
≤σn−2 n
i1
EZ2iIZi> Mn−→0.
2.29
Substituting2.29into2.28, we have that
σn−1σn−→1 asn−→ ∞. 2.30
Now we will applyTheorem 2.1to the random variableXn.By2.30andi, we get that
μnσn−1 n
i1
We also get that
μn
μn
−1 σn−1
n
i1EZiIZi ≤Mn
σ−1
n
n
i1EZi
σn−1
n
i1EZiIZi ≤Mn
σn−1
n i1EZi
1−σ −1
n
n
i1EZiIZi> Mn
EXn −→
1,
2.32
since EXn → ∞ by i and σn−1
n
i1EZiIZi > Mn ≤ σn−1M−n1
n
i1EZi2IZi > Mn
η−1σn−2
n
i1EZ2iIZi> Mn → 0 byii. From2.31and2.32,μn → ∞and so we have by
2.30that, for anys >0,
Mn
σnμsn
ησn
σnμsn −→
0. 2.33
Hence all conditions ofTheorem 2.1are satisfied. ByTheorem 2.1,
E
aσn−1 n
i1
Zi
−α
∼
aσn−1 n
i1
EZi
−α
∀a >0, α >0. 2.34
Note that the norming constants in2.34are different from those inXn.
To complete the proof, we will use Lemma 2.3. Since σ−1
n σn → 1, we have by
Lemma 2.3that
E
aσnσn−1σn−1 n
i1
Zi
−α
∼
aσn−1 n
i1
EZi
−α
. 2.35
Namely,
E
aσn−1 n
i1
Zi
−α
∼
aσnσn−1σn−1 n
i1
EZi
−α
. 2.36
Byiand2.30,
aσnσn−1σn−1 n
i1
EZi
−α
∼
aσn−1 n
i1
EZi
−α
. 2.37
Combining2.36with2.37gives the desired result.
only independent random variables but also NOD random variables. Hence Theorem 2.4 improves and extends the results of Wu et al.10and Wang et al.11to the more general random variables.
Theorem 2.6. Let{Zn, n ≥ 1}be a sequence of nonnegative random variables. Letμn EXnand
μnσn−1
n
i1EZiIZi≤Mn,whereXn andσnare defined by1.6, and{Mn, n≥1}is a sequence
of positive real numbers satisfying
Mn−→ ∞, MnO
ntfor some 0< t <1. 2.38
Assume that the Rosenthal-type inequality2.1holds for allq >2.Furthermore, assume that
i{Zn}is uniformly integrable,
iin−1n
i1EZi≥Cfor some positive constantC >0,
iiin−1/2σ
n≥Dfor some positive constantD >0.
Then1.5holds for all real numbersa >0andα >0.
Proof. We first note byiandiithat
Cn≤
n
i1
EZi≤nsup i≥1
EZiOn,
n−1
n
i1
EZiIZi> Mn≤sup i≥1
EZiIZi> Mn o1.
2.39
We next estimateσn.By2.39,
σn2≤
n
i1
EZi2IZi≤Mn≤Mn n
i1
EZiMnOn. 2.40
Combining2.40withiiigives
D2n≤σn2 ≤C1nMn for some constantC1>0. 2.41
Now we will applyTheorem 2.1to the random variableXn.Byii,2.41, and2.38, we get
that
μn
n i1EZi
σn ≥
Cn σn ≥
Cn
C1nMn
Cn1/2 C1O
We also get byiiand2.39that
μn
μn
σn−1
n
i1EZiIZi≤Mn
σ−1
n
n
i1EZi
n−1
n
i1EZiIZi≤Mn
n−1n i1EZi
1−n −1n
i1EZiIZi> Mn
n−1n i1EZi
−→1.
2.43
Since 0 < t < 1,we can takessuch that max{2t−1,0} < s < 1.Then we have byii,iii,
2.38, and2.43that
Mn
σnμsn
Mn
σn
μn
s
μn
−s
μs n
≤ Mn
σn
Cnσ−1
n
s
μn
−s
μs n
byii
≤ O
nt
CsD1−sns1−s/2μn−sμs n
−→0,
2.44
sinces 1−s/2 > tandμn−1μn → 1.Hence all conditions ofTheorem 2.1are satisfied.
The result follows fromTheorem 2.1.
Remark 2.7. The conditions ofTheorem 2.6are much weaker than those ofTheorem 1.2in the following three directions.
iIf{Zn, n≥1}is a sequence of independent random variables, then2.1is satisfied
from the Rosenthal inequality. Hence2.1is weaker than independence condition.
iiIf{Mn, n≥1}satisfies1.7, then it also satisfies2.38by the fact that2−δ/4−
r<2−δ/2<1.Hence2.38is weaker than1.7.
iiiThe condition supn≥1EZr
n < ∞ in Theorem 1.2 is not needed in Theorem 2.6.
ThereforeTheorem 2.6improves and extends Wu et al.10resultseeTheorem 1.2 to the more general random variables.
Acknowledgments
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