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On Inverse Moments for a Class of Nonnegative Random Variables
Journal of Inequalities and Applications volume 2010, Article number: 823767 (2010)
Abstract
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 be a sequence of nonnegative random variables with finite second moments. Let us denote

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

where and
means that
as
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.
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 Wesoowski [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 , (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
(
in the i.i.d. case) when
is a sequence of nonnegative independent random variables satisfying
and
(Lyapunov's condition of order 3), that is,
with
Hu et al. [9] generalized the result of Kaluszka and Okolewski [8] by considering
for some
instead of
Recently, Wu et al. [10] obtained the following result by using the truncation method and Bernstein's inequality.
Theorem 1.1.
Let be a sequence of nonnegative independent random variables such that
and
where
is defined by (1.1). Furthermore, assume that


Then (1.2) holds for all real numbers and
For a sequence of nonnegative independent random variables with only
th moments for some
Wu et al. [10] also obtained the following asymptotic approximation of the inverse moment:

for all real numbers and
. Here
is defined as

where is a sequence of positive constants satisfying

Specifically, Wu et al. [10] proved the following result.
Theorem 1.2.
Let be a sequence of nonnegative independent random variables. Suppose that, for some
,
(i) is uniformly integrable,
(ii)
(iii) for some positive constant
(iv) for some positive constant
where is the same as in (1.6) for some positive constants
satisfying (1.7). Then (1.5) holds for all real numbers
and
.
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 of independent random variables with
and
for some
Rosenthal [12] proved that there exists a positive constant
depending only on
such that

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 denotes a positive constant which is not necessarily the same one in each appearance, and
denotes the indicator function of the event
2. Main Results
Throughout this section, we assume that 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 be a sequence of nonnegative random variables. Let
and
where
and
are defined by (1.6), and
is a sequence of positive real numbers. Suppose that the following conditions hold:
(i)for any there exists a positive constant
depending only on
such that

where
(ii) as
(iii) as
(iv) for some
Then (1.5) holds for all real numbers and
.
Proof.
Let us decompose as

where and
Denote
Since
we have that
It follows by (ii) and (iii) that

Now, applying Jensen's inequality to the convex function yields
Therefore

Hence it is enough to show that

Since we can take
such that
and
Namely,
Let us write

where and
Since
we get that

which implies by (ii) and (2.3) that

It remains to show that

Observe by Markov's inequality and (i) that, for any

By the definition of we have that

Substituting (2.11) into (2.10), we have that

For we have by (iv) that

For we first note that

which entails by (iii) that

It follows by (iv) that

For we have by the definition of
that

For we have by (2.15) and (iv) that

Substituting (2.13) and (2.16)–(2.18) into (2.12), we get that

Since we can take
large enough such that
Then we have by (2.3) that

Hence all the terms in the second brace of (2.19) converge to 0 as Moreover, we have by (ii) and (iii) that

Therefore and so (2.9) is proved.
Remark 2.2.
In (2.1), are monotone transformations of
If
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
Examples include sequences of NOD random variables (see Asadian et al. [13]),
-mixing identically distributed random variables satisfying
(see Shao [14]),
-mixing identically distributed random variables satisfying
(see Shao [15]), negatively associated random variables (see Shao [16]), and
-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.
Lemma 2.3.
Let be a sequence of nonnegative random variables with
Let
be a sequence of positive real numbers satisfying
where
Assume that

Then
Proof.
Take such that
Since
there exists a positive integer
such that
if
We have by (2.22) that, for

It follows that

Similar to the above case, we get that, for


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 be a sequence of nonnegative random variables with
Let
and
be defined by (1.1). Assume that the Rosenthal-type inequality (2.1) with
holds for all
where
is the same as in (ii). Furthermore, assume that
(i) as
(ii)
Then (1.2) holds for all real numbers and
.
Proof.
Let and
where
and
are defined by (1.6). Note that

which implies that

But, we have by (ii) that

Substituting (2.29) into (2.28), we have that

Now we will apply Theorem 2.1 to the random variable By (2.30) and (i), we get that

We also get that

since by (i) and
by (ii). From (2.31) and (2.32),
and so we have by (2.30) that, for any

Hence all conditions of Theorem 2.1 are satisfied. By Theorem 2.1,

Note that the norming constants in (2.34) are different from those in
To complete the proof, we will use Lemma 2.3. Since we have by Lemma 2.3 that

Namely,

By (i) and (2.30),

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 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 be a sequence of nonnegative random variables. Let
and
where
and
are defined by (1.6), and
is a sequence of positive real numbers satisfying

Assume that the Rosenthal-type inequality (2.1) holds for all Furthermore, assume that
(i) is uniformly integrable,
(ii) for some positive constant
(iii) for some positive constant
Then (1.5) holds for all real numbers and
.
Proof.
We first note by (i) and (ii) that

We next estimate By (2.39),

Combining (2.40) with (iii) gives

Now we will apply Theorem 2.1 to the random variable By (ii), (2.41), and (2.38), we get that

We also get by (ii) and (2.39) that

Since we can take
such that
Then we have by (ii), (iii), (2.38), and (2.43) that

since and
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 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 satisfies (1.7), then it also satisfies (2.38) by the fact that
Hence (2.38) is weaker than (1.7).
(iii)The condition 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.
References
Wooff DA: Bounds on reciprocal moments with applications and developments in Stein estimation and post-stratification. Journal of the Royal Statistical Society. Series B 1985, 47(2):362–371.
Pittenger AO: Sharp mean-variance bounds for Jensen-type inequalities. Statistics & Probability Letters 1990, 10(2):91–94. 10.1016/0167-7152(90)90001-N
Marciniak E, Wesołowski J: Asymptotic Eulerian expansions for binomial and negative binomial reciprocals. Proceedings of the American Mathematical Society 1999, 127(11):3329–3338. 10.1090/S0002-9939-99-05105-9
Fujioka T: Asymptotic approximations of the inverse moment of the noncentral chi-squared variable. Journal of the Japan Statistical Society 2001, 31(1):99–109.
Jurlewicz A, Weron K: Relaxation of dynamically correlated clusters. Journal of Non-Crystalline Solids 2002, 305(1–3):112–121.
Ramsay CM: A note on random survivorship group benefits. ASTIN Bulletin 1993, 23: 149–156. 10.2143/AST.23.1.2005106
Garcia NL, Palacios JL: On inverse moments of nonnegative random variables. Statistics & Probability Letters 2001, 53(3):235–239. 10.1016/S0167-7152(01)00008-6
Kaluszka M, Okolewski A: On Fatou-type lemma for monotone moments of weakly convergent random variables. Statistics & Probability Letters 2004, 66(1):45–50. 10.1016/j.spl.2003.10.009
Hu SH, Chen GJ, Wang XJ, Chen EB: On inverse moments of nonnegative weakly convergent random variables. Acta Mathematicae Applicatae Sinica 2007, 30(2):361–367.
Wu T-J, Shi X, Miao B: Asymptotic approximation of inverse moments of nonnegative random variables. Statistics & Probability Letters 2009, 79(11):1366–1371. 10.1016/j.spl.2009.02.010
Wang X, Hu S, Yang W, Ling N: Exponential inequalities and inverse moment for NOD sequence. Statistics and Probability Letters 2010, 80(5–6):452–461. 10.1016/j.spl.2009.11.023
Rosenthal HP: On the subspaces of spanned by sequences of independent random variables. Israel Journal of Mathematics 1970, 8: 273–303. 10.1007/BF02771562
Asadian N, Fakoor V, Bozorgnia A: Rosenthal's type inequalities for negatively orthant dependent random variables. Journal of the Iranian Statistical Society 2006, 5: 69–75.
Shao QM: A moment inequality and its applications. Acta Mathematica Sinica 1988, 31(6):736–747.
Shao QM: Maximal inequalities for partial sums of -mixing sequences. The Annals of Probability 1995, 23(2):948–965. 10.1214/aop/1176988297
Shao Q-M: A comparison theorem on moment inequalities between negatively associated and independent random variables. Journal of Theoretical Probability 2000, 13(2):343–356. 10.1023/A:1007849609234
Utev S, Peligrad M: Maximal inequalities and an invariance principle for a class of weakly dependent random variables. Journal of Theoretical Probability 2003, 16(1):101–115. 10.1023/A:1022278404634
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).
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Sung, S. On Inverse Moments for a Class of Nonnegative Random Variables. J Inequal Appl 2010, 823767 (2010). https://doi.org/10.1155/2010/823767
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DOI: https://doi.org/10.1155/2010/823767