On Inverse Moments for a Class of Nonnegative Random Variables

Let be a sequence of nonnegative random variables with finite second moments. Let us denote

(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

(1.2)

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 Weso owski [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

(1.3)

(1.4)

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:

(1.5)

for all real numbers and . Here is defined as

(1.6)

where is a sequence of positive constants satisfying

(1.7)

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

(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 denotes a positive constant which is not necessarily the same one in each appearance, and denotes the indicator function of the event

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

(2.1)

where

(ii) as

(iii) as

(iv) for some

Then (1.5) holds for all real numbers and .

Proof.

Let us decompose as

(2.2)

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

(2.3)

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

(2.4)

Hence it is enough to show that

(2.5)

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

(2.6)

where and Since we get that

(2.7)

which implies by (ii) and (2.3) that

(2.8)

It remains to show that

(2.9)

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

(2.10)

By the definition of we have that

(2.11)

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

(2.12)

For we have by (iv) that

(2.13)

For we first note that

(2.14)

which entails by (iii) that

(2.15)

It follows by (iv) that

(2.16)

For we have by the definition of that

(2.17)

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

(2.18)

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

(2.19)

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

(2.20)

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

(2.21)

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

(2.22)

Then

Proof.

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

(2.23)

It follows that

(2.24)

Similar to the above case, we get that, for

(2.25)

(2.26)

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

(2.27)

which implies that

(2.28)

But, we have by (ii) that

(2.29)

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

(2.30)

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

(2.31)

We also get that

(2.32)

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

(2.33)

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

(2.34)

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

(2.35)

Namely,

(2.36)

By (i) and (2.30),

(2.37)

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

(2.38)

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

(2.39)

We next estimate By (2.39),

(2.40)

Combining (2.40) with (iii) gives

(2.41)

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

(2.42)

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

(2.43)

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

(2.44)

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.