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.