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.