A note on the complete convergence for weighted sums of negatively dependent random variables

The complete convergence theorems for weighted sums of arrays of rowwise negatively dependent random variables were obtained by Wu (Wu, Q: Complete convergence for weighted sums of sequences of negatively dependent random variables. J. Probab. Stat. 2011, Article ID 202015, 16 pages) and Wu (Wu, Q: A complete convergence theorem for weighted sums of arrays of rowwise negatively dependent random variables. J. Inequal. Appl. 2012, 50). In this paper, we complement the results of Wu.

MSC: 60F15.

The concept of complete convergence of a sequence of random variables was introduced by Hsu and Robbins [1]. A sequence of random variables converges completely to the constant θ if

By the Borel-Cantelli lemma, this implies that almost surely (a.s.). The converse is true if are independent random variables. Therefore, the complete convergence is a very important tool in establishing almost sure convergence. There are many complete convergence theorems for sums and weighted sums of independent random variables.

Volodin et al. [2] and Chen et al. [3] ( and , respectively) proved the following complete convergence for weighted sums of arrays of rowwise independent random elements in a real separable Banach space.

We recall that the array of random variables is said to be stochastically dominated by a random variable X if

where C is a positive constant.

Theorem 1.1 ([2,3])

Suppose that. Letbe an array of rowwise independent random elements in a real separable Banach space which are stochastically dominated by a random variableX. Letbe an array of constants satisfying

and

for someandμsuch thatand. Ifandin probability, then

If , then (1.3) is immediate. Hence Theorem 1.1 is of interest only for .

Recently, Wu [4] extended Theorem 1.1 to negatively dependent random variables when . Wu [4] also considered the case of (). But, the proof of Wu [4] does not work for the case of .

The concept of negatively dependent random variables was given by Lehmann [5]. A finite family of random variables is said to be negatively dependent (or negatively orthant dependent) if for each , the following two inequalities hold:

and

for all real numbers . An infinite family of random variables is negatively dependent if every finite subfamily is negatively dependent.

Theorem 1.2 (Wu [4])

Suppose that. Letbe an array of negatively dependent random variables which are stochastically dominated by a random variableX. Letbe an array of constants satisfying (1.1) for someand (1.2) for someθandμsuch thatand. Furthermore, assume thatfor allandif.

(i) Ifand, then

(ii) Ifand, then (1.4) holds.

Using the moment inequality of negatively dependent random variables, Wu [6] obtained a complete convergence result for weighted sums of identically distributed negatively dependent random variables.

Theorem 1.3 (Wu [6])

Suppose that. Letbe a sequence of identically distributed negatively dependent random variables. Letbe an array of constants satisfying

and

Furthermore, assume thatif. Then, for,

if and only if

For, (1.7) implies (1.8).

In (1.5), means that and . Theorem 1.3 extends the result of Liang and Su [7] for negatively associated random variables to negatively dependent case. The proof of the sufficiency part of Liang and Su [7] is mistakenly based on the fact that (1.8) implies that

The proof of the sufficiency is correct when . However, condition (1.5) does not hold, since the left-hand side of (1.5) goes to the limit as , but the right-hand side diverges. Hence, there are no arrays satisfying (1.5).

In this paper, we obtain complete convergence results for weighted sums of arrays of rowwise negatively dependent random variables. Our results complement the results of Wu [4,6].

Throughout this paper, the symbol C denotes a positive constant which is not necessarily the same one in each appearance. It proves convenient to define , where lnx denotes the natural logarithm.

In this section, we present some lemmas which will be used to prove our main results.

The following two lemmas are well known and their proofs are standard.

Lemma 2.1Letbe a sequence of random variables which are stochastically dominated by a random variableX. For anyand, the following statements hold:

(i) .

(ii) .

The following Lemma 2.2(i)-(iii) can be found in Sung [8].

Lemma 2.2LetXbe a random variable withfor some. For any, the following statements hold:

(i) for any.

(ii) for anysuch that.

(iii) .

(iv) .

The Marcinkiewicz-Zygmund and Rosenthal type inequalities play an important role in establishing complete convergence. Asadian et al. [9] proved the Marcinkiewicz-Zygmund and Rosenthal inequalities for negatively dependent random variables.

Lemma 2.3 (Asadian et al. [9])

Letbe a sequence of negatively dependent random variables withandfor someand all. Then there exist constantsanddepending only onpsuch that

The last lemma is a complete convergence theorem for an array of rowwise negatively dependent mean zero random variables.

Lemma 2.4 ([10,11])

Letbe an array of rowwise negatively dependent random variables withandfor alland. Letbe a sequence of nonnegative constants. Suppose that the following conditions hold.

(i) for all.

(ii) There existssuch that

Thenfor all.

In this section, we obtain two complete convergence results for weighted sums of arrays of rowwise negatively dependent random variables.

Theorem 3.1Suppose that. Letbe an array of rowwise negatively dependent mean zero random variables which are stochastically dominated by a random variableXsatisfyingfor some. Letbe an array of constants satisfying (1.1) for someand

Furthermore, assume that

if. Then

Proof Since , we may assume that . For and , define

Then and are still arrays of rowwise negatively dependent random variables, , and . Since , and are also arrays of rowwise negatively dependent random variables. In view of for all and , it suffices to show that

and

We will prove (3.3) and (3.4) with three cases.

Case 1 ().

For , we get by Markov’s inequality, Lemmas 2.1-2.3, (1.1), and (3.1) that

The sixth inequality follows from Lemma 2.2.

For , we first prove that

By Lemma 2.1, (1.1), and (3.1), is dominated by

Since and as , (3.5) holds.

Hence, to prove (3.4), it suffices to show that

Take such that . Since , we get by Markov’s inequality, Lemmas 2.1-2.2, (1.1), and (3.1) that

Case 2 ().

As in Case 1, we have that .

For , we take such that . Then we have by Markov’s inequality, Lemmas 2.1-2.3, (1.1), and (3.1) that

Case 3 ().

In this case, we will prove (3.3) and (3.4) by using Lemma 2.4. To prove (3.3), we take . Then we obtain by Markov’s inequality, Lemmas 2.1-2.2, (1.1), and (3.1) that

We also obtain that for such that ,

Hence (3.3) holds by Lemma 2.4.

To prove (3.4), we take such that . The proof of the rest is similar to that of (3.3) and is omitted. □

Remark 3.1 When , Theorem 3.1 holds without the condition of negative dependence (see Theorem 2(i) in Sung [8]). Theorem 3.1 extends the result of Sung [8] for independent random variables to negatively dependent case.

Remark 3.2 Theorem 1.2(i) follows from Theorem 3.1 by taking and , since

But, Theorem 1.2(i) does not deal with the case of .

Note that conditions (1.1) and (3.1) together imply

The following theorem shows that if the moment condition of Theorem 3.1 is replaced by a stronger condition , then condition (3.1) can be replaced by the weaker condition (3.7).

Theorem 3.2Suppose that. Letbe an array of rowwise negatively dependent mean zero random variables which are stochastically dominated by a random variableXsatisfyingfor some. Letbe an array of constants satisfying (1.1) and (3.7). Furthermore, assume that (3.2) holds for someif. Then

Proof As in the proof of Theorem 3.1, it suffices to prove (3.3) and (3.4). The proof of (3.3) is same as that of Theorem 3.1 except that q is replaced by p.

We now prove (3.4). When , we have by Markov’s inequality, Lemmas 2.1-2.3, and (3.7) that

When , we will prove (3.4) by using Lemma 2.4. We have by Markov’s inequality, Lemmas 2.1-2.2, and (3.7) that

We also have that for such that ,

Hence (3.4) holds by Lemma 2.4. □

Remark 3.3 If , then . Hence Theorem 1.2(ii) follows from Theorem 3.2 by taking . But, Theorem 1.2(ii) does not deal with the case of .

As mentioned in the Introduction, (1.5) does not hold. Hence it is of interest to find a complete convergence result similar to Theorem 1.3 without condition (1.5). The following corollary does not assume condition (1.5).

Corollary 3.1Suppose that. Letbe a sequence of identically distributed negatively dependent mean zero random variables. Letbe an array of constants satisfying (1.6) and. If (1.7) holds, then

Proof Let for and for . We will apply Theorem 3.2 with , , , and replaced by . Then

Furthermore, if , then

Hence the result follows from Theorem 3.2. □

Remark 3.4 When , Corollary 3.1 holds without the condition of negative dependence. Although (3.8) is weaker than (1.8), (3.8) can be strengthened to (1.8) if the negative dependence is replaced by the stronger condition of negative association.

The author declares that he has no competing interests.

The author would like to thank the referees for helpful comments and suggestions. 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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