# Moment Inequalities and Complete Moment Convergence

- Soo Hak Sung
^{1}Email author

**2009**:271265

https://doi.org/10.1155/2009/271265

© Soo Hak Sung. 2009

**Received: **22 August 2009

**Accepted: **26 September 2009

**Published: **11 October 2009

## Abstract

Let and be sequences of random variables. For any and , bounds for and are obtained. From these results, we establish general methods for obtaining the complete moment convergence. The results of Chow (1988), Zhu (2007), and Wu and Zhu (2009) are generalized and extended from independent (or dependent) random variables to random variables satisfying some mild conditions. Some applications to dependent random variables are discussed.

## 1. Introduction

Note that (1.3) and (1.4) imply (1.1) and (1.2), respectively. The above inequalities have been obtained for dependent random variables by many authors. Shao [3] proved that (1.3) and (1.4) hold for negatively associated random variables. Asadian et al. [4] proved that (1.1) and (1.2) hold for negatively orthant dependent random variables.

For a sequence of some mixing random variables, (1.4) holds. However, the constant depends on both and the sequence of mixing random variables. Shao [5] obtained (1.4) for -mixing identically distributed random variables satisfying . Shao [6] also obtained (1.4) for -mixing identically distributed random variables satisfying . Utev and Peligrad [7] obtained (1.4) for -mixing random variables.

In view of the Borel-Cantelli lemma, this implies that almost surely. Therefore the complete convergence is a very important tool in establishing almost sure convergence of summation of random variables. Hsu and Robbins [8] proved that the sequence of arithmetic means of i.i.d. random variables converges completely to the expected value if the variance of the summands is finite. Erdös [9] proved the converse.

where . Note that (1.7) implies (1.6) (see Remark 2.6).

Recently, Zhu [12] obtained a complete convergence for -mixing random variables. Wu and Zhu [13] obtained complete moment convergence results for negatively orthant dependent random variables.

In this paper, we give general methods for obtaining the complete moment convergence by using some moment inequalities. From these results, we generalize and extend the results of Chow [11], Zhu [12], and Wu and Zhu [13] from independent (or dependent) random variables to random variables satisfying some conditions similar to (1.1)–(1.4).

## 2. Complete Moment Convergence for Random Variables

In this section, we give general methods for obtaining the complete moment convergence by using some moment inequalities. The first two lemmas are simple inequalities for real numbers.

Lemma 2.1.

Proof.

The result follows by an elementary calculation.

The following lemma is a slight generalization of Lemma 2.1.

Lemma 2.2.

Proof.

The next two lemmas play essential roles in the paper. Lemma 2.3 gives a moment inequality for the sum of random variables.

Lemma 2.3.

Proof.

Substituting (2.6) into (2.5), we have the result.

The following lemma gives a moment inequality for the maximum partial sum of random variables.

Lemma 2.4.

Proof.

The rest of the proof is similar to that of Lemma 2.3 and is omitted.

Now we state and prove one of our main results. The following theorem gives a general method for obtaining the complete moment convergence for sums of random variables satisfying (2.9). The condition (2.9) is well known Marcinkiewicz-Zygmund inequality.

Theorem 2.5.

Let be an array of random variables with for , . Let and be sequences of positive real numbers. Suppose that the following conditions hold.

Proof.

The above two series converge by (ii) and (iii). Hence the result is proved.

Remark 2.6.

Hence complete moment convergence is more general than complete convergence.

When , we have the following theorem. Condition (2.15) is well-known Rosenthal inequality.

Theorem 2.7.

Let be an array of random variables with for , . Let and be sequences of positive real numbers. Suppose that the following conditions hold.

Then (2.10) holds.

Proof.

Corollary 2.8.

Let be a sequence of positive real numbers. Let be an array of random variables satisfying (2.15) for some . Suppose that the following conditions hold.

Proof.

Hence the result follows by Theorem 2.7.

The following theorem gives a general method for obtaining the complete moment convergence for maximum partial sums of random variables satisfying condition (2.20).

Theorem 2.9.

Let be an array of random variables with for , . Let and be sequences of positive real numbers. Suppose that the following conditions hold.

Proof.

Hence the result is proved.

Remark 2.10.

When , we have the following theorem.

Theorem 2.11.

Then (2.21) holds.

Proof.

The proof is similar to that of Theorem 2.9 and is omitted.

Corollary 2.12.

Let be a sequence of positive real numbers. Let be an array of random variables satisfying (2.24) for some . Suppose that the following conditions hold.

Proof.

## 3. Corollaries

In this section, we establish some complete moment convergence results by using the results obtained in the previous section.

To obtain complete moment convergence results, the following lemmas are needed.

Lemma 3.1.

Let be a random variable and a sequence of positive even functions satisfying (3.1) for some . Then for all and , the followings hold.

Proof.

So (ii) holds.

Lemma 3.2.

Then the followings hold.

Proof.

The result follows from Lemma 3.1.

By using Lemma 3.2, we can obtain Corollaries 3.3, 3.4, 3.5, 3.6 from Theorem 2.5, Corollary 2.8, Theorem 2.9, Corollary 2.12, respectively.

Corollary 3.3.

Let and be sequences of positive real numbers a sequence of positive even functions satisfying (3.1) for some . Assume that is an array of random variables satisfying (2.9) for and (3.4). Then (2.10) holds.

Corollary 3.4.

Then (2.17) holds and hence, (2.18) holds.

Corollary 3.5.

Let and be sequences of positive real numbers a sequence of positive even functions satisfying (3.1) for some . Assume that is an array of random variables satisfying (2.20) for and (3.4). Then (2.21) holds.

Corollary 3.6.

Let be a sequence of positive real numbers a sequence of positive even functions satisfying (3.1) for some . Assume that is an array of random variables satisfying (2.24) for some ( is the same as in (3.6)), (3.5), and (3.6). Then (2.25) holds and hence, (2.26) holds.

Remark 3.7.

- (1)
For an array of rowwise negatively associated random variables, condition (2.20) holds if , and (2.24) holds if by Shao's [3] results. Note that is still an array of rowwise negatively associated random variables. Hence Corollaries 3.3–3.6 hold for arrays of rowwise negatively associated random variables.

- (2)
For an array of rowwise negatively orthant dependent random variables, condition (2.9) holds if , and (2.15) holds if by the results of Asadian et al. [4]. Hence Corollaries 3.3 and 3.4 hold for arrays of rowwise negatively orthant dependent random variables. These results also were proved by Wu and Zhu [13]. Hence Corollaries 3.3 and 3.4 extend the results of Wu and Zhu [13] from an array of negatively orthant dependent random variables to an array of random variables satisfying (2.9) and (2.15).

- (3)
For an array of rowwise -mixing random variables, condition (2.24) does not necessarily hold if . As mentioned in Section 1, Utev and Peligrad [7] proved (1.4) for -mixing random variables. However, the constant depends on both and the sequence of -mixing random variables. Hence condition (2.24) holds for an array of rowwise -mixing random variables under the additional condition that depending on the sequence of random variables in each row are bounded. So Corollary 3.6 holds for arrays of rowwise -mixing random variables satisfying this additional condition. Zhu [12] obtained only (2.26) in Corollary 3.6 when the array is rowwise -mixing random variables satisfying the additional condition. This additional condition should be added in Zhu [12]. Hence Corollary 3.6 generalizes and extends Zhu's [12] result from -mixing random variables to more general random variables.

Finally, we apply the complete moment convergence results obtained in the previous section to a sequence of identically distributed random variables.

Corollary 3.8.

Proof.

Hence the result follows from Theorem 2.7.

Corollary 3.9.

Proof.

As in the proof of Corollary 3.8, (3.8) are satisfied. So the result follows from Theorem 2.11.

Remark 3.10.

If is a sequence of i.i.d. random variables, then conditions (3.7) and (3.9) are satisfied when . Hence Corollaries 3.8 and 3.9 generalize and extend the result of Chow [11]. There are many sequences of dependent random variables satisfying (3.7) for all . Examples include sequences of negatively orthant dependent random variables, negatively associated random variables, -mixing random variables, -mixing identically distributed random variables satisfying , and -mixing identically distributed random variables satisfying . The above sequences of dependent random variables except negatively orthant dependent random variables also satisfy (3.9) when . Hence Corollaries 3.8 and 3.9 hold for many dependent random variables as well as independent random variables.

## Declarations

### Acknowledgments

The author would like to thank the referees for the helpful comments and suggestions. This work was supported by the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government (MOST) (no. R01-2007-000-20053-0).

## Authors’ Affiliations

## References

- Marcinkiewicz J, Zygmund A:
**Sur les fonctions independantes.***Fundamenta Mathematicae*1937,**29:**60–90.MATHGoogle Scholar - Rosenthal HP:
**On the subspaces of spanned by sequences of independent random variables.***Israel Journal of Mathematics*1970,**8:**273–303. 10.1007/BF02771562MATHMathSciNetView ArticleGoogle Scholar - 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:1007849609234MATHMathSciNetView ArticleGoogle Scholar - 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.Google Scholar - Shao Q-M:
**A moment inequality and its applications.***Acta Mathematica Sinica*1988,**31**(6):736–747.MATHMathSciNetGoogle Scholar - Shao Q-M:
**Maximal inequalities for partial sums of -mixing sequences.***The Annals of Probability*1995,**23**(2):948–965. 10.1214/aop/1176988297MATHMathSciNetView ArticleGoogle Scholar - 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:1022278404634MATHMathSciNetView ArticleGoogle Scholar - Hsu PL, Robbins H:
**Complete convergence and the law of large numbers.***Proceedings of the National Academy of Sciences of the United States of America*1947,**33:**25–31. 10.1073/pnas.33.2.25MATHMathSciNetView ArticleGoogle Scholar - Erdös P:
**On a theorem of Hsu and Robbins.***Annals of Mathematical Statistics*1949,**20:**286–291. 10.1214/aoms/1177730037MATHMathSciNetView ArticleGoogle Scholar - Baum LE, Katz M:
**Convergence rates in the law of large numbers.***Transactions of the American Mathematical Society*1965,**120:**108–123. 10.1090/S0002-9947-1965-0198524-1MATHMathSciNetView ArticleGoogle Scholar - Chow YS:
**On the rate of moment convergence of sample sums and extremes.***Bulletin of the Institute of Mathematics. Academia Sinica*1988,**16**(3):177–201.MATHMathSciNetGoogle Scholar - Zhu M-H:
**Strong laws of large numbers for arrays of rowwise -mixing random variables.***Discrete Dynamics in Nature and Society*2007,**2007:**-6.Google Scholar - Wu Y-F, Zhu D-J: Convergence properties of partial sums for arrays of rowwise negatively orthant dependent random variables. Journal of the Korean Statistical Society. In press Journal of the Korean Statistical Society. In pressGoogle Scholar

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