- Research Article
- Open Access

# Doob's Type Inequality and Strong Law of Large Numbers for Demimartingales

- Wang Xuejun
^{1}, - Hu Shuhe
^{1}Email author, - Zhao Ting
^{1}and - Yang Wenzhi
^{1}

**2010**:838301

https://doi.org/10.1155/2010/838301

© Wang Xuejun et al. 2010

**Received:**9 October 2009**Accepted:**28 January 2010**Published:**21 February 2010

## Abstract

We establish some maximal inequalities for demimartingales which generalize the result of Wang (2004). The maximal inequality for demimartingales is used as a key inequality to establish other results including Doob's type maximal inequality, strong law of large numbers, strong growth rate, and integrability of supremum for demimartingales, which generalize and improve partial results of Christofides (2000) and Prakasa Rao (2007).

## Keywords

- Growth Rate
- Real Number
- Variable Sequence
- Partial Result
- Strong Growth

## 1. Introduction

Definition 1.1.

for all coordinatewise nondecreasing functions such that the expectation is defined. Then is called a demimartingale. If in addition the function is assumed to be nonnegative, then the sequence is called a demisubmartingale.

Definition 1.2.

for any two coordinatewise nondecreasing functions on such that the covariance is defined. An infinite sequence is associated if every finite subcollection is associated.

Definition 1.3.

An infinite sequence is strongly positive dependent if every finite subcollection is strongly positive dependent.

Remark 1.4.

Chow [1] proved a maximal inequality for submartingales. Newman and Wright [2] extended Doob's maximal inequality and upcrossing inequality to the case of demimartingales, and pointed out that the partial sum of a sequence of mean zero associated random variables is a demimartingale. Christofides [3] showed that the Chow's maximal inequality for (sub)martingales can be extended to the case of demi(sub)martingales. Wang [4] obtained Doob's type inequality for more general demimartingales. Hu et al. [5] gave a strong law of large numbers and growth rate for demimartingales. Prakasa Rao [6] established some maximal inequalities for demisubmartingales and N-demisuper-martingales.

It is easily seen that the partial sum of a sequence of mean zero strongly positive dependent random variables is also a demimartingale by the inequality (3) in Zheng [7], that is, for all ,

for all coordinatewise nondecreasing functions such that the expectation is defined. Therefore, the main results of this paper hold for the partial sums of sequences of mean zero associated random variables and strongly positive dependent random variables.

Let and be sequences of random variables defined on a fixed probability space and the indicator function of the event . Denote , , , , . The main results of this paper depend on the following lemmas.

Lemma 1.5 (see Wang [4, Theorem ]).

Lemma 1.6 (see Fazekas and Klesov [8, Theorem ] and Hu et al. [5, Lemma ]).

Lemma 1.8 (see Hu et al. [9, Theorem ]).

Using Lemma 1.5, Wang [4] obtained the following inequalities for demimartingales.

Theorem 1.10 (see Wang [4, Corollary ]).

We point out that there is a mistake in the proof of (1.17), that is,

should be replaced by

In fact, by Lemma 1.5 and Fubini Theorem, we can see that

The rest of the proof is similar to Corollary in Wang [4].

The same problem exists in Shiryaev [10, page 495, in the proof of Theorem ] and Krishna and Soumendra [11, page 414, in the proof of Theorem ]. For example, the following inequality

in Shiryaev [10, page 495] should be revised as

## 2. Main Results and Their Proofs

Theorem 2.1.

Proof.

Thus, (2.2) follows from (2.8) immediately. The proof is complete.

Remark 2.2.

If we take in Theorem 2.1, then Theorem 2.1 implies Corollary 2.1 in Wang [4].

Corollary 2.3.

Corollary 2.4 (Doob's type maximal inequality for demimartingales).

Theorem 2.5.

Proof.

Therefore, follows from Lemma 1.6, (2.16), and (2.17); (1.9), (1.10), (2.14), (2.15) hold. This completes the proof of the theorem.

In Theorem 2.5, if we assume that is a nonnegative and nondecreasing convex function on with , then the condition " for each " is satisfied.

Remark 2.6.

Theorem 2.5 generalizes and improves the results of Theorem in Christofides [3] and Theorem in Prakasa Rao [6].

Theorem 2.7.

Proof.

Theorem 2.8.

Proof.

Therefore, (2.22) follows from the above statements immediately.

Corollary 2.9.

By Corollary 2.9, we can get the following theorem.

Theorem 2.10.

Similar to the proof of Theorem 2.8 and using Lemma 1.9(i), we can get the following.

Theorem 2.11.

## Declarations

### Acknowledgments

The authors are most grateful to the Editor Andrei Volodin and an anonymous referee for the careful reading of the manuscript and valuable suggestions which helped in significantly improving an earlier version of this paper. This work was supported by the National Natural Science Foundation of China (Grant no. 10871001, 60803059), Talents Youth Fund of Anhui Province Universities (Grant no. 2010SQRL016ZD), Youth Science Research Fund of Anhui University (Grant no. 2009QN011A), Provincial Natural Science Research Project of Anhui Colleges and the Innovation Group Foundation of Anhui University.

## Authors’ Affiliations

## References

- Chow YS: A martingale inequality and the law of large numbers.
*Proceedings of the American Mathematical Society*1960, 11: 107–111. 10.1090/S0002-9939-1960-0112190-3MATHMathSciNetView ArticleGoogle Scholar - Newman CM, Wright AL: Associated random variables and martingale inequalities.
*Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete*1982, 59(3):361–371. 10.1007/BF00532227MATHMathSciNetView ArticleGoogle Scholar - Christofides TC: Maximal inequalities for demimartingales and a strong law of large numbers.
*Statistics & Probability Letters*2000, 50(4):357–363. 10.1016/S0167-7152(00)00116-4MATHMathSciNetView ArticleGoogle Scholar - Wang JF: Maximal inequalities for associated random variables and demimartingales.
*Statistics & Probability Letters*2004, 66(3):347–354. 10.1016/j.spl.2003.10.021MATHMathSciNetView ArticleGoogle Scholar - Hu SH, Chen GJ, Wang XJ: On extending the Brunk-Prokhorov strong law of large numbers for martingale differences.
*Statistics & Probability Letters*2008, 78(18):3187–3194. 10.1016/j.spl.2008.06.017MATHMathSciNetView ArticleGoogle Scholar - Prakasa Rao BLS: On some maximal inequalities for demisubmartingales and -demisuper martingales.
*Journal of Inequalities in Pure and Applied Mathematics*2007, 8(4, article 112):1–17.Google Scholar - Zheng YH: Inequalities of moment and convergence theorem of order statistics of partial sums for a class of strongly positive dependent stochastic sequence.
*Acta Mathematicae Applicatae Sinica*2001, 24(2):168–176.MATHMathSciNetGoogle Scholar - Fazekas I, Klesov O: A general approach rate to the strong law of large numbers.
*Statistics & Probability Letters*2001, 45(3):436–449.MathSciNetGoogle Scholar - Hu SH, Wang XJ, Yang WZ, Zhao T: The Hàjek-Rènyi-type inequality for associated random variables.
*Statistics & Probability Letters*2009, 79(7):884–888. 10.1016/j.spl.2008.11.014MATHMathSciNetView ArticleGoogle Scholar - Shiryaev AN:
*Probability, Graduate Texts in Mathematics*.*Volume 95*. 2nd edition. Springer, New York, NY, USA; 1996:xvi+623.Google Scholar - Athreya KB, Lahiri SN:
*Measure Theory and Probability Theory, Springer Texts in Statistics*. Springer, New York, NY, USA; 2006:xviii+618.Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.