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Doob's Type Inequality and Strong Law of Large Numbers for Demimartingales
Journal of Inequalities and Applications volumeÂ 2010, ArticleÂ number:Â 838301 (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).
1. Introduction
Definition 1.1.
Let be an sequence of random variables. Assume that for
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.
A finite collection of random variables is said to be associated if
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.
A finite collection of random variables is said to be strongly positive dependent if
for all Borel measurable and increasing (or decreasing) set pairs (A set is said increasing (or decreasing) if implies for any ), where
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 Ndemisupermartingales.
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 ]).
Let be a demimartingale and a nonnegative convex function on with and . Let be a nonincreasing sequence of positive numbers. Then for any ,
Lemma 1.6 (see Fazekas and Klesov [8, Theorem ] and Hu et al. [5, Lemma ]).
Let be a random variable sequence and for . Let be a nondecreasing unbounded sequence of positive numbers and nonnegative numbers. Let and be fixed positive numbers. Assume that for each ,
then
and with the growth rate
where
In addition,
If further assumes one that for infinitely many , then
Lemma 1.7 (see Christofides [3, Lemma , Corollary ]).

(i)
If is a demisubmartingale (or a demimartingale) and is a nondecreasing convex function such that , then is a demisubmartingale.

(ii)
If is a demimartingale, then is a demisubmartingale and is a demisubmartingale.
Lemma 1.8 (see Hu et al. [9, Theorem ]).
Let be a demimartingale and be a nonincreasing sequence of positive numbers. Let and for each , then for any and ,
Lemma 1.9 (see Christofides [3, Corollary , Theorem ]).

(i)
Let be a demisubmartingale. Then for any ,

(ii)
Let be a demisubmartingale and a nonincreasing sequence of positive numbers. Then for any ,
Using Lemma 1.5, Wang [4] obtained the following inequalities for demimartingales.
Theorem 1.10 (see Wang [4, Corollary ]).
Let be a demimartingale and a nonincreasing sequence of positive numbers. Then
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.
Let be a demimartingale and a nonnegative convex function on with . Let be a nonincreasing sequence of positive numbers. . Suppose that for each , then for every ,
Proof.
By Lemma 1.5 and HÃ¶lder's inequality, we have
where is a real number and satisfies Since for each , we can obtain
therefore,
Similar to the proof of (2.3) and using Lemma 1.5 again, we can see that
For constants and , it follows that
Combining (2.6) and (2.7), we have
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.
Let the conditions of Theorem 2.1 be satisfied with for each . Then for every ,
Corollary 2.4 (Doob's type maximal inequality for demimartingales).
Let and be a demimartingale. Suppose that for each , then for every ,
Theorem 2.5.
Let be a demimartingale and a nonnegative convex function on with . Let be a nondecreasing unbounded sequence of positive numbers. . Suppose that for each and
then , and (1.9)(1.10) hold ( is replaced by ), where
In addition,
If further one assumes that for infinitely many , then
Proof.
By the condition of the theorem, we can see that for all . Thus,
follows from (2.9) for each . By (2.12), we have
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.
Let and a demimartingale with for each . Let be a nondecreasing sequence of positive numbers. If
then for any ,
Proof.
Taking , and in Lemma 1.8, we have
Thus, by (2.20) and (2.18), we can get
Theorem 2.8.
Let be a demisubmartingale and a nondecreasing and nonnegative convex function on with and . Let be a nonincreasing sequence of positive numbers. Then for all and each ,
Proof.
By Fubini theorem, it is easy to check that
It follows from Lemma 1.7(i) and Lemma 1.9(ii) that
Therefore, (2.22) follows from the above statements immediately.
Corollary 2.9.
Let the conditions of Theorem 2.8 be satisfied with for each . Then for all and each ,
By Corollary 2.9, we can get the following theorem.
Theorem 2.10.
Let be a demisubmartingale and a nondecreasing and nonnegative convex function on with and . Let be a nondecreasing unbounded sequence of positive numbers. If there exists some such that
then a.s., and (1.9)(1.10) hold ( is replaced by ), where
In addition,
If further one assumes that for infinitely many , then
Similar to the proof of Theorem 2.8 and using Lemma 1.9(i), we can get the following.
Theorem 2.11.
Let be a nonnegative demisubmartingale. Then for all , .
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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.
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Xuejun, W., Shuhe, H., Ting, Z. et al. Doob's Type Inequality and Strong Law of Large Numbers for Demimartingales. J Inequal Appl 2010, 838301 (2010). https://doi.org/10.1155/2010/838301
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DOI: https://doi.org/10.1155/2010/838301