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Some results for demimartingales and N-demimartingales
Journal of Inequalities and Applications volume 2014, Article number: 489 (2014)
Abstract
In this paper, we obtain some results such as maximal and minimal type inequalities for demisubmartingales and demimartingales. Meanwhile, by giving an example, we point out that the Chow type maximal inequality of N-demimartingales is not true, which affects some maximal type inequalities for N-demimartingales.
MSC:60E15, 60F15.
1 Introduction
Let be a sequence of random variables defined on a probability space and .
Definition 1.1 Let be an sequence of random variables. Assume that for ,
for all coordinatewise nondecreasing functions f such that the expectation is defined. Then is called a demimartingale. If in addition the function f is assumed to be nonnegative, the sequence is called a demisubmartingale.
Definition 1.2 Let be an sequence of random variables. Assume that for ,
for all coordinatewise nondecreasing functions f such that the expectation is defined. Then is called an N-demimartingale. If in addition the function f is assumed to be nonnegative, the sequence is called an N-demisupermartingale.
The concepts of demimartingales and demisubmartingales were due to Newman and Wright [1]. It can be checked that a submartingale with the natural choice of σ-algebras is a demisubmartingale, but the converse statement cannot always be true. Newman and Wright [1] proved that the partial sums of mean zero associated random variables form a demimartingale. Similarly, the notion of N-demimartingales and N-demisupermartingales can be found in Christofides [2]. It is trivial to verify that the partial sums of mean zero negatively associated random variables form an N-demimartingale, and a supermartingale with the natural choice of σ-algebras is an N-demisupermartingale, but the converse statement cannot always be true (see Christofides [2]). Various results and examples of demisubmartingales and demimartingales have been obtained. For example, Newman and Wright [1] obtained Doob type maximal inequalities and upcrossing inequality for demisubmartingales; Wood [3] investigated more properties of demimartingales; Christofides [4] generalized the Chow type maximal inequalities for demisubmartingales; Prakasa Rao [5] investigated the Whittle type maximal inequality for demisubmartingales; Christofides [6] constructed some U-statistics based on associated random variables and proved them to be demimartingales; Wang [7] studied some maximal inequalities for associated random variables and demimartingales; Prakasa Rao [8] obtained more maximal and minimal type inequalities for demisubmartingales; Wang and Hu [9] also studied some maximal inequalities for demimartingales and their applications; Wang et al. [10] gave a Doob type inequality and a strong law of large numbers for demimartingales; Wang et al. [11] also studied the maximal and minimal type inequalities for demimartingales; Christofides and Hadjikyriakou [12] gave some maximal and moment inequalities for demimartingales; Hu et al. [13] investigated the Marshall type inequalities for demimartingales; Wang et al. [14] got some maximal inequalities for demimartingales based on concave Young functions. Meanwhile, for the results of N-demisupermartingales and N-demimartingales, Christofides [2] gave some maximal type inequalities for N-demimartingales; Prakasa Rao [15] studied the Chow type maximal inequality for N-demimartingales, Christofides and Hadjikyriakou [16] got some exponential inequalities for N-demimartingales; Hu et al. [17] gave a note on the inequalities for N-demimartingales; Hadjikyriakou [18] obtained a Marcinkiewicz-Zygmund type inequality for nonnegative N-demimartingales; Wang et al. [19] studied some maximal type inequalities for N-demimartingales and provided a strong law of large numbers as an application; Yang and Hu [20] investigated more maximal type inequalities for N-demimartingales, etc. For more results and examples of demimartingales and N-demimartingales, one can refer to Prakasa Rao [21] and Hadjikyriakou [22]. On the other hand, the conditional demimartingales and N-demimartingales have received more attention; we refer to Christofides and Hadjikyriakou [23], Wang and Wang [24], Prakasa Rao [21] and Hadjikyriakou [22], etc.
Inspired by the papers above, we investigate some maximal and minimal type inequalities for demisubmartingales and demimartingales. Meanwhile, by giving an example, we point out that the Chow type maximal inequality of N-demimartingales is not true, which affects some maximal type inequalities for N-demimartingales.
Throughout this paper, let denote the indicator function of the set A and ().
Lemma 1.1 (Christofides [[4], Lemma 2.1])
Let be a demisubmartingale (or a demimartingale) and g be a nondecreasing convex function such that , . Then is a demisubmartingale.
2 Main results
First, we provide a maximal type inequality for a sequence of demisubmartingales.
Theorem 2.1 Let be a demisubmartingale with and assume that is a nondecreasing sequence of positive numbers. Then, for any ,
Proof Following Christofides [4], we give the proof of Theorem 2.1. For fixed , let . Then A can be written as , where , , , and when . Therefore, one has
which is from the facts that , and is a nondecreasing sequence of positive numbers.
Let and . Then f and h are nonnegative nondecreasing functions. By the convexity of the function , we have
and then we can get
Since is a nonnegative nondecreasing function of and is a demisubmartingale, we have
So we can get
Since , one has . Thus we have
By the convexity of the function again,
then
Obviously, and is a nonnegative and componentwise nondecreasing function of , then is a nonnegative and componentwise nondecreasing function of . By the demisubmartingale property, the right-hand side of (2.6) is nonnegative. Thus
and the right-hand side of (2.4) is bounded by
Working in this manner we prove that
By the convexity of the function , we have
Hence
Since , is a nonnegative and componentwise nondecreasing function of . Then is a nonnegative and componentwise nondecreasing function of . As forms a demisubmartingale and is a sequence of positive numbers, we have
Consequently, it follows from (2.7), (2.8) and (2.9) that
So (2.1) is proved. □
Corollary 2.1 Assume that is a demisubmartingale or a demimartingale with . Let g be a nondecreasing convex function such that , and be a nondecreasing sequence of positive numbers. Then, for any ,
Proof By Lemma 1.1, is a demisubmartingale. By Theorem 2.1, we obtain the result of (2.10). □
Remark 2.1 Chow [25] proved a maximal inequality for submartingales, which contains the Hajek-Renyi inequality and other inequalities as special cases (see Theorem 1 of Chow [25]). Christofides [4] generalized Theorem 1 of Chow [25] and obtained a Chow type maximal inequality for demimartingales (see Theorem 2.1 of Christofides [4]). Wang [7] generalized Theorem 2.1 of Christofides [4] to the nonnegative convex functions (see Theorem 2.1 of Wang [7]). Based on Christofides [4] and Wang [7], Wang and Hu [9] obtained some similar maximal inequalities for demisubmartingales and demimartingales (see Theorem 2.1 and Theorem 2.2 of Wang and Hu [9]). Inspired by these papers, we also get some similar Chow type maximal inequality for demisubmartingales and demimartingales (see Theorem 2.1 and Corollary 2.1).
Second, we provide a minimal type inequalities for a sequence of nonnegative demimartingales.
Theorem 2.2 Let be a nonnegative demimartingale with and be a nonincreasing sequence of positive numbers. Then, for any ,
Proof Following Christofides [4], we let , . Then A can be written as , where , , , and when . Thus, similar to the proof of (2.2),
which is from the fact that and . Since is a nonincreasing function of , is a nondecreasing function of . By the definition of a demimartingale, one has
So we can get
Since , is a componentwise nonincreasing function of and is a componentwise nondecreasing function of . By the definition of a demimartingale,
It follows from (2.13) and (2.14) that
By iterations,
Since , is a componentwise nonincreasing function of and is a componentwise nondecreasing function of . By the fact that is a nonnegative demimartingale and is a nonincreasing sequence of positive numbers, it is checked that
Finally, by (2.15) and (2.16), we get
So (2.11) holds. □
Corollary 2.2 Let be a demimartingale. Then, for any ,
Proof By the proof of Theorem 2.2 with , we can get the minimal inequality for demimartingales without the assumption of nonnegativeness. □
Remark 2.2 Newman and Wright [1] obtained some inequalities for demisubmartingales and demimartingales, including maximal and minimal inequalities (see Theorem 3 of Newman and Wright [1]). Prakasa Rao [8] generalized some results of Newman and Wright [1] and got minimal type inequalities for demisubmartingales (see Theorems 2.8-2.10 of Prakasa Rao [8]). Wang et al. [11] also obtained some minimal inequalities for nonnegative demimartingales (see Theorem 2.1, Corollary 2.1 and Corollary 2.2 of Wang et al. [11]). Similar to Theorem 2.8 of Prakasa Rao [8] and Theorem 2.1 of Wang et al. [11], we get some minimal type inequalities for nonnegative demimartingales in Theorem 2.2 and Corollary 2.3. It is pointed out that Corollary 2.2 is not a new result (see Theorem 2.9 of Prakasa Rao [8], Corollary 2.1 of Hu et al. [17], Corollary 2.1 of Wang et al. [11]).
Third, we consider the Chow type maximal inequality for N-demimartingales. Similar to Chow [25] and Christofides [4], Prakasa Rao [15] obtained a Chow type maximal inequality for N-demimartingales.
Theorem 2.3 (see Theorem 3.1 of Prakasa Rao [15] or Theorem 3.5.1 of Prakasa Rao [21])
Assume that is an N-demimartingale with and is a nonnegative nondecreasing function on ℝ with . Let be a function on ℝ with and suppose that
for all x, y, where is a nonnegative and nondecreasing function. Further assume that is a sequence of positive numbers such that for . Define , , . Then
Let and define if and if . By Theorem 2.3,
(see (3.5.10) of Prakasa Rao [21]) was obtained. It can be seen that , , and , , satisfy the condition of (2.17) (see Prakasa Rao [15, 21]).
It is a fact that if is an N-demimartingale, then is also an N-demimartingale (see Christofides [2] or Prakasa Rao [21]). By using Theorem 2.3, Hadjikyriakou [22] got the following maximal inequality for N-demimartingales.
Corollary 2.3 (Hadjikyriakou [[22], Theorem 3.2.1])
Assume that is an N-demimartingale. Then, for every ,
But we find that the Chow type maximal inequality for N-demimartingales, i.e., Theorem 2.3, is not true. We give an example as follows.
An example for N -demimartingales Let , , , . Assume that and are independent random variables with probability distributions
In addition, let and
It is easy to check that for any nondecreasing function f,
Hence is an N-demimartingale. It follows from the distribution of that
Meanwhile,
so
It can be seen that if , then
Otherwise, for the case , one has
Consequently,
On the other hand, we can calculate that
and
But we have
which is contrary to (2.18). Therefore, Theorem 2.3 is not true. In fact, in the proof of Theorem 3.1 of Prakasa Rao [15] or the proof of Theorem 3.5.1 of Prakasa Rao [8], it was given that is a nondecreasing function of . But by checking the proof carefully, we find that one cannot find that is a nondecreasing function of under the conditions of Theorem 2.3.
Similarly, it can be checked that
and
Then
which is contrary to (2.19). So (2.19) is not true.
Meanwhile, one has
and
So
which is contrary to (2.20). Thus, (2.20) is not true. There are some problems of maximal type inequalities for N-demimartingales in the literature such as Wang et al. [19], Hu et al. [13], Wang et al. [14] and Yang and Hu [20]. It is interesting to investigate the maximal type inequalities of N-demimartingales for researchers in the future.
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Acknowledgements
The authors are deeply grateful to the editor and anonymous referees whose insightful comments and suggestions have contributed substantially to the improvement of this paper. This work is supported by the National Natural Science Foundation of China (11171001, 11201001, 11326172), Natural Science Foundation of Anhui Province (1408085QA02), Introduction Projects of Academic and Technology Leaders of Anhui University, Key Program of Research and Development Foundation of Hefei University (13KY05ZD), Key NSF of Anhui Educational Committee (KJ2014A255), Program of Student Science Research Training of Anhui University (KYXL2014015) and Doctoral Research Start-up Funds Projects of Anhui University.
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Dai, P., Shen, Y., Hu, S. et al. Some results for demimartingales and N-demimartingales. J Inequal Appl 2014, 489 (2014). https://doi.org/10.1186/1029-242X-2014-489
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DOI: https://doi.org/10.1186/1029-242X-2014-489