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# Some results for demimartingales and *N*-demimartingales

- Pingping Dai
^{1}, - Yan Shen
^{1}Email author, - Shuhe Hu
^{1}and - Wenzhi Yang
^{1}

**2014**:489

https://doi.org/10.1186/1029-242X-2014-489

© Dai et al.; licensee Springer. 2014

**Received:**11 July 2014**Accepted:**12 November 2014**Published:**12 December 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.

## Keywords

- maximal inequality
- demimartingales
*N*-demimartingales- minimal inequality

## 1 Introduction

Let ${S}_{1},{S}_{2},\dots ,{S}_{n},\dots $ be a sequence of random variables defined on a probability space $(\mathrm{\Omega},\mathcal{F},P)$ and ${S}_{0}=0$.

**Definition 1.1**Let $\{{S}_{j},j\ge 1\}$ be an ${L}^{1}$ sequence of random variables. Assume that for $j=1,2,\dots $ ,

for all coordinatewise nondecreasing functions *f* such that the expectation is defined. Then $\{{S}_{j},j\ge 1\}$ is called a demimartingale. If in addition the function *f* is assumed to be nonnegative, the sequence $\{{S}_{j},j\ge 1\}$ is called a demisubmartingale.

**Definition 1.2**Let $\{{S}_{j},j\ge 1\}$ be an ${L}^{1}$ sequence of random variables. Assume that for $j=1,2,\dots $ ,

for all coordinatewise nondecreasing functions *f* such that the expectation is defined. Then $\{{S}_{j},j\ge 1\}$ is called an *N*-demimartingale. If in addition the function *f* is assumed to be nonnegative, the sequence $\{{S}_{j},j\ge 1\}$ 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 $I(A)$ denote the indicator function of the set *A* and ${x}^{+}=I$ ($x\ge 0$).

**Lemma 1.1** (Christofides [[4], Lemma 2.1])

*Let* $\{{S}_{n},n\ge 1\}$ *be a demisubmartingale* (*or a demimartingale*) *and* *g* *be a nondecreasing convex function such that* $g({S}_{i})\in {L}^{1}$, $i\ge 1$. *Then* $\{g({S}_{n}),n\ge 1\}$ *is a demisubmartingale*.

## 2 Main results

First, we provide a maximal type inequality for a sequence of demisubmartingales.

**Theorem 2.1**

*Let*$\{{S}_{n},n\ge 1\}$

*be a demisubmartingale with*${S}_{0}=0$

*and assume that*$\{{c}_{n},n\ge 1\}$

*is a nondecreasing sequence of positive numbers*.

*Then*,

*for any*$\epsilon >0$,

*Proof*Following Christofides [4], we give the proof of Theorem 2.1. For fixed $n\ge 1$, let $A=\{{max}_{1\le k\le n}{c}_{k}{S}_{k}\ge \epsilon \}$. Then

*A*can be written as $A={\bigcup}_{j=1}^{n}{A}_{j}$, where ${A}_{1}=\{{c}_{1}{S}_{1}\ge \epsilon \}$, ${A}_{j}=\{{c}_{i}{S}_{i}<\epsilon ,1\le i<j,{c}_{j}{S}_{j}\ge \epsilon \}$, $1<j\le n$, and ${A}_{i}\cap {A}_{j}=\mathrm{\varnothing}$ when $i\ne j$. Therefore, one has

which is from the facts that ${A}_{1}\cap {A}_{2}=\mathrm{\varnothing}$, ${I}_{{A}_{2}}={I}_{{A}_{1}\cup {A}_{2}}-{I}_{{A}_{1}}$ and $\{{c}_{k},k\ge 1\}$ is a nondecreasing sequence of positive numbers.

*f*and

*h*are nonnegative nondecreasing functions. By the convexity of the function $f(x)={x}^{+}$, we have

So (2.1) is proved. □

**Corollary 2.1**

*Assume that*$\{{S}_{n},n\ge 1\}$

*is a demisubmartingale or a demimartingale with*${S}_{0}=0$.

*Let*

*g*

*be a nondecreasing convex function such that*$g({S}_{n})\in {L}^{1}$, $n\ge 1$

*and*$\{{c}_{n},n\ge 1\}$

*be a nondecreasing sequence of positive numbers*.

*Then*,

*for any*$\epsilon >0$,

*Proof* By Lemma 1.1, $\{g({S}_{n}),n\ge 1\}$ 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*$\{{S}_{n},n\ge 1\}$

*be a nonnegative demimartingale with*${S}_{0}=0$

*and*$\{{c}_{n},n\ge 1\}$

*be a nonincreasing sequence of positive numbers*.

*Then*,

*for any*$\epsilon >0$,

*Proof*Following Christofides [4], we let $A=\{{min}_{1\le k\le n}{c}_{k}{S}_{k}\le \epsilon \}$, $n\ge 1$. Then

*A*can be written as $A={\bigcup}_{j=1}^{n}{A}_{j}$, where ${A}_{1}=\{{c}_{1}{S}_{1}\le \epsilon \}$, ${A}_{j}=\{{c}_{i}{S}_{i}>\epsilon ,1\le i<j,{c}_{j}{S}_{j}\le \epsilon \}$, $1<j\le n$, and ${A}_{i}\cap {A}_{j}=\mathrm{\varnothing}$ when $i\ne j$. Thus, similar to the proof of (2.2),

So (2.11) holds. □

**Corollary 2.2**

*Let*$\{{S}_{n},n\ge 1\}$

*be a demimartingale*.

*Then*,

*for any*$\epsilon >0$,

*Proof* By the proof of Theorem 2.2 with ${c}_{k}\equiv 1$, 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*$\{{S}_{n},n\ge 1\}$

*is an*

*N*-

*demimartingale with*${S}_{0}=0$

*and*$m(\cdot )$

*is a nonnegative nondecreasing function on*ℝ

*with*$m(0)=0$.

*Let*$g(\cdot )$

*be a function on*ℝ

*with*$g(0)=0$

*and suppose that*

*for all*

*x*,

*y*,

*where*$h(\cdot )$

*is a nonnegative and nondecreasing function*.

*Further assume that*$\{{c}_{k},1\le k\le n\}$

*is a sequence of positive numbers such that*$({c}_{k}-{c}_{k+1})g({S}_{k})\ge 0$

*for*$1\le k\le n-1$.

*Define*${Y}_{k}={max}_{1\le j\le k}{c}_{j}g({S}_{j})$, $k\ge 1$, ${Y}_{0}=0$.

*Then*

(see (3.5.10) of Prakasa Rao [21]) was obtained. It can be seen that $g(x)=-\alpha x$, $\alpha \ge 0$, and $g(x)=-\alpha {x}^{+}$, $\alpha \ge 0$, satisfy the condition of (2.17) (see Prakasa Rao [15, 21]).

It is a fact that if ${\{{S}_{n}\}}_{n\ge 1}$ is an *N*-demimartingale, then ${\{-{S}_{n}\}}_{n\ge 1}$ 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*$\{{S}_{n},n\ge 1\}$

*is an*

*N*-

*demimartingale*.

*Then*,

*for every*$\epsilon >0$,

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 $g(x)=-x$, $m(x)={x}^{+}$, ${c}_{1}={c}_{2}=1$, ${S}_{0}=0$. Assume that ${S}_{1}$ and ${S}_{2}$ are independent random variables with probability distributions

*f*,

*N*-demimartingale. It follows from the distribution of ${S}_{1}$ that

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 $h({S}_{i})m({Y}_{i})$ is a nondecreasing function of ${S}_{1},{S}_{2},\dots ,{S}_{i}$. But by checking the proof carefully, we find that one cannot find that $h({S}_{i})m({Y}_{i})$ is a nondecreasing function of ${S}_{1},{S}_{2},\dots ,{S}_{i}$ under the conditions of Theorem 2.3.

which is contrary to (2.19). So (2.19) is not true.

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.

## Declarations

### 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.

## Authors’ Affiliations

## References

- Newman CM, Wright AL:
**Associated random variables and martingale inequalities.***Z. Wahrscheinlichkeitstheor. Verw. Geb.*1982,**59**(3):361–371. 10.1007/BF00532227MathSciNetView ArticleMATHGoogle Scholar - Christofides TC:
**Maximal inequalities for**N**-demimartingales.***Arch. Inequal. Appl.*2003,**50**(1):397–408.MathSciNetMATHGoogle Scholar - Wood TE:
**Sample paths of demimartingales. Lecture Notes in Mathematics 1080.**In*Probability Theory on Vector Spaces III*. Springer, New York; 1984:365–373.View ArticleGoogle Scholar - Christofides TC:
**Maximal inequalities for demimartingales and a strong law of large numbers.***Stat. Probab. Lett.*2000,**50**(4):357–363. 10.1016/S0167-7152(00)00116-4MathSciNetView ArticleMATHGoogle Scholar - Prakasa Rao BLS:
**Whittle type inequality for demisubmartingales.***Proc. Am. Math. Soc.*2002,**130**(12):3719–3724. 10.1090/S0002-9939-02-06517-6MathSciNetView ArticleMATHGoogle Scholar - Christofides TC: U
**-Statistics on associated random variables.***J. Stat. Plan. Inference*2004,**119**(1):1–15. 10.1016/S0378-3758(02)00418-4MathSciNetView ArticleMATHGoogle Scholar - Wang JF:
**Maximal inequalities for associated random variables and demimartingales.***Stat. Probab. Lett.*2004,**66**(3):347–354. 10.1016/j.spl.2003.10.021View ArticleMathSciNetMATHGoogle Scholar - Prakasa Rao BLS: On some maximal inequalities for demisubmartingales and N -demisupermartingales. J. Inequal. Pure Appl. Math. 2007.,8(4): Article ID 112Google Scholar
- Wang XJ, Hu SH:
**Maximal inequalities for demimartingales and their applications.***Sci. China Ser. A*2009,**52**(10):2207–2217. 10.1007/s11425-009-0067-xMathSciNetView ArticleMATHGoogle Scholar - Wang XJ, Hu SH, Zhao T, Yang WZ:
**Doob’s type inequality and strong law of large numbers for demimartingales.***J. Inequal. Appl.*2010.,**2010:**Article ID 838301Google Scholar - Wang XJ, Hu SH, Yang WZ, Shen Y:
**Some new results for demimartingales.***Appl. Math. J. Chin. Univ. Ser. B*2011,**26**(1):14–22. 10.1007/s11766-011-2303-8MathSciNetView ArticleMATHGoogle Scholar - Christofides TC, Hadjikyriakou M:
**Maximal and moment inequalities for demimartingales and**N**-demimartingales.***Stat. Probab. Lett.*2012,**82**(3):683–691. 10.1016/j.spl.2011.12.009MathSciNetView ArticleMATHGoogle Scholar - Hu SH, Wang XH, Yang WZ, Wang XJ:
**Some inequalities for demimartingales and**N**-demimartingales.***Stat. Probab. Lett.*2012,**82**(2):232–239. 10.1016/j.spl.2011.10.021MathSciNetView ArticleMATHGoogle Scholar - Wang XJ, Prakasa Rao BLS, Hu SH, Yang WZ:
**On some maximal inequalities for demimartingales and**N**-demimartingales based on concave Young functions.***J. Math. Anal. Appl.*2012,**396**(2):434–440. 10.1016/j.jmaa.2012.06.014MathSciNetView ArticleMATHGoogle Scholar - Prakasa Rao BLS:
**On some inequalities for**N**-demimartingales.***J. Indian Soc. Agric. Stat.*2004,**57:**208–216.MathSciNetMATHGoogle Scholar - Christofides TC, Hadjikyriakou M:
**Exponential inequalities for**N**-demimartingales and negatively associated random variables.***Stat. Probab. Lett.*2009,**79**(19):2060–2065. 10.1016/j.spl.2009.06.013MathSciNetView ArticleMATHGoogle Scholar - Hu SH, Yang WZ, Wang XJ, Shen Y:
**A note on the inequalities for**N**-demimartingales and demimartingales.***J. Syst. Sci. Math. Sci.*2010,**30**(8):1052–1058.MathSciNetMATHGoogle Scholar - Hadjikyriakou M:
**Marcinkiewicz-Zygmund inequality for nonnegative**N**-demimartingales and related results.***Stat. Probab. Lett.*2011,**81**(6):678–684. 10.1016/j.spl.2011.02.014MathSciNetView ArticleMATHGoogle Scholar - Wang XJ, Hu SH, Prakasa Rao BLS, Yang WZ:
**Maximal inequalities for**N**-demimartingale and strong law of large numbers.***Stat. Probab. Lett.*2011,**81**(9):1348–1353. 10.1016/j.spl.2011.04.018MathSciNetView ArticleMATHGoogle Scholar - Yang WZ, Hu SH:
**Some maximal type inequalities for**N**-demimartingales and related results.***Math. Inequal. Appl.*2014,**17**(2):731–747.MathSciNetMATHGoogle Scholar - Prakasa Rao BLS:
*Associated Sequences, Demimartingales and Nonparametric Inference*. Springer, Basel; 2012.View ArticleMATHGoogle Scholar - Hadjikyriakou, M: Probability and moment inequalities for demimartingales and associated random variables. Ph.D. dissertation, Department of Mathematics and Statistics, University of Cyprus, Nicosia (2010)Google Scholar
- Christofides TC, Hadjikyriakou M:
**Conditional demimartingales and related results.***J. Math. Anal. Appl.*2013,**398**(1):380–391. 10.1016/j.jmaa.2012.09.004MathSciNetView ArticleMATHGoogle Scholar - Wang XH, Wang XJ:
**Some inequalities for conditional demimartingales and conditional**N**-demimartingales.***Stat. Probab. Lett.*2013,**83**(3):700–709. 10.1016/j.spl.2012.11.017View ArticleMathSciNetMATHGoogle Scholar - Chow YS:
**A martingale inequality and the law of large numbers.***Proc. Am. Math. Soc.*1960,**11**(1):107–111. 10.1090/S0002-9939-1960-0112190-3View ArticleMathSciNetMATHGoogle Scholar

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