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# Some results for demimartingales and N-demimartingales

Journal of Inequalities and Applications20142014:489

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

• Received: 11 July 2014
• Accepted: 12 November 2014
• Published:

## 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 $\left(\mathrm{\Omega },\mathcal{F},P\right)$ and ${S}_{0}=0$.

Definition 1.1 Let $\left\{{S}_{j},j\ge 1\right\}$ be an ${L}^{1}$ sequence of random variables. Assume that for $j=1,2,\dots$ ,
$E\left\{\left({S}_{j+1}-{S}_{j}\right)f\left({S}_{1},\dots ,{S}_{j}\right)\right\}\ge 0$
(1.1)

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

Definition 1.2 Let $\left\{{S}_{j},j\ge 1\right\}$ be an ${L}^{1}$ sequence of random variables. Assume that for $j=1,2,\dots$ ,
$E\left\{\left({S}_{j+1}-{S}_{j}\right)f\left({S}_{1},\dots ,{S}_{j}\right)\right\}\le 0$
(1.2)

for all coordinatewise nondecreasing functions f such that the expectation is defined. Then $\left\{{S}_{j},j\ge 1\right\}$ is called an N-demimartingale. If in addition the function f is assumed to be nonnegative, the sequence $\left\{{S}_{j},j\ge 1\right\}$ is called an N-demisupermartingale.

The concepts of demimartingales and demisubmartingales were due to Newman and Wright . 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  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 . 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 ). Various results and examples of demisubmartingales and demimartingales have been obtained. For example, Newman and Wright  obtained Doob type maximal inequalities and upcrossing inequality for demisubmartingales; Wood  investigated more properties of demimartingales; Christofides  generalized the Chow type maximal inequalities for demisubmartingales; Prakasa Rao  investigated the Whittle type maximal inequality for demisubmartingales; Christofides  constructed some U-statistics based on associated random variables and proved them to be demimartingales; Wang  studied some maximal inequalities for associated random variables and demimartingales; Prakasa Rao  obtained more maximal and minimal type inequalities for demisubmartingales; Wang and Hu  also studied some maximal inequalities for demimartingales and their applications; Wang et al.  gave a Doob type inequality and a strong law of large numbers for demimartingales; Wang et al.  also studied the maximal and minimal type inequalities for demimartingales; Christofides and Hadjikyriakou  gave some maximal and moment inequalities for demimartingales; Hu et al.  investigated the Marshall type inequalities for demimartingales; Wang et al.  got some maximal inequalities for demimartingales based on concave Young functions. Meanwhile, for the results of N-demisupermartingales and N-demimartingales, Christofides  gave some maximal type inequalities for N-demimartingales; Prakasa Rao  studied the Chow type maximal inequality for N-demimartingales, Christofides and Hadjikyriakou  got some exponential inequalities for N-demimartingales; Hu et al.  gave a note on the inequalities for N-demimartingales; Hadjikyriakou  obtained a Marcinkiewicz-Zygmund type inequality for nonnegative N-demimartingales; Wang et al.  studied some maximal type inequalities for N-demimartingales and provided a strong law of large numbers as an application; Yang and Hu  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  and Hadjikyriakou . On the other hand, the conditional demimartingales and N-demimartingales have received more attention; we refer to Christofides and Hadjikyriakou , Wang and Wang , Prakasa Rao  and Hadjikyriakou , 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 $I\left(A\right)$ denote the indicator function of the set A and ${x}^{+}=I$ ($x\ge 0$).

Lemma 1.1 (Christofides [, Lemma 2.1])

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

## 2 Main results

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

Theorem 2.1 Let $\left\{{S}_{n},n\ge 1\right\}$ be a demisubmartingale with ${S}_{0}=0$ and assume that $\left\{{c}_{n},n\ge 1\right\}$ is a nondecreasing sequence of positive numbers. Then, for any $\epsilon >0$,
$\epsilon P\left\{\underset{1\le k\le n}{max}{c}_{k}{S}_{k}\ge \epsilon \right\}\le {c}_{n}E\left[{S}_{n}^{+}I\left(\underset{1\le k\le n}{max}{c}_{k}{S}_{k}\ge \epsilon \right)\right].$
(2.1)
Proof Following Christofides , we give the proof of Theorem 2.1. For fixed $n\ge 1$, let $A=\left\{{max}_{1\le k\le n}{c}_{k}{S}_{k}\ge \epsilon \right\}$. Then A can be written as $A={\bigcup }_{j=1}^{n}{A}_{j}$, where ${A}_{1}=\left\{{c}_{1}{S}_{1}\ge \epsilon \right\}$, ${A}_{j}=\left\{{c}_{i}{S}_{i}<\epsilon ,1\le i, $1, and ${A}_{i}\cap {A}_{j}=\mathrm{\varnothing }$ when $i\ne j$. Therefore, one has
$\begin{array}{rcl}\epsilon P\left(A\right)& =& \epsilon \sum _{j=1}^{n}P\left({A}_{j}\right)=\sum _{j=1}^{n}E\left(\epsilon {I}_{{A}_{j}}\right)\le \sum _{j=1}^{n}E\left({c}_{j}{S}_{j}{I}_{{A}_{j}}\right)=\sum _{j=1}^{n}E\left({c}_{j}{S}_{j}^{+}{I}_{{A}_{j}}\right)\\ =& E\left({c}_{1}{S}_{1}^{+}{I}_{{A}_{1}}\right)+E\left({c}_{2}{S}_{2}^{+}{I}_{{A}_{2}}\right)+\sum _{j=3}^{n}E\left({c}_{j}{S}_{j}^{+}{I}_{{A}_{j}}\right)\\ =& E\left({c}_{1}{S}_{1}^{+}{I}_{{A}_{1}}\right)+E\left[{c}_{2}{S}_{2}^{+}\left({I}_{{A}_{1}\cup {A}_{2}}-{I}_{{A}_{1}}\right)\right]+\sum _{j=3}^{n}E\left({c}_{j}{S}_{j}^{+}{I}_{{A}_{j}}\right)\\ =& E\left({c}_{2}{S}_{2}^{+}{I}_{{A}_{1}\cup {A}_{2}}\right)+E\left[\left({c}_{1}{S}_{1}^{+}-{c}_{2}{S}_{2}^{+}\right){I}_{{A}_{1}}\right]+\sum _{j=3}^{n}E\left({c}_{j}{S}_{j}^{+}{I}_{{A}_{j}}\right)\\ \le & E\left({c}_{2}{S}_{2}^{+}{I}_{{A}_{1}\cup {A}_{2}}\right)+{c}_{2}E\left[\left({S}_{1}^{+}-{S}_{2}^{+}\right){I}_{{A}_{1}}\right]+\sum _{j=3}^{n}E\left({c}_{j}{S}_{j}^{+}{I}_{{A}_{j}}\right)\\ =& E\left({c}_{2}{S}_{2}^{+}{I}_{{A}_{1}\cup {A}_{2}}\right)-{c}_{2}E\left[\left({S}_{2}^{+}-{S}_{1}^{+}\right){I}_{{A}_{1}}\right]+\sum _{j=3}^{n}E\left({c}_{j}{S}_{j}^{+}{I}_{{A}_{j}}\right),\end{array}$
(2.2)

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 $\left\{{c}_{k},k\ge 1\right\}$ is a nondecreasing sequence of positive numbers.

Let $h\left(y\right)={lim}_{x\to {y}^{-}}\left({x}^{+}-{y}^{+}\right)/\left(x-y\right)$ and $f\left(x\right)={x}^{+}=max\left\{0,x\right\}$. Then f and h are nonnegative nondecreasing functions. By the convexity of the function $f\left(x\right)={x}^{+}$, we have
${S}_{2}^{+}-{S}_{1}^{+}\ge \left({S}_{2}-{S}_{1}\right)h\left({S}_{1}\right),$
and then we can get
$E\left[\left({S}_{2}^{+}-{S}_{1}^{+}\right){I}_{{A}_{1}}\right]\ge E\left[\left({S}_{2}-{S}_{1}\right)h\left({S}_{1}\right){I}_{{A}_{1}}\right].$
Since $h\left({S}_{1}\right){I}_{{A}_{1}}$ is a nonnegative nondecreasing function of ${S}_{1}$ and $\left\{{S}_{n},n\ge 1\right\}$ is a demisubmartingale, we have
$E\left[\left({S}_{2}^{+}-{S}_{1}^{+}\right){I}_{{A}_{1}}\right]\ge E\left[\left({S}_{2}-{S}_{1}\right)h\left({S}_{1}\right){I}_{{A}_{1}}\right]\ge 0.$
So we can get
$\begin{array}{rl}\epsilon P\left(A\right)& \le E\left({c}_{2}{S}_{2}^{+}{I}_{{A}_{1}\cup {A}_{2}}\right)+\sum _{j=3}^{n}E\left({c}_{j}{S}_{j}^{+}{I}_{{A}_{j}}\right)\\ =E\left({c}_{2}{S}_{2}^{+}{I}_{{A}_{1}\cup {A}_{2}}\right)+E\left({c}_{3}{S}_{3}^{+}{I}_{{A}_{3}}\right)+\sum _{j=4}^{n}E\left({c}_{j}{S}_{j}^{+}{I}_{{A}_{j}}\right).\end{array}$
(2.3)
Since ${A}_{1}\cap {A}_{2}\cap {A}_{3}=\mathrm{\varnothing }$, one has ${I}_{{A}_{3}}={I}_{{A}_{1}\cup {A}_{2}\cup {A}_{3}}-{I}_{{A}_{1}\cup {A}_{2}}$. Thus we have
$\begin{array}{rcl}\epsilon P\left(A\right)& \le & E\left({c}_{2}{S}_{2}^{+}{I}_{{A}_{1}\cup {A}_{2}}\right)+E\left[{c}_{3}{S}_{3}^{+}\left({I}_{{A}_{1}\cup {A}_{2}\cup {A}_{3}}-{I}_{{A}_{1}\cup {A}_{2}}\right)\right]+\sum _{j=4}^{n}E\left({c}_{j}{S}_{j}^{+}{I}_{{A}_{j}}\right)\\ =& E\left({c}_{3}{S}_{3}^{+}{I}_{{A}_{1}\cup {A}_{2}\cup {A}_{3}}\right)+E\left[\left({c}_{2}{S}_{2}^{+}-{c}_{3}{S}_{3}^{+}\right){I}_{{A}_{1}\cup {A}_{2}}\right]+\sum _{j=4}^{n}E\left({c}_{j}{S}_{j}^{+}{I}_{{A}_{j}}\right)\\ \le & E\left({c}_{3}{S}_{3}^{+}{I}_{{A}_{1}\cup {A}_{2}\cup {A}_{3}}\right)+{c}_{3}E\left[\left({S}_{2}^{+}-{S}_{3}^{+}\right){I}_{{A}_{1}\cup {A}_{2}}\right]+\sum _{j=4}^{n}E\left({c}_{j}{S}_{j}^{+}{I}_{{A}_{j}}\right)\\ =& E\left({c}_{3}{S}_{3}^{+}{I}_{{A}_{1}\cup {A}_{2}\cup {A}_{3}}\right)-{c}_{3}E\left[\left({S}_{3}^{+}-{S}_{2}^{+}\right){I}_{{A}_{1}\cup {A}_{2}}\right]+\sum _{j=4}^{n}E\left({c}_{j}{S}_{j}^{+}{I}_{{A}_{j}}\right).\end{array}$
(2.4)
By the convexity of the function $f\left(x\right)={x}^{+}$ again,
${S}_{3}^{+}-{S}_{2}^{+}\ge \left({S}_{3}-{S}_{2}\right)h\left({S}_{2}\right),$
(2.5)
then
$E\left[\left({S}_{3}^{+}-{S}_{2}^{+}\right){I}_{{A}_{1}\cup {A}_{2}}\right]\ge E\left[\left({S}_{3}-{S}_{2}\right)h\left({S}_{2}\right){I}_{{A}_{1}\cup {A}_{2}}\right].$
(2.6)
Obviously, ${A}_{1}\cup {A}_{2}=\left\{max\left({c}_{1}{S}_{1},{c}_{2}{S}_{2}\right)\ge \epsilon \right\}$ and ${I}_{{A}_{1}\cup {A}_{2}}$ is a nonnegative and componentwise nondecreasing function of $\left\{{S}_{1},{S}_{2}\right\}$, then $h\left({S}_{2}\right){I}_{{A}_{1}\cup {A}_{2}}$ is a nonnegative and componentwise nondecreasing function of $\left\{{S}_{1},{S}_{2}\right\}$. By the demisubmartingale property, the right-hand side of (2.6) is nonnegative. Thus
$E\left[\left({S}_{3}^{+}-{S}_{2}^{+}\right){I}_{{A}_{1}\cup {A}_{2}}\right]\ge 0$
and the right-hand side of (2.4) is bounded by
$E\left({c}_{3}{S}_{3}^{+}{I}_{{A}_{1}\cup {A}_{2}\cup {A}_{3}}\right)+\sum _{j=4}^{n}E\left({c}_{j}{S}_{j}^{+}{I}_{{A}_{j}}\right).$
Working in this manner we prove that
$\begin{array}{rcl}\epsilon P\left(A\right)& \le & E\left({c}_{n-1}{S}_{n-1}^{+}{I}_{{A}_{1}\cup {A}_{2}\cup \cdots \cup {A}_{n-1}}\right)+E\left({c}_{n}{S}_{n}^{+}{I}_{{A}_{n}}\right)\\ =& E\left({c}_{n-1}{S}_{n-1}^{+}{I}_{{A}_{1}\cup {A}_{2}\cup \cdots \cup {A}_{n-1}}\right)+E\left[{c}_{n}{S}_{n}^{+}\left({I}_{{A}_{1}\cup {A}_{2}\cup \cdots \cup {A}_{n}}-{I}_{{A}_{1}\cup {A}_{2}\cup \cdots \cup {A}_{n-1}}\right)\right]\\ \le & {c}_{n}E\left({S}_{n}^{+}{I}_{A}\right)-{c}_{n}E\left[\left({S}_{n}^{+}-{S}_{n-1}^{+}\right){I}_{{A}_{1}\cup {A}_{2}\cup \cdots \cup {A}_{n-1}}\right].\end{array}$
(2.7)
By the convexity of the function $f\left(x\right)={x}^{+}$, we have
${S}_{n}^{+}-{S}_{n-1}^{+}\ge \left({S}_{n}-{S}_{n-1}\right)h\left({S}_{n-1}\right).$
Hence
$E\left[\left({S}_{n}^{+}-{S}_{n-1}^{+}\right){I}_{{A}_{1}\cup {A}_{2}\cup \cdots \cup {A}_{n-1}}\right]\ge E\left[\left({S}_{n}-{S}_{n-1}\right)h\left({S}_{n-1}\right){I}_{{A}_{1}\cup {A}_{2}\cup \cdots \cup {A}_{n-1}}\right].$
(2.8)
Since ${A}_{1}\cup {A}_{2}\cup \cdots \cup {A}_{n-1}=\left\{max\left({c}_{1}{S}_{1},{c}_{2}{S}_{2},\dots ,{c}_{n-1}{S}_{n-1}\right)\ge \epsilon \right\}$, ${I}_{{A}_{1}\cup {A}_{2}\cup \cdots \cup {A}_{n-1}}$ is a nonnegative and componentwise nondecreasing function of $\left\{{S}_{1},{S}_{2},\dots ,{S}_{n-1}\right\}$. Then $h\left({S}_{n-1}\right){I}_{{A}_{1}\cup {A}_{2}\cup \cdots \cup {A}_{n-1}}$ is a nonnegative and componentwise nondecreasing function of $\left\{{S}_{1},{S}_{2},\dots ,{S}_{n-1}\right\}$. As $\left\{{S}_{n},n\ge 1\right\}$ forms a demisubmartingale and $\left\{{c}_{n},n\ge 1\right\}$ is a sequence of positive numbers, we have
${c}_{n}E\left[\left({S}_{n}-{S}_{n-1}\right)h\left({S}_{n-1}\right){I}_{{A}_{1}\cup {A}_{2}\cup \cdots \cup {A}_{n-1}}\right]\ge 0.$
(2.9)
Consequently, it follows from (2.7), (2.8) and (2.9) that
$\epsilon P\left(A\right)\le E\left({c}_{n}{S}_{n}^{+}{I}_{A}\right).$

So (2.1) is proved. □

Corollary 2.1 Assume that $\left\{{S}_{n},n\ge 1\right\}$ is a demisubmartingale or a demimartingale with ${S}_{0}=0$. Let g be a nondecreasing convex function such that $g\left({S}_{n}\right)\in {L}^{1}$, $n\ge 1$ and $\left\{{c}_{n},n\ge 1\right\}$ be a nondecreasing sequence of positive numbers. Then, for any $\epsilon >0$,
$\epsilon P\left\{\underset{1\le k\le n}{max}{c}_{k}g\left({S}_{k}\right)\ge \epsilon \right\}\le E\left[{c}_{n}{g}^{+}\left({S}_{n}\right)I\left(\underset{1\le k\le n}{max}{c}_{k}g\left({S}_{k}\right)\ge \epsilon \right)\right].$
(2.10)

Proof By Lemma 1.1, $\left\{g\left({S}_{n}\right),n\ge 1\right\}$ is a demisubmartingale. By Theorem 2.1, we obtain the result of (2.10). □

Remark 2.1 Chow  proved a maximal inequality for submartingales, which contains the Hajek-Renyi inequality and other inequalities as special cases (see Theorem 1 of Chow ). Christofides  generalized Theorem 1 of Chow  and obtained a Chow type maximal inequality for demimartingales (see Theorem 2.1 of Christofides ). Wang  generalized Theorem 2.1 of Christofides  to the nonnegative convex functions (see Theorem 2.1 of Wang ). Based on Christofides  and Wang , Wang and Hu  obtained some similar maximal inequalities for demisubmartingales and demimartingales (see Theorem 2.1 and Theorem 2.2 of Wang and Hu ). 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 $\left\{{S}_{n},n\ge 1\right\}$ be a nonnegative demimartingale with ${S}_{0}=0$ and $\left\{{c}_{n},n\ge 1\right\}$ be a nonincreasing sequence of positive numbers. Then, for any $\epsilon >0$,
$\epsilon P\left\{\underset{1\le k\le n}{min}{c}_{k}{S}_{k}\le \epsilon \right\}\ge {c}_{n}E\left[{S}_{n}I\left(\underset{1\le k\le n}{min}{c}_{k}{S}_{k}\le \epsilon \right)\right].$
(2.11)
Proof Following Christofides , we let $A=\left\{{min}_{1\le k\le n}{c}_{k}{S}_{k}\le \epsilon \right\}$, $n\ge 1$. Then A can be written as $A={\bigcup }_{j=1}^{n}{A}_{j}$, where ${A}_{1}=\left\{{c}_{1}{S}_{1}\le \epsilon \right\}$, ${A}_{j}=\left\{{c}_{i}{S}_{i}>\epsilon ,1\le i, $1, and ${A}_{i}\cap {A}_{j}=\mathrm{\varnothing }$ when $i\ne j$. Thus, similar to the proof of (2.2),
$\begin{array}{rcl}\epsilon P\left(A\right)& =& \epsilon \sum _{j=1}^{n}P\left({A}_{j}\right)=\sum _{j=1}^{n}E\left(\epsilon {I}_{{A}_{j}}\right)\ge \sum _{j=1}^{n}E\left({c}_{j}{S}_{j}{I}_{{A}_{j}}\right)\\ =& E\left({c}_{1}{S}_{1}{I}_{{A}_{1}}\right)+E\left({c}_{2}{S}_{2}{I}_{{A}_{2}}\right)+\sum _{j=3}^{n}E\left({c}_{j}{S}_{j}{I}_{{A}_{j}}\right)\\ =& E\left({c}_{1}{S}_{1}{I}_{{A}_{1}}\right)+E\left[{c}_{2}{S}_{2}\left({I}_{{A}_{1}\cup {A}_{2}}-{I}_{{A}_{1}}\right)\right]+\sum _{j=3}^{n}E\left({c}_{j}{S}_{j}{I}_{{A}_{j}}\right)\\ =& E\left({c}_{2}{S}_{2}{I}_{{A}_{1}\cup {A}_{2}}\right)+E\left[\left({c}_{1}{S}_{1}-{c}_{2}{S}_{2}\right){I}_{{A}_{1}}\right]+\sum _{j=3}^{n}E\left({c}_{j}{S}_{j}{I}_{{A}_{j}}\right)\\ \ge & E\left({c}_{2}{S}_{2}{I}_{{A}_{1}\cup {A}_{2}}\right)+{c}_{2}E\left[\left({S}_{1}-{S}_{2}\right){I}_{{A}_{1}}\right]+\sum _{j=3}^{n}E\left({c}_{j}{S}_{j}{I}_{{A}_{j}}\right)\\ =& E\left({c}_{2}{S}_{2}{I}_{{A}_{1}\cup {A}_{2}}\right)+{c}_{2}E\left[\left({S}_{2}-{S}_{1}\right)\left(-{I}_{{A}_{1}}\right)\right]+\sum _{j=3}^{n}E\left({c}_{j}{S}_{j}{I}_{{A}_{j}}\right),\end{array}$
which is from the fact that ${A}_{1}\cap {A}_{2}=\mathrm{\varnothing }$ and ${I}_{{A}_{2}}={I}_{{A}_{1}\cup {A}_{2}}-{I}_{{A}_{1}}$. Since ${I}_{{A}_{1}}$ is a nonincreasing function of ${S}_{1}$, $-{I}_{{A}_{1}}$ is a nondecreasing function of ${S}_{1}$. By the definition of a demimartingale, one has
$E\left[\left({S}_{2}-{S}_{1}\right)\left(-{I}_{{A}_{1}}\right)\right]\ge 0.$
(2.12)
So we can get
$\begin{array}{rl}\epsilon P\left(A\right)& \ge E\left({c}_{2}{S}_{2}{I}_{{A}_{1}\cup {A}_{2}}\right)+\sum _{j=3}^{n}E\left({c}_{j}{S}_{j}{I}_{{A}_{j}}\right)\\ =E\left({c}_{2}{S}_{2}{I}_{{A}_{1}\cup {A}_{2}}\right)+E\left({c}_{3}{S}_{3}{I}_{{A}_{3}}\right)+\sum _{j=4}^{n}E\left({c}_{j}{S}_{j}{I}_{{A}_{j}}\right)\\ =E\left({c}_{2}{S}_{2}{I}_{{A}_{1}\cup {A}_{2}}\right)+E\left[{c}_{3}{S}_{3}\left({I}_{{A}_{1}\cup {A}_{2}\cup {A}_{3}}-{I}_{{A}_{1}\cup {A}_{2}}\right)\right]+\sum _{j=4}^{n}E\left({c}_{j}{S}_{j}{I}_{{A}_{j}}\right)\\ =E\left({c}_{3}{S}_{3}{I}_{{A}_{1}\cup {A}_{2}\cup {A}_{3}}\right)+E\left[\left({c}_{2}{S}_{2}-{c}_{3}{S}_{3}\right){I}_{{A}_{1}\cup {A}_{2}}\right]+\sum _{j=4}^{n}E\left({c}_{j}{S}_{j}{I}_{{A}_{j}}\right)\\ \ge E\left({c}_{3}{S}_{3}{I}_{{A}_{1}\cup {A}_{2}\cup {A}_{3}}\right)+{c}_{3}E\left[\left({S}_{2}-{S}_{3}\right){I}_{{A}_{1}\cup {A}_{2}}\right]+\sum _{j=4}^{n}E\left({c}_{j}{S}_{j}{I}_{{A}_{j}}\right)\\ =E\left({c}_{3}{S}_{3}{I}_{{A}_{1}\cup {A}_{2}\cup {A}_{3}}\right)+{c}_{3}E\left[\left({S}_{3}-{S}_{2}\right)\left(-{I}_{{A}_{1}\cup {A}_{2}}\right)\right]+\sum _{j=4}^{n}E\left({c}_{j}{S}_{j}{I}_{{A}_{j}}\right).\end{array}$
(2.13)
Since ${A}_{1}\cup {A}_{2}=\left\{min\left({c}_{1}{S}_{1},{c}_{2}{S}_{2}\right)\le \epsilon \right\}$, ${I}_{{A}_{1}\cup {A}_{2}}$ is a componentwise nonincreasing function of $\left\{{S}_{1},{S}_{2}\right\}$ and $-{I}_{{A}_{1}\cup {A}_{2}}$ is a componentwise nondecreasing function of $\left\{{S}_{1},{S}_{2}\right\}$. By the definition of a demimartingale,
$E\left[\left({S}_{3}-{S}_{2}\right)\left(-{I}_{{A}_{1}\cup {A}_{2}}\right)\right]\ge 0.$
(2.14)
It follows from (2.13) and (2.14) that
$\epsilon P\left(A\right)\ge E\left({c}_{3}{S}_{3}{I}_{{A}_{1}\cup {A}_{2}\cup {A}_{3}}\right)+\sum _{j=4}^{n}E\left({c}_{j}{S}_{j}{I}_{{A}_{j}}\right).$
By iterations,
$\begin{array}{rcl}\epsilon P\left(A\right)& \ge & E\left({c}_{n-1}{S}_{n-1}{I}_{{A}_{1}\cup {A}_{2}\cup \cdots \cup {A}_{n-1}}\right)+E\left({c}_{n}{S}_{n}{I}_{{A}_{n}}\right).\\ =& E\left({c}_{n-1}{S}_{n-1}{I}_{{A}_{1}\cup {A}_{2}\cup \cdots \cup {A}_{n-1}}\right)+E\left[{c}_{n}{S}_{n}\left({I}_{{A}_{1}\cup {A}_{2}\cup \cdots \cup {A}_{n}}-{I}_{{A}_{1}\cup {A}_{2}\cup \cdots \cup {A}_{n-1}}\right)\right]\\ \ge & {c}_{n}E\left({S}_{n}{I}_{A}\right)+{c}_{n}E\left[\left({S}_{n}-{S}_{n-1}\right)\left(-{I}_{{A}_{1}\cup {A}_{2}\cup \cdots \cup {A}_{n-1}}\right)\right].\end{array}$
(2.15)
Since ${A}_{1}\cup {A}_{2}\cup \cdots \cup {A}_{n-1}=\left\{min\left({c}_{1}{S}_{1},{c}_{2}{S}_{2},\dots ,{c}_{n-1}{S}_{n-1}\right)\le \epsilon \right\}$, ${I}_{{A}_{1}\cup {A}_{2}\cup \cdots \cup {A}_{n-1}}$ is a componentwise nonincreasing function of $\left\{{S}_{1},{S}_{2},\dots ,{S}_{n-1}\right\}$ and $-{I}_{{A}_{1}\cup {A}_{2}\cup \cdots \cup {A}_{n-1}}$ is a componentwise nondecreasing function of $\left\{{S}_{1},{S}_{2},\dots ,{S}_{n-1}\right\}$. By the fact that $\left\{{S}_{n},n\ge 1\right\}$ is a nonnegative demimartingale and $\left\{{c}_{k},k\ge 1\right\}$ is a nonincreasing sequence of positive numbers, it is checked that
${c}_{n}E\left[\left({S}_{n}-{S}_{n-1}\right)\left(-{I}_{{A}_{1}\cup {A}_{2}\cup \cdots \cup {A}_{n-1}}\right)\right]\ge 0.$
(2.16)
Finally, by (2.15) and (2.16), we get
$\epsilon P\left(A\right)\ge {c}_{n}E\left({S}_{n}{I}_{A}\right).$

So (2.11) holds. □

Corollary 2.2 Let $\left\{{S}_{n},n\ge 1\right\}$ be a demimartingale. Then, for any $\epsilon >0$,
$\epsilon P\left\{\underset{1\le k\le n}{min}{S}_{k}\le \epsilon \right\}\ge {\int }_{\left\{{min}_{1\le k\le n}{S}_{k}\le \epsilon \right\}}{S}_{n}\phantom{\rule{0.2em}{0ex}}dP.$

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  obtained some inequalities for demisubmartingales and demimartingales, including maximal and minimal inequalities (see Theorem 3 of Newman and Wright ). Prakasa Rao  generalized some results of Newman and Wright  and got minimal type inequalities for demisubmartingales (see Theorems 2.8-2.10 of Prakasa Rao ). Wang et al.  also obtained some minimal inequalities for nonnegative demimartingales (see Theorem 2.1, Corollary 2.1 and Corollary 2.2 of Wang et al. ). Similar to Theorem 2.8 of Prakasa Rao  and Theorem 2.1 of Wang et al. , 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 , Corollary 2.1 of Hu et al. , Corollary 2.1 of Wang et al. ).

Third, we consider the Chow type maximal inequality for N-demimartingales. Similar to Chow  and Christofides , Prakasa Rao  obtained a Chow type maximal inequality for N-demimartingales.

Theorem 2.3 (see Theorem 3.1 of Prakasa Rao  or Theorem 3.5.1 of Prakasa Rao )

Assume that $\left\{{S}_{n},n\ge 1\right\}$ is an N-demimartingale with ${S}_{0}=0$ and $m\left(\cdot \right)$ is a nonnegative nondecreasing function on with $m\left(0\right)=0$. Let $g\left(\cdot \right)$ be a function on with $g\left(0\right)=0$ and suppose that
$g\left(x\right)-g\left(y\right)\ge \left(y-x\right)h\left(y\right)$
(2.17)
for all x, y, where $h\left(\cdot \right)$ is a nonnegative and nondecreasing function. Further assume that $\left\{{c}_{k},1\le k\le n\right\}$ is a sequence of positive numbers such that $\left({c}_{k}-{c}_{k+1}\right)g\left({S}_{k}\right)\ge 0$ for $1\le k\le n-1$. Define ${Y}_{k}={max}_{1\le j\le k}{c}_{j}g\left({S}_{j}\right)$, $k\ge 1$, ${Y}_{0}=0$. Then
$E\left({\int }_{0}^{{Y}_{n}}u\phantom{\rule{0.2em}{0ex}}dm\left(u\right)\right)\le \sum _{i=1}^{n}{c}_{i}E\left[\left(g\left({S}_{i}\right)-g\left({S}_{i-1}\right)\right)m\left({Y}_{n}\right)\right].$
(2.18)
Let $\epsilon >0$ and define $m\left(t\right)=1$ if $t\ge \epsilon$ and $m\left(t\right)=0$ if $t<\epsilon$. By Theorem 2.3,
$\epsilon P\left({Y}_{n}\ge \epsilon \right)\le \sum _{i=1}^{n}{c}_{i}E\left[\left(g\left({S}_{i}\right)-g\left({S}_{i-1}\right)\right)I\left({Y}_{n}\ge \epsilon \right)\right]$
(2.19)

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

It is a fact that if ${\left\{{S}_{n}\right\}}_{n\ge 1}$ is an N-demimartingale, then ${\left\{-{S}_{n}\right\}}_{n\ge 1}$ is also an N-demimartingale (see Christofides  or Prakasa Rao ). By using Theorem 2.3, Hadjikyriakou  got the following maximal inequality for N-demimartingales.

Corollary 2.3 (Hadjikyriakou [, Theorem 3.2.1])

Assume that $\left\{{S}_{n},n\ge 1\right\}$ is an N-demimartingale. Then, for every $\epsilon >0$,
$\epsilon P\left(\underset{1\le k\le n}{max}{S}_{k}\ge \epsilon \right)\le E\left({S}_{n}I\left(\underset{1\le k\le n}{max}{S}_{k}\ge \epsilon \right)\right).$
(2.20)

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 $g\left(x\right)=-x$, $m\left(x\right)={x}^{+}$, ${c}_{1}={c}_{2}=1$, ${S}_{0}=0$. Assume that ${S}_{1}$ and ${S}_{2}$ are independent random variables with probability distributions
${S}_{1}\sim \left(\begin{array}{cc}-2& 2\\ \frac{1}{2}& \frac{1}{2}\end{array}\right),\phantom{\rule{2em}{0ex}}{S}_{2}\sim \left(\begin{array}{cc}-1& 1\\ \frac{1}{2}& \frac{1}{2}\end{array}\right).$
In addition, let ${Y}_{1}={c}_{1}g\left({S}_{1}\right)=-{S}_{1}$ and
${Y}_{2}=max\left\{{c}_{1}g\left({S}_{1}\right),{c}_{2}g\left({S}_{2}\right)\right\}=max\left\{-{S}_{1},-{S}_{2}\right\}.$
It is easy to check that for any nondecreasing function f,
$\begin{array}{r}E\left[\left({S}_{2}-{S}_{1}\right)f\left({S}_{1}\right)\right]\\ \phantom{\rule{1em}{0ex}}=\left[-1-\left(-2\right)\right]f\left(-2\right)×\frac{1}{4}+\left(-1-2\right)f\left(2\right)×\frac{1}{4}+\left[1-\left(-2\right)\right]f\left(-2\right)\\ \phantom{\rule{2em}{0ex}}×\frac{1}{4}+\left(1-2\right)f\left(2\right)×\frac{1}{4}\\ \phantom{\rule{1em}{0ex}}=f\left(-2\right)-f\left(2\right)\le 0.\end{array}$
Hence $\left\{{S}_{1},{S}_{2}\right\}$ is an N-demimartingale. It follows from the distribution of ${S}_{1}$ that
${Y}_{1}\sim \left(\begin{array}{cc}-2& 2\\ \frac{1}{2}& \frac{1}{2}\end{array}\right).$
Meanwhile,
$\begin{array}{c}P\left({Y}_{2}=1\right)=P\left({S}_{1}=2,{S}_{2}=-1\right)=\frac{1}{4},\hfill \\ P\left({Y}_{2}=-1\right)=P\left({S}_{1}=2,{S}_{2}=1\right)=\frac{1}{4},\hfill \\ P\left({Y}_{2}=2\right)=P\left({S}_{1}=-2,{S}_{2}=1\right)+P\left({S}_{1}=-2,{S}_{2}=-1\right)=\frac{1}{4}+\frac{1}{4}=\frac{1}{2},\hfill \end{array}$
so
${Y}_{2}\sim \left(\begin{array}{ccc}-1& 1& 2\\ \frac{1}{4}& \frac{1}{4}& \frac{1}{2}\end{array}\right).$
It can be seen that if ${Y}_{2}\left(\omega \right)\ge 0$, then
${\int }_{0}^{{Y}_{2}}u\phantom{\rule{0.2em}{0ex}}dm\left(u\right)={\int }_{0}^{{Y}_{2}}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{2}{Y}_{2}^{2}.$
Otherwise, for the case ${Y}_{2}\left(\omega \right)<0$, one has
${\int }_{0}^{{Y}_{2}}u\phantom{\rule{0.2em}{0ex}}dm\left(u\right)=0.$
Consequently,
${\int }_{0}^{{Y}_{2}}u\phantom{\rule{0.2em}{0ex}}dm\left(u\right)=\frac{1}{2}{Y}_{2}^{2}I\left({Y}_{2}\ge 0\right)\sim \left(\begin{array}{ccc}0& \frac{1}{2}& 2\\ \frac{1}{4}& \frac{1}{4}& \frac{1}{2}\end{array}\right).$
On the other hand, we can calculate that
$E\left[{\int }_{0}^{{Y}_{2}}u\phantom{\rule{0.2em}{0ex}}dm\left(u\right)\right]=\frac{9}{8}$
and
$\begin{array}{rcl}\sum _{k=1}^{2}{c}_{k}E\left\{\left[g\left({S}_{k}\right)-g\left({S}_{k-1}\right)\right]m\left({Y}_{2}\right)\right\}& =& E\left[g\left({S}_{2}\right)m\left({Y}_{2}\right)\right]=-E\left({S}_{2}{Y}_{2}^{+}\right)\\ =& -\left[-1×1×\frac{1}{4}+1×2×\frac{1}{4}+\left(-1\right)×2×\frac{1}{4}\right]\\ =& \frac{1}{4}.\end{array}$
But we have
$\frac{9}{8}=E\left[{\int }_{0}^{{Y}_{2}}u\phantom{\rule{0.2em}{0ex}}dm\left(u\right)\right]>\sum _{k=1}^{2}{c}_{k}E\left\{\left[g\left({S}_{k}\right)-g\left({S}_{k-1}\right)\right]m\left({Y}_{2}\right)\right\}=\frac{1}{4},$

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  or the proof of Theorem 3.5.1 of Prakasa Rao , it was given that $h\left({S}_{i}\right)m\left({Y}_{i}\right)$ 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 $h\left({S}_{i}\right)m\left({Y}_{i}\right)$ is a nondecreasing function of ${S}_{1},{S}_{2},\dots ,{S}_{i}$ under the conditions of Theorem 2.3.

Similarly, it can be checked that
$P\left({Y}_{2}\ge 1\right)=\frac{1}{2}+\frac{1}{4}=\frac{3}{4},$
and
$\begin{array}{rl}\sum _{k=1}^{2}E\left[\left(g\left({S}_{k}\right)-g\left({S}_{k-1}\right)\right)I\left({Y}_{2}\ge 1\right)\right]& =-E\left[{S}_{2}I\left({Y}_{2}\ge 1\right)\right]\\ =-\left[-1×1×\frac{1}{4}+1×1×\frac{1}{4}+\left(-1\right)×1×\frac{1}{4}\right]\\ =\frac{1}{4}.\end{array}$
Then
$P\left({Y}_{2}\ge 1\right)>-E\left[{S}_{2}I\left({Y}_{2}\ge 1\right)\right],$

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

Meanwhile, one has
$P\left(\underset{1\le k\le 2}{max}{S}_{k}\ge 1\right)=P\left({S}_{1}=2,{S}_{2}=-1\right)+P\left({S}_{1}=2,{S}_{2}=1\right)+P\left({S}_{1}=-2,{S}_{2}=1\right)=\frac{3}{4}$
and
$E\left[{S}_{2}I\left(max\left({S}_{1},{S}_{2}\right)\right)\ge 1\right]=-1×\frac{1}{4}+1×\frac{1}{4}+1×\frac{1}{4}=\frac{1}{4}.$
So
$\frac{3}{4}=P\left(\underset{1\le k\le 2}{max}{S}_{k}\ge 1\right)>E\left[{S}_{2}I\left(max\left({S}_{1},{S}_{2}\right)\right)\ge 1\right]=\frac{1}{4},$

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. , Hu et al. , Wang et al.  and Yang and Hu . 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

(1)
School of Mathematical Science, Anhui University, Hefei, 230039, P.R. China

## References 