Note on complete convergence and complete moment convergence for negatively dependent random variables under sub-linear expectations

1.   Introduction

Peng ^[1,2] introduced seminal concepts of the sub-linear expectations space to study the uncertainty in probability. The works of Peng ^[1,2] stimulate many scholars to investigate the results under sub-linear expectations space, extending those in classic probability space. Zhang ^[3,4] got exponential inequalities and Rosenthal's inequality under sub-linear expectations. For more limit theorems under sub-linear expectations, the readers could refer to Zhang ^[5], Xu and Zhang ^[6,7], Wu and Jiang ^[8], Zhang and Lin ^[9], Zhong and Wu ^[10], Chen ^[11], Chen and Wu ^[12], Zhang ^[13], Hu et al. ^[14], Gao and Xu ^[15], Kuczmaszewska ^[16], Xu and Cheng ^[17,18,19], Xu et al. ^[20] and references therein.

In probability space, Shen et al. ^[21] obtained equivalent conditions of complete convergence and complete moment convergence for extended negatively dependent random variables. For references on complete moment convergence and complete convergence in probability space, the reader could refer to Hsu and Robbins ^[22], Chow ^[23], Ko ^[24], Meng et al. ^[25], Hosseini and Nezakati ^[26], Meng et al. ^[27] and refercences therein. Inspired by the work of Shen et al. ^[21], we try to investigate complete convergence and complete moment convergence for negatively dependent (ND) random variables under sub-linear expectations, and the Marcinkiewicz-Zygmund type result for ND random variables under sub-linear expectations, which complements the relevant results in Shen et al. ^[21].

Recently, Srivastava et al. ^[28] introduced and studied comcept of statistical probability convergence. Srivastava et al. ^[29] investigated the relevant results of statistical probability convergence via deferred Nörlund summability mean. For more recent works, the interested reader could refer to Srivastava et al. ^[30,31,32], Paikary et al. ^[33] and references therein. We conjecture the relevant notions and results of statistical probability convergence could be extended to that under sub-linear expectation.

We organize the remainders of this article as follows. We cite relevant basic notions, concepts and properties, and present relevant lemmas under sub-linear expectations in Section 2. In Section 3, we give our main results, Theorems 3.1 and 3.2, the proofs of which are given in Section 4.

2.   Preliminary

In this article, we use notions as in the works by Peng ^[2], Zhang ^[4]. Suppose that $(\Omega, {\mathcal F})$ is a given measurable space. Assume that ${\mathcal H}$ is a collection of all random variables on $(\Omega, {\mathcal F})$ satisfying $\varphi(X_1, \cdots, X_n)\in {\mathcal H}$ for $X_1, \cdots, X_n\in {\mathcal H}$, and each $\varphi\in {\mathcal C}_{l, Lip}(\mathbb{R}^n)$, where ${\mathcal C}_{l, Lip}(\mathbb{R}^n)$ represents the space of $\varphi$ fulfilling

for some $C > 0$, $m\in \mathbb{N}$ relying on $\varphi$.

Definition 2.1. A sub-linear expectation $\mathbb{E}$ on ${\mathcal H}$ is a functional $\mathbb{E}:{\mathcal H}\mapsto \bar{ \mathbb{R}}: = [-\infty, \infty]$ fulfilling the following: for every $X, Y\in {\mathcal H}$,

$\rm (a)$ $X\ge Y$ yields $\mathbb{E}[X]\ge \mathbb{E}[Y]$;

$\rm (b)$ $\mathbb{E}[c] = c$, $\forall c\in \mathbb{R}$;

$\rm (c)$ $\mathbb{E}[\lambda X] = \lambda \mathbb{E}[X]$, $\forall \lambda\ge 0$;

$\rm (d)$ $\mathbb{E}[X+Y]\le \mathbb{E}[X]+ \mathbb{E}[Y]$ whenever $\mathbb{E}[X]+ \mathbb{E}[Y]$ is not of the form $\infty-\infty$ or $-\infty+\infty$.

We name a set function $V:{\mathcal F}\mapsto[0, 1]$ a capacity if

$\rm (a)$$V(\emptyset) = 0$, $V(\Omega) = 1$;

$\rm (b)$$V(A)\le V(B)$, $A\subset B$, $A, B\in {\mathcal F}$.

Moreover, if $V$ is continuous, then $V$ obey

$\rm (c)$ $A_n\uparrow A$ concludes $V(A_n)\uparrow V(A)$;

$\rm (d)$ $A_n\downarrow A$ concludes $V(A_n)\downarrow V(A)$.

$V$ is named to be sub-additive when $V(A+B)\le V(A)+V(B)$, $A, B\in {\mathcal F}$.

Under the sub-linear expectation space $(\Omega, {\mathcal H}, \mathbb{E})$, set $\mathbb{V}(A): = \inf\{ \mathbb{E}[\xi]:I_A\le \xi, \xi\in {\mathcal H}\}$, $\forall A\in {\mathcal F}$ (cf. Zhang ^[3,4,9,13], Chen and Wu ^[12], Xu et al. ^[20]). $\mathbb{V}$ is a sub-additive capacity. Set

By Definition 4.2 and Lemma 4.3 of Zhang ^[34], if $\mathbb{E} = \mathbf{E}$ is linear expectation, $\mathbb{V}^{*}$ coincide with the probability measure introduced by the linear expectation $\mathbf{E}$. As in Zhang ^[3], $\mathbb{V}^{*}$ is countably sub-additive, $\mathbb{V}^{*}(A)\le \mathbb{V}(A)$. Hence, in Theorem 3.1, Corollary 3.1, $\mathbb{V}$ could be replaced by $\mathbb{V}^{*}$, implying that the results here could be considered as natural extensions of the corresponding ones in classic probability space. Write

Assume $\mathbf{X} = (X_1, \cdots, X_m)$, $X_i\in{\mathcal H}$ and $\mathbf{Y} = (Y_1, \cdots, Y_n)$, $Y_i\in {\mathcal H}$ are two random vectors on $(\Omega, {\mathcal H}, \mathbb{E})$. $\mathbf{Y}$ is called to be negatively dependent to $\mathbf{X}$, if for $\psi_1$ on ${\mathcal C}_{l, Lip}(\mathbb{R}^m)$, $\psi_2$ on ${\mathcal C}_{l, Lip}(\mathbb{R}^n)$, we have $\mathbb{E}[\psi_1(\mathbf{X})\psi_2(\mathbf{Y})]\le \mathbb{E}[\psi_1(\mathbf{X})] \mathbb{E}[\psi_2(\mathbf{Y})]$ whenever $\psi_1(\mathbf{X})\ge 0$, $\mathbb{E}[\psi_2(\mathbf{Y})]\ge 0$, $\mathbb{E}[|\psi_1(\mathbf{X})\psi_2(\mathbf{Y})|] < \infty$, $\mathbb{E}[|\psi_1(\mathbf{X})|] < \infty$, $\mathbb{E}[|\psi_2(\mathbf{Y})|] < \infty$, and either $\psi_1$ and $\psi_2$ are coordinatewise nondecreasing or $\psi_1$ and $\psi_2$ are coordinatewise nonincreasing (cf. Definition 2.3 of Zhang ^[3], Definition 1.5 of Zhang ^[4]).

$\{X_n\}_{n = 1}^{\infty}$ is called to be negatively dependent, if $X_{n+1}$ is negatively dependent to $(X_{1}, \cdots, X_{n})$ for each $n\ge 1$. The existence of negatively dependent random variables $\{X_n\}_{n = 1}^{\infty}$ under sub-linear expectations could be yielded by Example 1.6 of Zhang ^[4] and Kolmogorov's existence theorem in classic probabililty space. We below give an concrete example.

Example 2.1. Let ${\mathcal P} = \{Q_1, Q_2\}$ be a family of probability measures on $(\Omega, {\mathcal F})$. Suppose that $\{X_n\}_{n = 1}^{\infty}$ are independent, identically distributed under each $Q_i$, $i = 1, 2$ with $Q_1(X_1 = -1) = Q_1(X_1 = 1) = 1/2$, $Q_2(X_1 = -1) = 1$. Define $\mathbb{E}[\xi] = \sup_{Q\in {\mathcal P}}\mathbf{E}_{Q}[\xi]$, for each random variable $\xi$. Here $\mathbb{E}[\cdot]$ is a sub-linear expectation. By the discussion of Example 1.6 of Zhang ^[4], we see that $\{X_n\}_{n = 1}^{\infty}$ are negatively dependent random variables under $\mathbb{E}$.

Assume that $\mathbf{X}_1$ and $\mathbf{X}_2$ are two $n$-dimensional random vectors in sub-linear expectation spaces $(\Omega_1, {\mathcal H}_1, \mathbb{E}_1)$ and $(\Omega_2, {\mathcal H}_2, \mathbb{E}_2)$ repectively. They are named identically distributed if for every $\psi\in{\mathcal C}_{l, Lip}(\mathbb{R}^n)$,

$\{X_n\}_{n = 1}^{\infty}$ is called to be identically distributed if for every $i\ge 1$, $X_i$ and $X_1$ are identically distributed.

In this article we assume that $\mathbb{E}$ is countably sub-additive, i.e., $\mathbb{E}(X)\le \sum_{n = 1}^{\infty} \mathbb{E}(X_n)$ could be implied by $X\le \sum_{n = 1}^{\infty}X_n$, $X, X_n\in {\mathcal H}$, and $X\ge 0$, $X_n\ge 0$, $n = 1, 2, \ldots$. Write $S_n = \sum_{i = 1}^{n}X_i$, $n\ge 1$. Let $C$ denote a positive constant which may vary in different occasions. $I(A)$ or $I_A$ represent the indicator function of $A$. The notion $a_x\approx b_x$ means that there exist two positive constants $C_1$, $C_2$ such that $C_1 |b_x|\le |a_x|\le C_2 |b_x|$.

As in Zhang ^[4], by definition, if $X_1, X_2, \ldots, X_n$ are negatively dependent random variables and $f_1$, $f_2, \ldots, f_n$ are all non increasing (or non decreasing) functions, then $f_1(X_1)$, $f_2(X_{2}), \ldots, f_n(X_{n})$ are still negatively dependent random variables.

We cite the useful inequalities under sub-linear expectations.

Lemma 2.1. (See Lemma 4.5 (III) of Zhang ^[3]) If $\mathbb{E}$ is countably sub-additive under $(\Omega, {\mathcal H}, \mathbb{E})$, then for $X\in {\mathcal H}$,

Lemma 2.2. (See Lemmas 2.3, 2.4 of Xu et al. ^[20] and Theorem 2.1 of Zhang ^[4]) Assume that $p\ge 1$ and $\{X_n; n\ge 1\}$ is a sequence of negatively dependent random varables under $(\Omega, {\mathcal H}, \mathbb{E})$. Then there exists a positive constant $C = C(p)$ relying on $p$ such that

Lemma 2.3. Assume that $X\in{\mathcal H}$, $\alpha > 0$, $\gamma > 0$, $C_{ \mathbb{V}}\left(|X|^{\alpha}\right) < \infty$. Then there exists a positive constant $C$ relying on $\alpha, \gamma$ such that

Proof. By the method of substitution of definite integral, letting $\gamma y = z^{1/\alpha}$, we get

Lemma 2.4. Let $Y_n, Z_n\in {\mathcal H}$. Then for any $q > 1$, $\varepsilon > 0$ and $a > 0$,

Proof. By Markov' inequality under sub-linear expectations, Lemma 2.1, and the similar proof of Lemma 2.4 of Sung ^[35], we could finish the proof. Hence, the proof is omitted here.

3.   Main results

Our main results are below.

Theorem 3.1. Suppose $\alpha > \frac12$ and $\alpha p > 1$. Assume that $\{X_n, n\ge 1\}$ is a sequence of negatively dependent random variables, and for each $n\ge1$, $X_n$ is identically distributed as $X$ under sub-linear expectation space $(\Omega, {\mathcal H}, \mathbb{E})$. Moreover, assume $\mathbb{E}(X) = \mathbb{E}(-X) = 0$ if $p\ge 1$. Suppose $C_{ \mathbb{V}}\left(|X|^p\right) < \infty$. Then for all $\varepsilon > 0$,

Remark 3.1. By Example 2.1, the assumption $\mathbb{E}(X) = \mathbb{E}(-X) = 0$ if $p\ge 1$ in Theorem 3.1 can not be weakened to $\mathbb{E}(X) = 0$ if $p\ge 1$. In fact, in the case of Example 2.1, if $\frac12 < \alpha\le 1$, $\alpha p > 1$, then for any $0 < \varepsilon < 1$,

which implies that Theorems 3.1, 3.2, Corollary 3.1 do not hold. However, by Example 1.6 of Zhang ^[4], the assumptions of Theorem 3.3, Corollary 3.2 hold for random variables in Example 2.1, hence Theorem 3.3, Corollary 3.2 are valid in this example.

By Theorem 3.1, we could get the Marcinkiewicz-Zygmund strong law of large numbers for negatively dependent random variables under sub-linear expectations below.

Corollary 3.1. Let $\alpha > \frac12$ and $\alpha p > 1$. Assume that under sub-linear expectation space $(\Omega, {\mathcal H}, \mathbb{E})$, $\{X_n\}$ is a sequence of negatively dependent random variables and for each $n$, $X_n$ is identically distributed as $X$. Moreover, assume $\mathbb{E}(X) = \mathbb{E}(-X) = 0$ if $p\ge 1$. Assume that $\mathbb{V}$ induced by $\mathbb{E}$ is countably sub-additive. Suppose $C_{ \mathbb{V}}\{|X|^p\} < \infty$. Then

Theorem 3.2. If the assumptions of Theorem 3.1 hold for $p\ge1$ and $C_{ \mathbb{V}}\{|X|^p\log^{\theta}|X|\} < \infty$ for some $\theta > \max\{\frac{\alpha p-1}{\alpha -\frac12}, p\}$, then for any $\varepsilon > 0$,

By the similar proof of Theorem 3.1, with Theorem 2.1 (b) for negative dependent random variables of Zhang ^[4] (cf. the proof of Theorem 2.1 (c) there) in place of Lemma 2.2 here, we could obtain the following result.

Theorem 3.3. Suppose $\alpha > \frac12$, $p\ge1$, and $\alpha p > 1$. Assume that $X_k$ is negatively dependent to $(X_{k+1}, \ldots, X_n)$, for each $k = 1, \ldots, n$, $n\ge 1$. Suppose for each $n$, $X_n$ is identically distributed as $X$ under sub-linear expectation space $(\Omega, {\mathcal H}, \mathbb{E})$. Suppose $C_{ \mathbb{V}}\left(|X|^p\right) < \infty$. Then for all $\varepsilon > 0$,

By the similar proof of Corollary 3.1, with Theorem 3.3 in place of Theorem 3.1, we get the following result.

Corollary 3.2. Let $\alpha > \frac12$, $p\ge 1$, and $\alpha p > 1$. Assume that $X_k$ is negatively dependent to $(X_{k+1}, \ldots, X_n)$, for each $k = 1, \ldots, n$, $n\ge 1$. Suppose for each $n$, $X_n$ is identically distributed as $X$ under sub-linear expectation space $(\Omega, {\mathcal H}, \mathbb{E})$. Assume that $\mathbb{V}$ induced by $\mathbb{E}$ is countably sub-additive. Suppose $C_{ \mathbb{V}}\{|X|^p\} < \infty$. Then

By the similar proof of Theorem 3.1 and Corollary 3.1, and adapting the proof of (4.10), we could obtain the following result.

Corollary 3.3. Suppose $\alpha > 1$ and $p\ge 1$. Assume that $\{X_n, n\ge 1\}$ is a sequence of negatively dependent random variables, and for each $n\ge1$, $X_n$ is identically distributed as $X$ under sub-linear expectation space $(\Omega, {\mathcal H}, \mathbb{E})$. Suppose $C_{ \mathbb{V}}\left(|X|^p\right) < \infty$. Then for all $\varepsilon > 0$,

Moreover assume that $\mathbb{V}$ induced by $\mathbb{E}$ is countably sub-additive. Then

By the discussion below Definition 4.1 of Zhang ^[34], and Corollary 3.2, we conjecture the following.

Conjecture 3.1. Suppose $\frac12 < \alpha\le 1$ and $\alpha p > 1$. Assume that $\{X_n, n\ge 1\}$ is a sequence of negatively dependent random variables, and for each $n\ge1$, $X_n$ is identically distributed as $X$ under sub-linear expectation space $(\Omega, {\mathcal H}, \mathbb{E})$. Assume that $\mathbb{V}$ induced by $\mathbb{E}$ is continuous. Suppose $C_{ \mathbb{V}}\{|X|^p\} < \infty$. Then

4.   Proofs of main results, Theorems 3.1, 3.2, Corollary 3.1

Proof of Theorem 3.1. We investigate the following cases.

Case 1. $0 < p < 1$.

For fixed $n\ge 1$, for $1\le i\le n$, write

Observing that $X_i = Y_{ni}+Z_{ni}$, we see that for all $\varepsilon > 0$,

By Markov's inequality under sub-linear expectations, $C_r$ inequality, and Lemmas 2.1, 2.3, we conclude that

and

Therefore, by (4.1)–(4.3), we deduce that (3.1) holds.

Case 2. $p\ge 1$.

Observing that $\alpha p > 1$, we choose a suitable $q$ such that $\frac{1}{\alpha p} < q < 1$. For fixed $n\ge 1$, for $1\le i\le n$, write

and $X_n^{(1)}$, $X_n^{(2)}$, $X_{n}^{(3)}$ is defined as $X_{ni}^{(1)}$, $X_{ni}^{(2)}$, $X_{ni}^{(3)}$ only with $X$ in place of $X_i$ above. Observing that $\sum_{i = 1}^{j}X_i = \sum_{i = 1}^{j}X_{ni}^{(1)}+\sum_{i = 1}^{j}X_{ni}^{(2)}+\sum_{i = 1}^{j}X_{ni}^{(3)}$, for $1\le j\le n$, we see that for all $\varepsilon > 0$,

Therefore, to establish (3.1), it is enough to prove that $II_1 < \infty$, $II_2 < \infty$, $II_3 < \infty$.

For $II_1$, we first establish that

By $\mathbb{E}(X) = 0$, Markov's inequality under sub-linear expectations, Lemma 2.1, we conclude that

which results in (4.5) by $C_{ \mathbb{V}}\{|X|^p\} < \infty$ and $1/(\alpha p) < q < 1$. Thus, from (4.5), it follows that

For fixed $n\ge 1$, we note that $\{X_{ni}^{(1)}- \mathbb{E} X_{ni}^{(1)}, 1\le i\le n\}$ are negatively dependent random variables. By (4.6), Markov's inequality under sub-linear expectations, and Lemma 2.2, we see that for any $\beta\ge2$,

Taking $\beta > \max\left\{\frac{\alpha p-1}{\alpha-1/2}, 2, p, \frac{\alpha p-1}{\alpha qp-\alpha q+\alpha-1}\right\}$, we obtain

and

By $C_r$ inequality, Markov's inequality under sub-linear expectations, Lemma 2.1, we see that

and

Therefore, combining (4.7)–(4.10) results in $II_1 < \infty$.

Next, we will establish that $II_{2} < \infty$. Let $g_{\mu}(x)$ be a non-increasing Lipschitz function such that $I\{x\le \mu\}\le g_{\mu}(x)\le I\{x\le 1\}$, $\mu\in (0, 1)$. Obviously, $I\{x > \mu\} > 1-g_{\mu}(x) > I\{x > 1\}$. For fixed $n\ge 1$, for $1\le i\le n$, write

and

We see that

which results in

By $C_{ \mathbb{V}}\{|X|^p\} < \infty$, we conclude that

Observing that $\frac{1}{\alpha p} < q < 1$, from the definition of $X_{ni}^{(2)}$, follows that

By $X_{ni}^{(4)} > 0$, (4.11)–(4.13), we see that

For fixed $n\ge 1$, we know that $\{X_{ni}^{(4)}- \mathbb{E} X_{ni}^{(4)}, 1\le i\le n\}$ are negatively dependent random variables under sub-linear expectations. By Markov's inequality under sub-linear expectations, $C_r$-inequality, Lemma 2.2, we obtain

By $C_r$ inequality, Lemma 2.3, we have

As in the proof of (4.9) and (4.16), we can deduce that $II_{22} < \infty$.

By Lemma 2.1, we see that

By (4.15)–(4.17), we deduce that $II_2 < \infty$.

As in the proof of $II_2 < \infty$, we also can obtain $II_3 < \infty$. Therefore, combining (4.5), $II_1 < \infty$, $II_2 < \infty$, and $II_3 < \infty$ results in (3.1). This finishes the proof.

Proof of Corollary 3.1. By $C_{ \mathbb{V}}\{|X|^p\} < \infty$, and Theorem 3.1, we deduce that for all $\varepsilon > 0$,

By (4.18), we conclude that for any $\varepsilon > 0$,

which, combined with Borel-Cantell lemma under sub-linear expectations, yields that

For all positive integers $n$, $\exists$ a positive integer $k$ satisfying $2^{k-1}\le n < 2^k$, we see that

which yields (3.2). This completes the proof.

Proof of Theorem 3.2. For fixed $n\ge 1$, for $1\le i\le n$, write

and

From Lemma 2.4 follows that for any $\beta > 1$,

Noticing that $Z_n\le (|X|-n^{\alpha})I(|X| > n^{\alpha})\le |X|I(|X| > n^{\alpha})$, by Lemma 2.3, we see that

Now, we will establish $III_1 < \infty$. Observing that $\theta > p\ge 1$, we can choose $\beta = \theta$. We analysize the following two cases.

Case 1. $1 < \theta\le 2$. From (2.2) of Lemma 2.2, Lemma 2.1, $\mathbb{E}(Y) = \mathbb{E}(-Y) = 0$, and Markov's inequality under sub-linear expectations follows that

Case 2. $\theta > 2$. Observe that $\theta > \frac{\alpha p-1}{\alpha -\frac12}$, we conclude that $\alpha p-2-\alpha\theta+\frac{\theta}{2} < -1$. As in the proof of (4.22), by Lemma 2.2, and $C_r$ inequality, we see that

By Lemma 2.3, we see that

By Lemma 2.1, we deduce that

And the proof of $III_{13} < \infty$ is similar to that of (4.22). This finishes the proof.

5.   Conclusions

We have obtained new results about complete convergence and complete moment convergence for maximum partial sums of negatively dependent random variables under sub-linear expectations. Results obtained in our article extend those for negatively dependent random variables under classical probability space, and Theorems 3.1, 3.2 here are different from Theorems 3.1, 3.2 of Xu et al. ^[20], and the former can not be deduced from the latter. Corollary 3.1 complements Theorem 3.1 in Zhang ^[9] in the case $p\ge 2$, Corollaries 3.2, 3.3 complement Theorem 3.3 in Zhang ^[4] in the case $p > 1$ in some sense.

Acknowledgments

This study was supported by Science and Technology Research Project of Jiangxi Provincial Department of Education of China (No. GJJ2201041), Academic Achievement Re-cultivation Project of Jingdezhen Ceramic University (No. 215/20506135), Doctoral Scientific Research Starting Foundation of Jingdezhen Ceramic University (No. 102/01003002031).

Conflict of interest

All authors state no conflicts of interest in this article.