Complete convergence and complete moment convergence for maximal weighted sums of extended negatively dependent random variables under sub-linear expectations

1.   Introduction

Peng ^[1,2] introduced important concepts of the sub-linear expectations space to describe the uncertainty in probability. Stimulated by the works of Peng ^[1,2], many scholars tried to discover the results under sub-linear expectations space, similar to those in classic probability space. Zhang ^[3,4] proved exponential inequalities and Rosenthal's inequality under sub-linear expectations. Xu et al. ^[5], Xu and Kong ^[6] obtained complete convergence and complete moment convergence of weighted sums of negatively dependent random variables under sub-linear expectations. For more limit theorems under sub-linear expectations, the readers could refer to Zhang ^[7], Xu and Zhang ^[8,9], Wu and Jiang ^[10], Zhang and Lin ^[11], Zhong and Wu ^[12], Hu et al. ^[13], Gao and Xu ^[14], Kuczmaszewska ^[15], Xu and Cheng ^[16,17], Zhang ^[18], Chen ^[19], Zhang ^[20], Chen and Wu ^[21], Xu et al. ^[5], Xu and Kong ^[6], and references therein.

In classic probability space, Yan ^[22] established complete convergence and complete moment convergence for maximal weighted sums of 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 ^[23], Chow ^[24], Hosseini and Nezakati^[25], Meng et al. ^[26] and refercences therein. Stimulated by the works of Yan ^[22], Xu et al. ^[5], Xu and Kong ^[6], we try to prove complete convergence and complete moment convergence for maximal weighted sums of extended negatively dependent random variables under sub-linear expectations, and the corresponding Marcinkiewicz-Zygmund strong law of large number, which extends the corresponding results in Yan ^[22]. Another main innovation point here is Rosenthal-type inequality for extended negatively dependent random variables, provided by Lemma 2.3.

We organize the remainders of this article as follows. We present relevant basic notions, concepts and properties, and give relevant lemmas under sub-linear expectations in Section 2. In Section 3, we give our main results, Theorems 3.1–3.3, the proofs of which are presented in Section 4.

2.   Preliminary

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

for $C > 0$, $m\in \mathbb{N}$ 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 $X, Y\in {\mathcal H}$,

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

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

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

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

Definition 2.2. We say that $\{X_n; n\ge 1\}$ is stochastically dominated by a random variable $X$ under $(\Omega, {\mathcal H}, \mathbb{E})$, if there exist a constant $C$ such that $\forall n\ge 1$, for all non-negative $h\in {\mathcal C}_{l, Lip}(\mathbb{R})$, $\mathbb{E}(h(X_n))\le C \mathbb{E}(h(X)).$

    $V:{\mathcal F}\mapsto[0, 1]$ is named to be a capacity if

${\rm (a)}$$V(\emptyset) = 0$, $V(\Omega) = 1$.

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

    Furthermore, if $V$ is continuous, then $V$ obeys

${\rm (c)}$ $A_n\uparrow A$ yields $V(A_n)\uparrow V(A)$.

${\rm (d)}$ $A_n\downarrow A$ yields $V(A_n)\downarrow V(A)$.

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

Under $(\Omega, {\mathcal H}, \mathbb{E})$, set $\mathbb{V}(A): = \inf\{ \mathbb{E}[\xi]:I_A\le \xi, \xi\in {\mathcal H}\}$, $\forall A\in {\mathcal F}$ (cf. Zhang ^[3]). $\mathbb{V}$ is a sub-additive capacity. Write

Under sub-linear expectation space $(\Omega, {\mathcal H}, \mathbb{E})$, $\{X_n; n\ge 1\}$ are called to be upper (resp. lower) extended negatively dependent if there is a constant $K\ge 1$ fulfilling

whenever the non-negative functions $g_i\in C_{b, Lip}(\mathbb{R})$, $i = 1, 2, \ldots$ are all non-decreasing (resp. all non-increasing) (cf. Definition 2.4 of Zhang ^[18]). They are named extended negatively dependent (END) if they are both upper extended negatively dependent and lower extended negatively dependent.

Suppose ${\bf{X}}_1$ and ${\bf{X}}_2$ are two $n$-dimensional random vectors under $(\Omega_1, {\mathcal H}_1, \mathbb{E}_1)$ and $(\Omega_2, {\mathcal H}_2, \mathbb{E}_2)$ respectively. They are said to be identically distributed if for every $\psi\in{\mathcal C}_{l, Lip}(\mathbb{R}^n)$,

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

Throughout this paper, we suppose 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 change from line to line. $I(A)$ or $I_A$ is the indicator function of $A$. The symbol $a_x\approx b_x$ means that there exists two positive constants $C_1$, $C_2$ fulfilling $C_1 |b_x|\le |a_x|\le C_2 |b_x|$, $x^{+}$ stands for $\max\{x, 0\}$, $x^{-} = (-x)^{+}$, for $x\in \mathbb{R}$.

As in Zhang ^[18], if $X_1, X_2, \ldots, X_n$ are extended negatively dependent random variables and $f_1(x), f_2(x), \ldots, f_n(x)\in {\mathcal C}_{l, Lip}(\mathbb{R})$ are all non increasing (or non decreasing) functions, then $f_1(X_1)$, $f_2(X_{2}), \ldots, f_n(X_{n})$ are extended negatively dependent random variables.

We cite the following under sub-linear expectations.

Lemma 2.1. (Cf. 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. Assume that $p > 1$ and $\{X_n; n\ge 1\}$ is a sequence of upper extended negatively dependent random varables with $\mathbb{E}[X_k]\le 0$, $k\ge 0$, under $(\Omega, {\mathcal H}, \mathbb{E})$. Then for every $n\ge 1$, there exists a positive constant $C = C(p)$ relying on $p$ such that for $p\ge2$,

By (2.1) of Lemma 2.2 and similar proof of Lemma 2.4 of Xu et al. ^[5], we could get the following.

Lemma 2.3. Assume that $p > 1$ and $\{X_n; n\ge 1\}$ is a sequence of upper extended negatively dependent random varables with $\mathbb{E}[X_k]\le 0$, $k\ge 0$, under $(\Omega, {\mathcal H}, \mathbb{E})$. Then for every $n\ge 1$, there exists a positive constant $C = C(p)$ relying on $p$ such that for $p\ge2$,

We first give two lemmas.

Lemma 2.4. Suppose that $0 < \alpha < 2$, $\gamma > 0$, and $\{a_{ni}, 1\le i\le n, n\ge 1\}$ is an array of real numbers fulfilling

Assume that $\{X_{ni}, i\ge 1, n\ge 1\}$ is stochastically dominated by a random variable $X$ with $C_{ \mathbb{V}}\{|X|^{\alpha}\} < \infty$. Moreover, suppose that $\mathbb{E}(a_{ni}X_{ni}) = 0$ for $1 < \alpha < 2$ and $b_{n} = n^{1/\alpha}(\log n)^{3/\gamma}$ for some $\gamma > 0$. Then

where $Y_{ni} = a_{ni}X_{ni}I\left(|a_{ni}X_{ni}|\le b_n\right)+b_nI\left(a_{ni}X_{ni} > b_n\right)-b_nI\left(a_{ni}X_{ni} < b_n\right)$ for each $1\le i\le n$, $n\ge 1$.

Lemma 2.5. Assume that $\{a_{ni}, 1\le i\le n, n\ge 1\}$ is an array of real numbers fuffilling (2.3) and $X\in {\mathcal H}$ under sub-linear expectation space $(\Omega, {\mathcal H}, \mathbb{E})$. Suppose $b_n$ is as in Lemma 2.4.

${\rm (i)}$ If $p > \max\{\alpha, \gamma (\beta+1)/3\}$ for some $\beta$, then

${\rm (ii)}$ If $p = \alpha$, $\beta = 2$, then

3.   Main results

Below are our main results.

Theorem 3.1. Suppose $\{X_{ni}, i\ge 1, n\ge 1\}$ is an array of rowwise END random variables, which is stochastic dominated by $X$ under $(\Omega, {\mathcal H}, \mathbb{E})$. Assume that for some $0 < \alpha < 2$, $0 < \gamma < 2$, $\{a_{ni}, 1\le i\le n, n\ge 1\}$ is an array of real numbers, being all non-negative or all non-positive, fulfilling (2.3) and $b_n$ is as in Lemma 2.5. Moreover, suppose that $\mathbb{E}(a_{ni}X_{ni}) = 0$ for $1 < \alpha < 2$. If

then for all $\varepsilon > 0$,

Similarly, moreover, with the condition that $\mathbb{E}(-a_{ni}X_{ni}) = 0$ for $1 < \alpha < 2$ in place of that $\mathbb{E}(a_{ni}X_{ni}) = 0$ for $1 < \alpha < 2$, we have for all $\varepsilon > 0$,

Moreover, suppose that $\mathbb{E}(X_{ni}) = \mathbb{E}(-X_{ni}) = 0$ for $1 < \alpha < 2$. Then (3.1) implies

Theorem 3.2. Suppose $\{X_{ni}, i\ge 1, n\ge 1\}$ is an array of rowwise END random variables, which is stochastic dominated by $X$ under $(\Omega, {\mathcal H}, \mathbb{E})$. Assume that for some $0 < \alpha < 2$, $0 < \gamma < 2$, $\{a_{ni}, 1\le i\le n, n\ge 1\}$ is an array of real numbers, being all non-negative or all non-positive, fulfilling (2.3) and $b_n$ is as in Lemma 2.5. Suppose (3.1) holds. Moreover, suppose that $\mathbb{E}(a_{ni}X_{ni}) = 0$ for $1 < \alpha < 2$. Then for $0 < \tau < \alpha$,

Similarly, moreover, with the condition that $\mathbb{E}(-a_{ni}X_{ni}) = 0$ for $1 < \alpha < 2$ in place of that $\mathbb{E}(a_{ni}X_{ni}) = 0$ for $1 < \alpha < 2$, we have for $0 < \tau < \alpha$,

Moreover, if $\mathbb{E}(X_{ni}) = \mathbb{E}(-X_{ni}) = 0$ for $1 < \alpha < 2$, then

Remark 3.1. From (3.7) follows that (3.4) holds. Hence we know that the complete moment convergence implies the complete convergence.

Theorem 3.3. Under the same conditions of Theorem 3.1, and assume that $\mathbb{V}$ induced by $\mathbb{E}$ is countably sub-additive, we have

Moreover, if $\mathbb{E}(X_{ni}) = \mathbb{E}(-X_{ni}) = 0$ for $1 < \alpha < 2$, then

4.   Proofs of main results

Proof of Lemma 2.4. We will investigate (2.4) in two cases.

Case I. $0 < \alpha\le 1$.

By Markov's inequality under sub-linear expectations and $C_{ \mathbb{V}}\{|X|^{\alpha}\} < \infty$, we see that

where $Y_{ni}'\equiv a_{ni}XI\{|a_{ni}X|\le b_n\}+b_nI\{a_{ni}X > b_n\}-b_nI\left(a_{ni}X_ < b_n\right)$.

Case II. $1 < \alpha < 2$.

For all $1\le i\le n$, $n\ge 1$, write

Then $0 < Z_{ni} = a_{ni}X_{ni}-b_n < a_{ni}X_{ni}$ for $a_{ni}X_{ni} > b_n$, $a_{ni}X_{ni} < Z_{ni} = a_{ni}X_{ni}+b_n < 0$ for $a_{ni}X_{ni} < -b_n$. Therefore, $|Z_{ni}|\le|a_{ni}X_{ni}|I\{|a_{ni}X_{ni}| > b_n\}$.

From $\mathbb{E}(a_{ni}X_{ni}) = 0$ for $1 < \alpha < 2$ and $C_{ \mathbb{V}}\{|X|^{\alpha}\} < \infty$ follows that

where $Z_{ni}'\equiv (a_{ni}X-b_n)I\{a_{ni}X > b_n\}+(a_{ni}X+b_n)I\{a_{ni}X < -b_n\}$. Combining (4.1) and (4.2) results in (2.4). □

Proof of Lemma 2.5. Without loss of generality, we suppose that

Write

Then by Lemma 2.4 of Yan ^[22], we have

First, for $I_1$, when $\alpha > \frac{\gamma (\beta+1)}{3}$, from Lemma 2.4 of Yan ^[22] and $p > \alpha$ follows that

When $\alpha\le \frac{\gamma (\beta+1)}{3}$, from Lemma 2.4 of Yan ^[22], and $p > \frac{\gamma (\beta+1)}{3}$ follows that

Next for $I_2$, from Lemma 2.4 of Yan ^[22] and $p > \alpha$ follows that

Noting that

we obtain

Hence,

The proof is completed. □

Proof of Theorem 3.1. By (2.3) and $a_{ni} = a_{ni}^{+}-a_{ni}^{-}$, without loss of generality, we suppose that $a_{ni}\ge 0$ and $\sum_{i = 1}^{n}a_{ni}^{\alpha}\le n$. We need only to prove (3.2).

For fixed $n\ge 1$, write $Z_{ni}$ as in the proof of Lemma 2.4, and

We easily see that for all $\varepsilon > 0$,

Then from Lemma 2.4, for $n$ sufficiently large follows that

To establish (4.3), we only need to prove that

For $J_1$, we see that $\{Y_{ni}- \mathbb{E}(Y_{ni}), 1\le i\le n, n\ge 1\}$ is an array of rowwise END random variables under sub-linear expectations. Therefore, by Markov's inequality under sub-linear expectations, Lemmas 2.1, 2.3, and similar proof of (2.8) of Zhang ^[20], we obtain

By Lemma 2.5(i) (for $p = \beta = 2$) and its proof, we have $J_{11}<\infty$.

For $J_{12}$, when $0<\alpha\le 1$, by Lemma 2.4 of Yan ^[22] and its proof, we see that

where for $0 < \gamma < 2$,

When $1 < \alpha < 2$, by $\mathbb{E}(a_{ni}X_{ni}) = 0$, we have

By $\#{I_{nj}}\le (j+1)$, we see that

By $\sum_{j = 1}^{k}\#{I_{nj}}j^{-1}\le C$, $\sum_{j = 1}^{k}\#{I_{nj}}j^{-1/\alpha}\le \sum_{j = 1}^{k}\#{I_{nj}}j^{-1/\alpha}j^{1-1/\alpha}\le Ck^{1-1/\alpha}$, we see that

Hence, we prove that $J_1 < \infty$.

Define $g_{\mu}(x)\in {\mathcal C}_{l, Lip}(\mathbb{R})$ such that $I\{|x|\le \mu\}\le g_{\mu}(|x|) < I\{|x|\le 1\}$, for some $0 < \mu < 1$. Then $I\{|x| > \mu\}\ge 1-g_{\mu}(|x|)\ge I\{|x| > 1\}$. For $J_2$, by $\sum_{j = 1}^{\infty}\frac{\#{I_{nj}}}{j+1}\le 1$, we see that

The proof of Theorem 3.1 is finished. □

Proof of Theorem 3.2. We only prove (3.5). For all $\varepsilon > 0$, we see that

To establish (3.5), it is enough to prove that $K_1 < \infty$ and $K_2 < \infty$. By Theorem 3.1, we know $K_1 < \infty$. For $K_2$, for each $1\le i\le n$, $n\ge 1$, and $t\ge 1$, write

By Lemma 2.4, for all $t\ge 1$ and $n$ large sufficiently, we obtain

To establish $K_2 < \infty$, we only need to prove that

We know that $\{Y_{ni}'- \mathbb{E}(Y_{ni}'), 1\le i\le n, n\ge 1\}$ is an array of rowwise END random variables under sub-linear expectations. Therefore, by Markov's inequality under sub-linear expectations, Lemmas 2.1, 2.3, and the similar proof of (2.8) of Zhang ^[20], we obtain

By $0 < \tau < \alpha < 2$ and Lemma 2.5 and its proof, we have

By using $t = x^{\tau}$, Markov's inequality under sub-linear expectations, Lemmas 2.1 and 2.5, we see that

For $K_{213}$, for $0 < \alpha\le 1$, by similar proof of $J_{12}$ of Theorem 3.1, we see that

where

When $1 < \alpha < 2$, by $\mathbb{E}(a_{ni}X_{ni}) = 0$ and the similar proof of $J_{12}$ for $1 < \alpha < 2$ in Theorem 3.1, we have

Combining (4.10)–(4.14) results in (4.8). By the similar proof of $J_2$ of Theorem 3.1, for $0 < \mu < 1$, we have

(4.15) together with (4.6)–(4.9) completes the proof of Theorem 3.2. □

Proof of Theorem 3.3. The proof here is similar to that of Corollary 3.1 of Xu and Kong ^[6] with Theorem 3.1 here in place of Theorem 3.1 of Xu and Kong ^[6] (also cf. the proof of Theorem 2.11 of Yan ^[22]), hence the proof here is omitted. This completes the proof. □

5.   Conclusions

We have obtained new results about complete convergence and complete moment convergence for maximal weighted sums of extended negatively dependent random variables under sub-linear expectations. Results obtained in our article generalize those for extended negatively dependent random variables in probability space, and Theorems 3.1–3.3 complement the results of Xu et al. ^[5], Xu and Kong ^[6] in some sense. In addition, Lemma 2.3 is the Rosenthal-type inequality for extended negatively dependent random variables, which is another main innovation point of this article.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This study was supported by Science and Technology Research Project of Jiangxi Provincial Department of Education of China (No. GJJ2201041), Doctoral Scientific Research Starting Foundation of Jingdezhen Ceramic University (No. 102/01003002031), Academic Achievement Re-cultivation Project of Jingdezhen Ceramic University (Grant No. 215/20506277).

Conflict of interest

The author states no conflict of interest in this article.