Complete integration convergence for arrays of rowwise extended negatively dependent random variables under the sub-linear expectations

1.   Introduction

The study of sub-linear expectation is a new direction of probability limit theory and has attracted great attention in the field in recent years. Probability limit theory is mainly good at dealing with those situations where the corresponding probability model can be determined by mathematical statistical methods and data analysis. However, most random variables in the real world have different degrees of uncertainty. How to analyze and calculate the financial and economic problems under uncertainty has become an important concern at present. Compared with the classical linear model, sub-linear expectation theory is more complex and more challenging. By relaxing the linear property of classical expectation to sub-additivity and positive homogeneity, we can obtain many interesting properties. Peng ^[1,2] first introduced the concept and framework of sub-linear expectation, and provided the corresponding basic properties. Under the sub-linear expectation, Peng ^[3,4] developed the law of weak large numbers and the central limit theorem for independent identically distributed random variables. Subsequently, Zhang ^[5,6,7] did a series of studies on the basis of the framework established by Peng, proved the exponential inequality and Rosenthal's inequality, and obtained Kolmogorov's strong law of larger numbers and Hartman-Wintner's law of iterated logarithm. Zhang's work provides us with a powerful tool for studying theorems under sub-linear expectations.

Complete convergence and complete moment convergence are two very important ideas in probability limit theory. Chow ^[8] first proposed the concept of complete convergence for sequences of independent random variables. Complete moment convergence is a more accurate convergence than complete convergence. A large number of scholars have studied the theorems of complete moment convergence for random variable sequences, and obtained many related results in the classical probability space. Wang ^[9,10] studied the complete moment convergence of difference sequences and random variables satisfying the Rosenthal-type inequality. Qiu ^[11] and Yi ^[12] extended the sequence suitable for the complete moment convergence again, and both obtained the complete moment convergence of END sequence. In recent years, Some scholars have also begun to study the complete integral convergence, but the results are relatively few. Lu ^[13] proved the complete integral convergence of wise widely negative dependent random variables under sub-linear expectation. Zhong and Wu ^[14] studied the complete integral convergence of the weighted sum of the END variable under sub-linear expectation space context. Liang and Wu ^[15] extend complete integral convergence theorems for END sequences of random variables. Li and Wu ^[16] discuss the property of complete integral convergence and obtain the result of q-th integral convergence of arrays of rowwise extended negatively dependent random variables under sub-linear expectation. Based on the above results, the study of complete integral convergence needs to be improved.

The content of this article is as follows. In Section 2, we introduce some basic notation, concepts and related properties. In Section 3, complete integral convergence theorems for weighted sums of arrays of rowwise extended negatively dependent random variables are established. Our results generalize Ge's ^[17] conclusions from probability space to sub-linear expectation. In Section 4, we give lemmas that are useful for proving the main results, and use these lemmas to prove the main results of this paper.

2.   Preliminaries

We use the framework and notions of Peng ^[1,2]. Let $(\Omega, \mathcal{F})$ be a given measurable space and let $\mathcal{H}$ be a linear space of real functions defined on $(\Omega, \mathcal{F})$ such that if $X_{1}, X_{2}, \ldots, X_{n} \in \mathcal{H}$ then $\varphi(X_{1}, \ldots, X_{n})\in \mathcal{H}$ for each $\varphi \in \mathrm{C}_{l, Lip}(\mathbb{R}_{n})$, where $\mathrm{C}_{l, Lip}(\mathbb{R}_{n})$ denotes the linear space of (local Lipschitz) functions $\varphi$ satisfying

for some $c > 0, \ m\in\mathbb{N}$ depending on $\varphi.\ \mathcal{H}$ is considered as a space of random variables. In this case we denote $X \in \mathcal{H}.$

Definition 2.1. A sub-linear expectation $\mathbb{\hat{E}}$ on $\mathcal{H}$ is a function $\mathbb{\hat{E}} : \mathcal{H} \rightarrow \bar{\mathbb{R}}$ satisfying the following properties: for all $X, Y \in \mathcal{H},$ we have

(a) Monotonicity: If$\ X \geq Y$ then $\mathbb{\hat{E}} X \geq \mathbb{\hat{E}} Y;$

(b) Constant preserving: $\mathbb{\hat{E}}c = c;$

(c) Sub-additivity: $\mathbb{\hat{E}}(X+Y) \leq \mathbb{\hat{E}}X + \mathbb{\hat{E}}Y;$ whenever $\mathbb{\hat{E}}X + \mathbb{\hat{E}}Y$ is not of the form $+\infty - \infty$ or $-\infty + \infty;$

(d) Positive homogeneity: $\mathbb{\hat{E}}(\lambda X) = \lambda \mathbb{\hat{E}}X, \lambda \geq 0.$

Here $\bar{\mathbb{R}} = [-\infty, \infty].$ The triple $(\Omega, \mathcal{H}, \mathbb{\hat{E}})$ is called a sub-linear expectation space.

Given a sub-linear expectation $\mathbb{\hat{E}}$, let us denote the conjugate expectation $\hat{\varepsilon}$ of $\mathbb{\hat{E}}$ by

From the definition, it is easily shown that for all $X, Y \in \mathcal{H}$,

If $\mathbb{\hat{E}}Y = \hat{\varepsilon}Y,$ then $\mathbb{\hat{E}}(X + aY) = \mathbb{\hat{E}}X + a\mathbb{\hat{E}}Y$ for any $a \in \mathbb{R}.$

Next, we consider the capacities corresponding to the sub-linear expectations. Let $\mathcal{G} \subset \mathcal{F}.$ A function $V : \mathcal{G} \rightarrow [0, 1]$ is called a capacity if

It is called to be sub-additive if $V(A \cup B) \leq V(A) + V(B)$ for all $A, B \in \mathcal{G}$ with $A \cup B \in \mathcal{G}$. In the sub-linear space $(\Omega, \mathcal{H}, \mathbb{\hat{E}})$, we denote a pair $(\mathbb{V}, \mathcal{V})$ of capacities by

where $\mathbb{V}(A^{c})$ is the complement set of A. By definition of $\mathbb{V}$ and $\mathcal{V}$, it is obvious that $\mathbb{V}$ is sub-additive, and

Definition 2.2. We define the Choquet integrals/expectations $(C_{\mathbb{V}}, C_{\mathcal{V}})$ by

with V being replaced by $\mathbb{V}$ and $\mathcal{V}$ respectively.

Definition 2.3. (i) (Identical distribution) Let $X_{1}$ and $X_{2}$ be two random variables defined severally in sub-linear expectation spaces $(\Omega_{1}, \mathcal{H}_{1}, \mathbb{\hat{E}}_{1})$ and $(\Omega_{2}, \mathcal{H}_{2}, \mathbb{\hat{E}}_{2})$. They are called identically distributed if

whenever the sub-expectations are finite. A sequence $\{X_{n}; n \geq 1\}$ of random variables is said to be identically distributed if $X_{i}$ and $X_{1}$ are identically distributed for each $i\ge 1$.

(ii) (Extended negatively dependence) A sequence of random variables $\{X_{n}; n \geq 1\}$ is said to be upper (resp.lower) extended negatively dependent if there is some dominating constant $K \geq 1$ such that

whenever the non-negative functions $\varphi_{i}(x) \in C_{l, Lip}(\mathbb{R}), i = 1, 2, \ldots,$ are all non-decreasing (resp. all non-increasing). They are said to be extended negatively dependent (END) if they are both upper extended negatively dependent and lower extended negatively dependent.

It is obvious that, if $\{X_{n}; n \geq 1\}$ is a sequence of END random variables and $f_{1}(x), f_{2}(x), \ldots \in C_{l, \mathrm{Lip}}(\mathbb{R})$ are non-descreasing (resp. non-increasing) functions, then $\left\{f_{n}\left(X_{n}\right); n \geqq 1\right\}$ is also a sequence of END random variables.

For $0 < \mu < 1$, let $g_{\mu}\left (x \right) \in C_{l, Lip}(\mathbb{R})$ be an even function and it is decreasing in $x > 0$ such that $0\leq g_{\mu}\left (x \right)\leq 1$ for all $x$, $g_{\mu}\left (x \right) = 1$ if $\left | x \right |\leq \mu$, $g_{\mu}\left (x \right) = 0$ if $\left | x \right | > 1$. Then

Let $A_{n}\ll B_{n}$ $\left (A_{n}\gg B_{n} \right)$ denote that there exists a constant $c > 0$ such that $A_{n}\leq cB_{n}$ $\left (A_{n}\geq cB_{n} \right)$ for sufficiently large $n$, and the symbol $I\left (\cdot \right)$ will be used to signify the indicator function. $\left \{ a \right \}_{+}$ means $\max \left \{ a, 0 \right \}$.

3.   Main results

Theorem 3.1. Let $\left\{X_{n j}; 1\le j\le b_{n}, n \geq 1\right\}$ be an array of rowwise END random variables, $\left\{a_{n j}, j \geq 1, n \geq 1\right\}$ be an array of positive numbers, $\left\{b_{n}, n \geq 1\right\}$ be a non-decreasing sequence of positive integers and $\left\{c_{n}, n \geq 1\right\}$ be a non-decreasing sequence of positive numbers. Suppose that for any $\varepsilon > 0$,

and

Then

and

Particularly, if $\hat{\mathbb{E}} X_{n j} g_{\mu}\left(\frac{a_{n j} X_{n j}}{\varepsilon b_{n}^{1 / t}}\right) = \hat{\varepsilon} X_{n j} g_{\mu}\left(\frac{a_{n j} X_{n j}}{\varepsilon b_{n}^{1 / t}}\right)$, then

Further, if the following condition also hold:

and

Then

Remark 3.1. The results obtained from Theorem 3.1 is very extensive, we can obtain different forms of complete integration convergence theorems by taking different forms of $c_{n}$ and $b_{n}$.

Taking $c_{n} = n^{2\left (\alpha -1 \right) }$, $b_{n} = n$, $t = 1/ \alpha$ or $c_{n} = n^{\alpha-2}h\left (n \right)$, $b_{n} = n$, $a_{nj} = 1$ in Theorem 3.1, $h\left (n \right)$ be a slowly varying function here, then we obtain two complete integration convergence theorems such as Theorems 3.2 and 3.3.

Theorem 3.2. Let $\{X_{n j}; j \geq 1, n \geq 1\}$ be an array of rowwise END random variables, $\{a_{n j}; j \geq 1, n \geq 1\}$ be an array of positive numbers satisfying $\hat{\mathbb{E}} X_{n j} g_{\mu}\left(\frac{a_{n j} X_{n j}}{\varepsilon n^{\alpha }}\right) = \hat{\varepsilon} X_{n j} g_{\mu}\left(\frac{a_{n j} X_{n j}}{\varepsilon n^{\alpha }}\right)$, and

Then, for any $\varepsilon > 0$,

Particularly, if $\mathbb{\hat{E}}X_{n j} = 0$ and $\alpha > \delta /2$, then

Theorem 3.3. Let $\{X, X_{n j}; j \geq 1, n \geq 1\}$ be an array of rowwise identically distributed END random variables with $\mathbb{\hat{E}}X = 0$, and $h\left (x \right) > 0$ be a slowly varying function as $x\longrightarrow \infty$. If $\hat{\mathbb{E}} X_{n j} g_{\mu}\left(\frac{ X_{n j}}{\varepsilon n^{1/t}}\right) = \hat{\varepsilon} X_{n j} g_{\mu}\left(\frac{ X_{n j}}{\varepsilon n^{1/t}}\right)$, $\hat{\mathbb{E}}\left (\left | X \right |^{\alpha t} h\left (\left | X \right |^{t} \right) \right) \le C_{\mathbb{V}}\left (\left | X \right |^{\alpha t} h\left (\left | X \right |^{t} \right) \right) < \infty$ for $\alpha > 1$, $1 < \alpha t < 2$, then

Remark 3.2. Theorem 3.1 extends Theorem 2.1 of Ge ^[16] from the conventional probability space to sub-linear expectation space. Theorems 3.2 and 3.3 generalize Corollary 2.6 and Theorem 2.4 of Ge ^[16].

Remark 3.3. Under sub-linear expectation, Li and Wu ^[16] study q-th integral convergence of arrays of rowwise extended negatively dependent random variables. Wu and Jiang ^[18] study complete convergence and complete integral convergence for negatively dependent random variables. Wu ^[19] establish precise asymptotics for complete integral convergence. In this paper, we study complete integral convergence of weighted sums, and the negatively dependent random variable expands to arrays of rowwise extended negatively dependent random variables.

4.   Proof of main results

To prove our results, we need the following lemmas.

Lemma 4.1. Zhong and Wu ^[14] suppose $X \in \mathcal{H}$, $\alpha > 0$, $t > 0$, and $h(x)$ is a slow varying function.

(i) Then

(ii) If $C_{\mathbb{V}}\left (\left | X \right |^{\alpha t} h\left (\left | X \right |^{t} \right) \right) < \infty$, then for $\forall c > 0$ and $\theta > 1$,

Lemma 4.2. Zhang ^[5] Let $X_{1}, X_{2}, \ldots, X_{n}$ be a sequence of upper extended negatively dependent random variables in with $\mathbb{\hat{E}}\left [X_{k} \right]\leq 0$. Set $S_{n} = \sum_{k = 1}^{n}X_{k}$, then

Proof of Theorem 3.1. For array of rowwise END random variables $\left\{X_{n j}; 1\le j\le b_{n}, \right.$ $\left. n \geq 1\right\}$, we need that truncated function belong to $C_{l, L i p}(\mathbb{R})$ and is non-decreasing to make the truncated random variables are also END. For any $c > 0, 1\leq j\leq b_{n}, n \geq 1$,

There are $X_{nj} = Y_{nj} + Z_{nj}$. Note that

Hence, to prove (3.3), it suffices to verify that

By (3.1), we have

For any $r > 0$, combining the $Cr$ inequality and (2.1),

thus,

Through definition 2.3 (ii), we know that $\left \{ a_{nj}\left (Y_{n j}-\hat{\mathbb{E}}Y_{n j} \right), 1\le j\le b_{n}, n\ge 1\right \}$ is still END with $\hat{\mathbb{E}}a_{nj}\left (Y_{n j}-\hat{\mathbb{E}}Y_{n j} \right) = 0$, by (4.3), (3.2) and (4.5), it is easy to prove that

Next, we verify $b_{n}^{-1/t}\left | I_{3} \right | \longrightarrow 0 \ \ \mathrm{as} \ \ n\longrightarrow \infty$. We can obtain $\sum_{j = 1}^{b_{n}} \mathbb{V}\left(\left|a_{n j} X_{n j}\right| \geq \varepsilon b_{n}^{1 / t}\right)\longrightarrow 0$ $\mathrm{as}$ $n\longrightarrow \infty$ from (3.1) and $\left\{c_{n}, n \geq 1\right\}$ is a non-decreasing sequence of positive numbers, then, by $g_{\mu}\left (x \right) \downarrow$ in $x > 0$,

Combining (4.4), (4.6) and (4.7), we obtain (3.3). Obviously, $\left\{-X_{n j}; 1 \leq j \leq b_{n}, n \geq 1\right\}$ also satisfies the conditions of Theorem 3.1, replacing $\left\{X_{n j}; 1 \leq j \leq b_{n}, n \geq 1\right\}$ by $\left\{-X_{n j}; 1 \leq j \leq b_{n}, n \geq 1\right\}$ in (3.3), by $g_{\mu}(x)$ is an even function, we obtain (3.4). According to the $\hat{\mathbb{E}} X_{n j} g_{\mu}\left(\frac{a_{n j} X_{n j}}{\varepsilon b_{n}^{1 / t}}\right) = \hat{\varepsilon} X_{n j} g_{\mu}\left(\frac{a_{n j} X_{n j}}{\varepsilon b_{n}^{1 / t}}\right)$, (3.3) and (3.4), we get

Therefore (3.5) holds.

Next, we prove (3.8). For $\forall \varepsilon > 0$,

By (3.5), we can get $H_{1} < \infty$. Furthermore, the proof of $H_{2} < \infty$ is similar to the considerations in (4.8), by $\hat{\mathbb{E}} X_{n j} g_{\mu}\left(\frac{a_{n j} X_{n j}}{\varepsilon b_{n}^{1 / t}}\right) = \hat{\varepsilon} X_{n j} g_{\mu}\left(\frac{a_{n j} X_{n j}}{\varepsilon b_{n}^{1 / t}}\right)$, we only need to prove

For any $1\leq j\leq b_{n}$, $u\geq \varepsilon$, $\delta > 0$, let

There are $X_{nj} = Y_{nj}^{'} + Z_{nj}^{'}$. Noting that

By (3.6), we have

The proof of is $H_{32} < \infty$ similar to considerations in (4.6), by Lemma 4.2 we have

By (3.2), we have

By (3.7), we obtain

The proof of $H_{323} < \infty$ is similar to the proof of the convergence of $H_{31}$, by (3.6), we have

Therefore, we get $H_{32} < \infty$. Next, we consider $H_{33} < \infty$. For $u > \varepsilon$, $1-g_{\mu}\left (\frac{a_{nj}X_{nj}}{\mu u b_{n}^{1/t }} \right) < 1-g_{\mu}\left (\frac{a_{nj}X_{nj}}{\mu \varepsilon b_{n}^{1/t }} \right)$, the way is similar to the proof of (4.7), we have

Thereby, for sufficiently large $n$, we can get

By (4.9), we get $H_{33} < \infty$. Then $H_{3} < \infty$. The proof of Theorem 3.1 is completed.

Proof of Theorem 3.2. In Theorem 3.1, let $c_{n} = n^{2\left (\alpha -1 \right) }$, $b_{n} = n$, $t = 1/ \alpha$, by (3.9) and $0 < \delta < 1$, we have

Similarly, we can obtain

And for $\forall c > 0$, we have

Hence, (4.10) and (4.11) satisfy conditions (3.2) and (3.7) of Theorem 3.1. By (4.12), we can get $H_{31} < \infty$, then we obtain (3.10). Finally, we show that (3.11). We only need to verify that

For $\alpha > \delta/2$, by $\mathbb{\hat{E}}X_{n j} = 0$, we have

Therefore the proof of Theorem 3.2 is completed.

Proof of Theorem 3.3. For $j \geq 1$, let $g_{j}(x) \in C_{l, L i p}(\mathbb{R})$ be an even function, such that $0 \leq g_{j}(x) \leq 1$ for all $x$; $g_{j}\left(\frac{x}{2^{j/t}}\right) = 1$ if $2^{(j-1)/t} < |x| \leq 2^{j/t}$ and $g_{j}\left(\frac{x}{2^{j/t}}\right) = 0$ if $|x| \leq \mu 2^{(j-1)/t }$ or $|x| > \left (1+\mu \right)2^{j/t }$. Then

Let $c_{n} = n^{\alpha -2}h\left (n \right)$, $b_{n} = n$, $a_{nj} = 1$. By Theorem 3.1, we can obtain

By (4.2), (4.13), (4.14) and $\alpha t < 2$, we have

By (4.1), we have

And by (4.2), (4.13) and (4.15), we have

thus (3.2), (3.6), (3.7) are satisfied. In order to prove (3.12), it remain to show that $n^{-1/t }\left|\sum_{j = 1}^{n} \hat{\mathbb{E}} X_{n j} g_{\mu}\left(\frac{X_{n j}}{\varepsilon n^{1/t }}\right)\right| \longrightarrow 0 \ \mathrm{as} \ n\longrightarrow \infty$. First of all, by the properties of the slowly varying function, we obtain $\left | X \right | ^{\alpha t} h\left (\left | X \right | ^{t} \right) \uparrow$ when $\alpha t > 1$. By (2.1), $\alpha t > 1$, $\alpha > 1$ and $\left | X \right | ^{\alpha t} h\left (\left | X \right | ^{t} \right) \uparrow$, we can get

By $\mathbb{\hat{E}}X = 0$, (4.17), $\alpha t > 1$, $\alpha > 1$ and $\hat{\mathbb{E}}\left (\left | X \right |^{\alpha t} h\left (\left | X \right |^{t} \right) \right) \le C_{\mathbb{V}}\left (\left | X \right |^{\alpha t} h\left (\left | X \right |^{t} \right) \right) < \infty$, we have

Combining (4.16) and (4.18), we can get (3.12). Then Theorem 3.3 holds.

Acknowledgments

This paper was supported by the National Natural Science Foundation of China (12061028), the Support Program of the Guangxi China Science Foundation (2018GXNSFAA281011), and Guangxi Colleges and Universities Key Laboratory of Applied Statistics.

Conflict of interest

All authors declare no conflict of interest in this paper.