Periodic traveling wave solutions of the Nicholson's blowflies model with delay and advection

1.   Introduction

The classical limit theorem considers additive probability and additive expectation, which is suitable for the case of model determination, but this assumption of additivity is not feasible in many fields of practice. As a mathematical theory, nonlinear expectation can be analyzed and calculated under the uncertainty of the mathematical model. In its research, sub-linear expectation plays a special role and is the most studied. Peng ^[1,2,3] put forward the concept of generalization of sublinear expectation space in 2006, which transforms the probability in probability space into the capacity in sublinear expectation space, which enriches the theoretical part of probability space. Then, after Zhang's ^[4,5,6,7] research in sublinear expectation space, some important inequalities are obtained. These inequalities are a powerful tool for us to study sublinear expectation space. In addition, Zhang also studies the law of iterated logarithm and the strong law of large numbers in sublinear expectation space. After further extension, Wu and Jiang ^[8] obtained the Marcinkiewicz type strong law of numbers and the Chover type iterated logarithm law for more general cases in sublinear expectation space.

In probability space, the complete convergence and complete moment convergence are two very important research parts. The notion of complete convergence was proposed by Hsu and Robbins ^[9] in 1947. In 1988, Chow ^[10] introduced the concept of complete moment convergence. The complete moment convergence is stronger than the complete convergence. The complete convergence and complete moment convergence in probability space have been relatively mature. For example, Qiu ^[11], Wu ^[12], and Shen ^[13] respectively obtained the complete convergence and complete moment convergence for independent identically distributed (i.i.d.), negatively associated (NA), extended negatively dependent (END) random variables sequence in probability space. Due to many methods and tools in probability space, sublinear expectation space can not be used, which increases the difficulty of studying sublinear expectation space, but many scholars have done the research, such as Wu ^[14] pushed the theorem in Wu ^[12] from probability space to sublinear expectation space. Feng ^[15] and Liang ^[16] obtained the complete convergence and complete integral convergence for arrays of row-wise ND and END random variables respectively. Zhong ^[17] studied the complete convergence and complete integral convergence for the weighted sum of END random variables. Lu ^[18] obtained more extensive conditions and conclusions than Zhong ^[17] in sublinear expectation space. The exponential inequality used in this article was proposed by Anna ^[19] in 2020. In the inequality, it is assumed that the truncated random variable sequence is a WA random variable sequence. Because it was proposed later, there is little research on WA random variable sequence in sublinear expectation space. Hu ^[20] proved the complete convergence for weighted sums of WA random variables in 2021.

The organizational structure of this paper is as follows. In Section 2, we summarize some basic symbols and concepts, as well as the related properties in sublinear expectation space, and give a preliminary lemma which is helpful to obtain the main results. In Section 3, We deduce ^[21] from probability space to sublinear expectation space, obtain the corresponding conclusions, and prove the complete convergence and complete integral convergence for the weighted sums of WA random variables in sublinear expectation space.

2.   Preliminaries

We use the framework and notations of Peng ^[1,2,3]. 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 ${X_1}, {X_2}, \cdots, {X_n} \in \mathcal{H}$ then $\varphi \left({{X_1}, {X_2}, \cdots, {X_n}} \right) \in \mathcal{H}$ for each $\varphi \in {C_{l, Lip}}\left({{\mathbb{R}_n}} \right),$ where ${C_{l, Lip}}\left({{\mathbb{R}_n}} \right)$ denotes the linear space of (local Lipschitz) functions $\varphi$ satisfying

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

Definition 2.1. A sub-linear expectation ${\hat{\mathbb{E}}},$ on $\mathcal{H}$ is a function ${\hat{\mathbb{E}}}{\text{:}}\mathcal{H} \to [- \infty, + \infty]$ satisfying the following properties: For all $X, Y \in \mathcal{H},$ we have

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

(b) Constant preserving: ${\hat{\mathbb{E}}}\left(c \right) = c$, for $c \in \mathbb{R}$;

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

(d) Positive homogeneity: ${\hat{\mathbb{E}}}\left({\lambda X} \right) = \lambda {\hat{\mathbb{E}}}X, \lambda \geqslant 0.$ Convention: when $\lambda {\text{ = }}0$ and ${\hat{\mathbb{E}}}X{\text{ = + }}\infty,$ ${\hat{\mathbb{E}}}\left({\lambda X} \right) = \lambda {\hat{\mathbb{E}}}X{\text{ = }}0.$

The triple $(\Omega, \mathcal{H}, {\hat{\mathbb{E}}})$ is called a sub-linear expectation space.

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

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

If ${\hat{\mathbb{E}}}Y = \hat \varepsilon Y,$ then ${\hat{\mathbb{E}}}\left({X + aY} \right) = {\hat{\mathbb{E}}}X + a{\hat{\mathbb{E}}}Y$ 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 $V:\mathcal{G} \to [0, 1]$ is called a capacity if

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

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

If $f \leqslant {\rm I}(A) \leqslant g, {\text{ }}f, g \in \mathcal{H},$ then

This implies Markov inequality: $\forall X \in \mathcal{H},$

From $I\left({\left| X \right| > x} \right) \leqslant {{{{\left| X \right|}^p}} \mathord{\left/ {\vphantom {{{{\left| X \right|}^p}} {{x^p}}}} \right. } {{x^p}}} \in \mathcal{H}.$ From Lemma 4.1 in Zhang ^[5], we have Hӧlder inequality: $\forall X, Y \in \mathcal{H}, p, q > 1,$ satisfying ${p^{ - 1}} + {q^{ - 1}} = 1,$

whenever

Particularly, Jensen inequality:

We define the Choquet integrals $({C_\mathbb{V}}, {C_\mathcal{V}})$ by

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

Definition 2.2.

(i) ${\hat{\mathbb{E}}}$ countably sub-additive: ${\hat{\mathbb{E}}}$ is called to be countably sub-additive if it satisfies ${\hat{\mathbb{E}}}(X) \leqslant \sum\limits_{n = 1}^\infty {{\hat{\mathbb{E}}}({X_n})},$ where $X \leqslant \sum\limits_{n = 1}^\infty {{X_n}}, {\text{ }}X, {X_n} \in \mathcal{H}, X \geqslant 0, {X_n} \geqslant 0, n \geqslant 1.$

(ii) $V$ is called to be countably sub-additive if

Definition 2.3. (Identical distribution) Let ${X_1}$ and ${X_2}$ be two random variables defined severally in sub-linear expectation space $({\Omega _1}, {\mathcal{H}_1}, {{\hat{\mathbb{E}}}_1})$ and $({\Omega _2}, {\mathcal{H}_2}, {{\hat{\mathbb{E}}}_2}).$ They are called identically distributed if

whenever the sub-linear expectations are finite. A sequence $\left\{ {{X_n}; n \geqslant 1} \right\}$ of random variables is said to be identical distribution if ${X_i}$ and ${X_1}$ are identical distribution for each $i \geqslant 1.$

Definition 2.4. (WA) Let $\left\{ {{Y_n}; n \geqslant 1} \right\}$ be a sequence of random variables in a sub-linear expectation space $(\Omega, \mathcal{H}, {\hat{\mathbb{E}}}).$ The sequence $\left\{ {{Y_n}; n \geqslant 1} \right\}$ is called WA if for $t \geqslant 0$ and for all $n \in \mathbb{N}$

where $0 < g(n) < \infty.$

Definition 2.5. ^[22] A function $L:(0, \infty) \to (0, \infty)$ is:

(i) A slowly varying function (at infinity), if for any $a > 0$

(ii) A regularly varying function with index $\alpha > 0,$ if for any $a > 0$

Lemma 2.6. ^[22] Every regularly varying function (with index $\alpha > 0$) $l:(0, \infty) \to (0, \infty)$ is of the form

where $L$ is a slowly varying function.

In the following, let $\left\{ {{X_n}; n \geqslant 1} \right\}$ be a sequence of random variables in $(\Omega, \mathcal{H}, {\hat{\mathbb{E}}}).$ The symbol $c$ stands for a generic positive constant which may differ from one place to another. Let ${a_x} \sim {b_x}$ denote ${\lim _{x \to \infty }}{{{a_x}} \mathord{\left/ {\vphantom {{{a_x}} {{b_x}}}} \right. } {{b_x}}} = 1.$ ${a_n} \ll {b_n}$ denote that there exists a constant $c > 0$ such that ${a_n} \leqslant c{b_n}$ for sufficiently large $n,$ and $I(\cdot)$ denotes an indicator function. $a \vee b$ means to take the maximum value of $a$ and $b,$ while $a \wedge b$ means to take the minimum value of $a$ and $b.$

To prove our results, we need the following lemmas.

In ^[17], we can get the following lemma.

Lemma 2.7. ^[17] Suppose $X \in \mathcal{H}, \alpha > 0, p > 0,$ and $l(x)$ is a slow varying function.

(i) Then, for $\forall c > 0$,

taking $l(x) = 1$ and $\log x,$ respectively, we can get that for $\forall c > 0$,

(ii) If ${C_\mathbb{V}}\left({{{\left| X \right|}^p}l\left({{{\left| X \right|}^{{1 \mathord{\left/ {\vphantom {1 \alpha }} \right. } \alpha }}}} \right)} \right) < \infty,$ then for any $\theta > 1$ and $c > 0,$

taking $l(x) = 1$ and $\log x,$ respectively, we have

The last one is the exponential inequality for WA random variables, which was can be found in ^[19].

Lemma 2.8. ^[19] Let $\left\{ {{X_1}, {X_2}, \cdots, {X_n}} \right\}$ be a sequence of random variables in $(\Omega, \mathcal{H}, {\hat{\mathbb{E}}}),$ with ${\hat{\mathbb{E}}}{X_i} \leqslant 0$ for $1 \leqslant i \leqslant n.$ Let $d > 0$ be a real number, we define ${X^{(d)}} = \min \left\{ {X, d} \right\}.$ Assume that ${Y_i}: = X_i^{(d)},$ $1 \leqslant i \leqslant n$ satisfy (2.2) for all $t > 0.$ Then, for all $x > 0,$ we have

3.   Results and proof

Next, we give the theorems and proof in this article.

Let $\left\{ {{X_n}; n \geqslant 1} \right\}$ be a sequence of random variables in sub-linear expectation space $(\Omega, \mathcal{H}, {\hat{\mathbb{E}}}), \alpha > {1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}, \alpha p > 1, \varepsilon > 0, \delta > 0$ and ${\beta _1} = \frac{{[\alpha (p \wedge 2) -1]\varepsilon }}{{4(\alpha p - 1 + \delta)}} > 0.$ For fixed $n \geqslant 1,$ denote for $1 \leqslant i \leqslant n$ that

Theorem 3.1 Let $\alpha > {1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}, \alpha p > 1$ and $\left\{ {{X_n}; n \geqslant 1} \right\}$ be a sequence of random variables in $(\Omega, \mathcal{H}, {\hat{\mathbb{E}}})$ with ${\hat{\mathbb{E}}}{X_i} = \hat \varepsilon {X_i} = 0$ if $p > 1$ such that sequence $\left\{ {{Y_i}; 1 \leqslant i \leqslant n} \right\}$ of truncated random variables is WA and control coefficient $g(n)$ in (2.2) is regularly varying function with index $\delta$ for some $\delta > 0$. Assume that $\left\{ {{a_{ni}}; 1 \leqslant i \leqslant n, n \geqslant 1} \right\}$ is an array of real numbers and there exist some $q$ with $q > \max \left\{ {2, p} \right\}$ have

and there also exist a random variable $X \in \mathcal{H}$ and a constant $c$ satisfying

then

implies that for all $\varepsilon > 0$

Let $0 < {\beta _2} < \min \left\{ {\frac{{2 \wedge (p \vee r)}}{{2r}}, \frac{{\alpha [2 \wedge (p \vee r)] - 1}}{{2(\alpha p - 1 + \delta)}}} \right\}.$ For any $1 \leqslant i \leqslant n, n \geqslant 1,$ and $t \geqslant {n^{\alpha r}},$ denote

Theorem 3.2. Let $r > 0, \alpha > {1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}, \alpha (p \vee r) > 1$ and $\left\{ {{X_n}; n \geqslant 1} \right\}$ be a sequence of random variables in $(\Omega, \mathcal{H}, {\hat{\mathbb{E}}})$ with ${\hat{\mathbb{E}}}{X_i} = \hat \varepsilon {X_i} = 0$ if $p \vee r > 1$ such that sequence $\left\{ {{{Y'}_i}; 1 \leqslant i \leqslant n} \right\}$ of truncated random variables is WA and control coefficient $g(n)$ in (2.2) is regularly varying function with index $\delta$ for some $\delta > 0$. Assume that $\left\{ {{a_{ni}}; 1 \leqslant i \leqslant n, n \geqslant 1} \right\}$ is an array of real numbers and condition (3.2) holds for $q > \max \left\{ {2, p \vee r} \right\}$, moreover, the condition (3.3) is also true, then

implies that for any $\varepsilon > 0,$

Remark. In Theorem 3.2, we extend the complete moment convergence for the weighted sums of random variables in the probability space of article ^[21] to the complete integral convergence for the weighted sums of WA random variables in sublinear expectation space.

Proof of Theorem 3.1. Since $\sum\limits_{i = 1}^n {{a_{ni}}{X_i}} = \sum\limits_{i = 1}^n {a_{ni}^ + {X_i}} - \sum\limits_{i = 1}^n {a_{ni}^ - {X_i}},$ we have

So, without loss of generality, we can assume that ${a_{ni}} \geqslant 0$ for $1 \leqslant i \leqslant n$ and $n \geqslant 1.$

If we want to prove (3.5), we just need to prove

Because of considering $\left\{ { - {X_n}; n \geqslant 1} \right\}$ still satisfies the conditions in the Theorem 3.1, we can obtain

Form (3.9) and (3.10), we can get (3.5). The following proves that (3.9) is established. The definition of $\left\{ {{Y_i}; 1 \leqslant i \leqslant n} \right\}$ is (3.1). For fixed $n \geqslant 1,$ denote for $1 \leqslant i \leqslant n$ that

It is easily checked that for $\forall \varepsilon > 0,$

So, we have

To prove (3.9), it suffices to show ${I_1} < \infty$ and ${I_2} < \infty.$

For $0 < \mu < 1,$ let $g(x) \in {C_{l, Lip}}(\mathbb{R})$ be a decreasing function when $x \geqslant 0$ such that $0 \leqslant g(x) \leqslant 1$ for all $x$ and $g(x) = 1$ if $\left| x \right| \leqslant \mu,$ $g(x) = 0$ if $\left| x \right| \geqslant 1.$ Then

By (3.12), Lemma 2.7 (i) and (3.3), we can get that

In the following, we prove that ${I_2} < \infty.$ Firstly, we will show that

By (3.2) and Hӧlder inequality, we have for any $0 < \rho < q$ that

For any $\lambda > 0,$ by (3.12) and ${C_r}$ inequality, we have

Thus

By Lemma 2.7 (i), we can get that

and $\mathbb{V}\left({\left| X \right| > c{n^\alpha }} \right) \downarrow,$ so we get $n\mathbb{V}\left({\left| X \right| > c{n^\alpha }} \right) \to 0$ as $n \to \infty.$

When $0 < p \leqslant 1.$ Since $q > \max \left\{ {2, p} \right\} > 1,$ by (3.13), (3.14) and $\alpha p > 1,$ we have that

When $p > 1.$ Since $q > p > 1,$ by (3.13), (3.14) and ${\hat{\mathbb{E}}}{X_i} = 0,$ we have

Hence, ${n^{ - \alpha }}\left| {\sum\limits_{i = 1}^n {{a_{ni}}{\hat{\mathbb{E}}}{Y_i}} } \right| \leqslant {\varepsilon \mathord{\left/ {\vphantom {\varepsilon 2}} \right. } 2}$ for all $n$ large enough, which implies that

According to assume that sequence $\left\{ {{Y_i}; 1 \leqslant i \leqslant n} \right\}$ of truncated random variables is WA and ${a_{ni}} \geqslant 0,$ by (2.2), we have

Because $\exp \left({ - \sum\limits_{i = 1}^n {t{a_{ni}}{\hat{\mathbb{E}}}{Y_i}} } \right) \geqslant 0,$ we can get that

which means that ${a_{ni}}({Y_i} - {\hat{\mathbb{E}}}{Y_i})$ are WA random variables. Without loss of generality, according to (3.2), we assume that ${a_{ni}} \leqslant {1 \mathord{\left/ {\vphantom {1 2}} \right. } 2},$ then

We can verify that ${a_{ni}}({Y_i} - {\hat{\mathbb{E}}}{Y_i}) = \min \left\{ {{a_{ni}}({Y_i} - {\hat{\mathbb{E}}}{Y_i}), {\text{ }}{\beta _1}{n^\alpha }} \right\}.$

So $\left\{ {{a_{ni}}({Y_i} - {\hat{\mathbb{E}}}{Y_i}); 1 \leqslant i \leqslant n, n \geqslant 1} \right\}$ satisfy the conditions in Lemma 2.8 with ${\hat{\mathbb{E}}}\left({{a_{ni}}({Y_i} - {\hat{\mathbb{E}}}{Y_i})} \right) = 0.$ Taking $x = \frac{{\varepsilon {n^\alpha }}}{2}, d = {\beta _1}{n^\alpha } = \frac{{[\alpha (p \wedge 2) -1]\varepsilon {n^\alpha }}}{{4(\alpha p - 1 + \delta)}}$ in Lemma 2.8, we obtain

Let $\beta > 0, {\text{ }}{g_{j\beta }}(x) \in {C_{l, Lip}}(\mathbb{R}), {\text{ }}j \geqslant 1,$ suppose ${g_{j\beta }}(x)$ is an even function, such that $0 \leqslant {g_{j\beta }}(x) \leqslant 1$ for all $x$ and ${g_{j\beta }}(x) = 1$ if ${{\beta {2^{\left({j - 1} \right)\alpha }}} \mathord{\left/ {\vphantom {{\beta {2^{\left({j - 1} \right)\alpha }}} \mu }} \right. } \mu } \leqslant \left| x \right| \leqslant {{\beta {2^{j\alpha }}} \mathord{\left/ {\vphantom {{\beta {2^{j\alpha }}} \mu }} \right. } \mu },$ ${g_{j\beta }}(x) = 0$ if $\left| x \right| < \beta {2^{\left({j - 1} \right)\alpha }}$ or $\left| x \right| > {{(1 + \mu)\beta {2^{j\alpha }}} \mathord{\left/ {\vphantom {{(1 + \mu)\beta {2^{j\alpha }}} \mu }} \right. } \mu }.$ Then for any $l > 0,$

The truncation that defines $Y$ as $X$ is as follows

According to Markov inequality, ${C_r}$ inequality, (3.2), (3.3), (3.14), (3.15), Lemma 2.7, $q > p$ and $g(x) \downarrow$ when $x \geqslant 0.$ Then

Next, we prove ${I_{22}} < \infty.$ If $p \geqslant 2,$ then $d = \frac{{(2\alpha - 1)\varepsilon {n^\alpha }}}{{4(\alpha p - 1 + \delta)}},$ by (3.3), (3.4), (3.13), $\alpha p > 1,$ ${C_r}$ inequality, the condition of $g(n),$ there exist a slowly varying function $L(n),$ such that $g(n){\text{ = }}{n^\delta }L(n),$ we have

If $p < 2,$ then $d = \frac{{(\alpha p - 1)\varepsilon {n^\alpha }}}{{4(\alpha p - 1 + \delta)}},$ by (3.3), (3.4), (3.13), $\alpha p > 1,$ ${C_r}$ inequality, we have

Hence, the proof of Theorem 3.1 is completed.

Proof of Theorem 3.2. Without loss of generality, we also can assume that ${a_{ni}} \geqslant 0$ for $1 \leqslant i \leqslant n$ and $n \geqslant 1.$ Where, the definitions of $g(x)$ and ${g_{j\beta }}(x)$ are the same as in the proof of Theorem 3.1. For $\forall \varepsilon > 0,$ we have that

According to Theorem 3.1, we have ${J_1} < \infty.$ So if we want to prove (3.8), we just need to prove ${J_2} < \infty.$ Hence, we first to prove

The definition of $\left\{ {{{Y'}_i}; 1 \leqslant i \leqslant n} \right\}$ is (3.6). For any $1 \leqslant i \leqslant n, n \geqslant 1,$ and $t \geqslant {n^{\alpha r}},$ denote

We have

In order to prove $H < \infty,$ it suffices to show ${H_1} < \infty$ and ${H_2} < \infty.$ Firstly, we prove ${H_1} < \infty,$ by (3.7), (3.12), Lemma 2.7 (i), $g(x) \downarrow$ when $x \geqslant 0,$ we have

Then, we prove ${H_2} < \infty.$ Firstly, we will show that

Similar to (3.14), by (3.12), (3.17), ${C_r}$ inequality, for any $\lambda > 0,$ we can get that

The truncation that defines $Y'$ as $X$ is as follows

When $0 < p \vee r \leqslant 1.$ Since $t \geqslant {n^{\alpha r}},$ ${\hat{\mathbb{E}}}{\left| X \right|^{p \vee r}} < \infty,$ and $\alpha (p \vee r) > 1,$ we get

When $p \vee r > 1.$ Since ${\hat{\mathbb{E}}}{X_i} = 0$ and $t \geqslant {n^{\alpha r}},$ we can get that

It follows that for all $n$ large enough,

which imply that

For fixed $t \geqslant {n^{\alpha r}}$ and $n \geqslant 1,$ through the definition in Theorem 3.2 and assume that ${a_{ni}} \leqslant {1 \mathord{\left/ {\vphantom {1 2}} \right. } 2},$ we know that $\left\{ {{a_{ni}}({{Y'}_i} - {\hat{\mathbb{E}}}{{Y'}_i}); 1 \leqslant i \leqslant n, n \geqslant 1} \right\}$ are WA random variables with ${\hat{\mathbb{E}}}\left({{a_{ni}}({{Y'}_i} - {\hat{\mathbb{E}}}{{Y'}_i})} \right){\text{ = }}0$ and ${a_{ni}}({Y_i}^\prime - {\hat{\mathbb{E}}}{Y'_i}) = \min \left\{ {{a_{ni}}({Y_i}^\prime - {\hat{\mathbb{E}}}{{Y'}_i}), {\text{ }}{\beta _2}{t^{{1 \mathord{\left/ {\vphantom {1 r}} \right. } r}}}} \right\}$. Use Lemma 2.8 for $\mathbb{V}\left({\sum\limits_{i = 1}^n {{a_{ni}}\left({{{Y'}_i} - {\hat{\mathbb{E}}}{{Y'}_i}} \right) > {{{t^{1/r}}} \mathord{\left/ {\vphantom {{{t^{1/r}}} 2}} \right. } 2}} } \right),$ taking $0 < {\beta _2} < \min \left\{ {\frac{{2 \wedge (p \vee r)}}{{2r}}, \frac{{\alpha [2 \wedge (p \vee r)] - 1}}{{2(\alpha p - 1 + \delta)}}} \right\},$ $d = {\beta _2}{t^{{1 \mathord{\left/ {\vphantom {1 r}} \right. } r}}},$ $x = {{{t^{{1 \mathord{\left/ {\vphantom {1 r}} \right. } r}}}} \mathord{\left/ {\vphantom {{{t^{{1 \mathord{\left/ {\vphantom {1 r}} \right. } r}}}} 2}} \right. } 2},$ we have

thus

So, to prove ${H_2} < \infty,$ we first need to prove ${H_{21}} < \infty.$ By Markov inequality, ${C_r}$ inequality, (3.12), (3.15), (3.17), Lemma 2.7 (ii), $q > p \vee r$ and ${H_1} < \infty$, we have that

Then, we prove ${H_{22}} < \infty.$ Similar to previous proof, we consider the following two situations.

If $(p \vee r) \geqslant 2.$ By ${\beta _2} < \frac{1}{r}, \alpha p - 2 + \delta + \frac{{1 - 2\alpha }}{{2{\beta _2}}} < - 1,$ (3.3), (3.7), (3.13), ${C_r}$ inequality, we have

If $(p \vee r) < 2.$ By ${\beta _2} < \frac{{p \vee r}}{{2r}}, \alpha p - 2 + \delta + \frac{{1 - (p \vee r)\alpha }}{{2{\beta _2}}} < - 1,$ ${\hat{\mathbb{E}}}{\left| X \right|^{p \vee r}} < \infty,$ ${C_r}$ inequality, we have

We have proved (3.16). Because of considering $\left\{ { - {X_n}; n \geqslant 1} \right\}$ instead of $\left\{ {{X_n}; n \geqslant 1} \right\}$ in Theorem 3.2, Theorem 3.2 still holds. Then we can obtain

According to (3.16) and (3.19), we can get ${J_2} < \infty.$ Hence, the finishes the proof of Theorem 3.2.

In conclusion, we prove the complete convergence and complete integral convergence for weighted sums of WA random variables under the sub-linear expectations.

4.   Conclusions

In this paper, we extend the conclusion in probability space to sublinear expectation space and obtain the complete convergence and complete integral convergence for weighted sums of WA random variables under the sub-linear expectations, which enriches the limit theory research of WA random variable sequence in sublinear expectation space. In the future work, we will establish the corresponding inequalities in the sublinear expectation space according to the existing important inequalities and moment inequalities in the probability space, overcome the problems caused by the sub-additive of $V$ and ${\hat{\mathbb{E}}}$, and generalize the complete convergence and complete integral convergence in the sublinear expectation to obtain a conclusion similar to that in the original probability space.

Acknowledgments

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

Conflict of interest

All authors declare no conflicts of interest in this paper.