Complete convergence for weighted sums of negatively dependent random variables under sub-linear expectations

1.   Introduction

Peng ^[1,2] firstly introduced the important concepts of the sub-linear expectations space to study the uncertainty in probability. Inspired by the important works of Peng ^[1,2], many scholars try to investigate the results under sub-linear expectations space, extending the corresponding ones in classic probability space. Zhang ^[3,4,5] established Donsker's invariance principle, exponential inequalities and Rosenthal's inequality under sub-linear expectations. Wu ^[6] obtained precise asymptotics for complete integral convergence under sub-linear expectations. Under sub-linear expectations, Xu and Cheng ^[7] investigated how small the increments of $G$-Brownian motion are. For more limit theorems under sub-linear expectations, the interested readers could refer to Xu and Zhang ^[8,9], Wu and Jiang ^[10], Zhang and Lin ^[11], Zhong and Wu ^[12], Hu and Yang ^[13], Chen ^[14], Chen and Wu ^[15], Zhang ^[16], Hu, Chen and Zhang ^[17], Gao and Xu ^[18], Kuczmaszewska ^[19], Xu and Cheng ^{[7,20,21,22,23]} and references therein.

In classic probability space, Hsu and Robbins ^[24] introduced concept of complete convergence, Chow ^[25] investigated complete moment convergence for independent random variables, Zhang and Ding ^[26] proved the complete moment convergence of the partial sums of moving average processes under some proper assumptions, Meng et al. ^[27] established complete convergence and complete moment convergence for weighted sums of extended negatively dependent random variables. For references on complete moment convergence in linear expectation space, the interested reader could refer to Ko ^[28], Meng et al. ^[29], Hosseini and Nezakati ^[30] and references therein. Encouraged by the work of Meng et al. ^[27], since the fact that $X$ is independent to $Y$ under sub-linear expectations implies that $X$ is negatively dependent to $Y$ under sub-linear expectations, we try to study the complete convergence and complete moment convergence for weighted sums of identically distributed, negatively dependent random variables under sub-linear expectations, which extends the corresponding results in Meng et al. ^[27].

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

2.   Preliminaries

As in Xu and Cheng ^[22], we use similar notations as in the work by Peng ^[2], Chen ^[14], Zhang ^[5]. Suppose that $(\Omega, {\mathcal F})$ is a given measurable space. Assume that ${\mathcal H}$ is a subset of all random variables on $(\Omega, {\mathcal F})$ such that $I_A\in {\mathcal H}$ (cf. Chen ^[14]), where $I(A)$ or $I_A$ represent the indicator function of $A$ throughout this paper, $A\in{\mathcal F}$, and $X_1, \cdots, X_n\in {\mathcal H}$ implies $\varphi(X_1, \cdots, X_n)\in {\mathcal H}$ for each $\varphi\in {\mathcal C}_{l, Lip}(\mathbb{R}^n)$, where ${\mathcal C}_{l, Lip}(\mathbb{R}^n)$ represents the linear space of (local lipschitz) function $\varphi$ fulfilling

for some $C > 0$, $m\in \mathbb{N}$ both depending 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 properties: for all $X, Y\in {\mathcal H}$, we have

$\rm (a)$ Monotonicity: If $X\ge Y$, then $\mathbb{E}[X]\ge \mathbb{E}[Y];$

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

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

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

Remark 2.1. In ${\rm(c)}$ of Definition 2.1, positive homogeneity could be understood by Theorem 1.2.1 of Peng ^[2], which says that a sub-linear expectation could be represented as a supremum of linear expectations. In Theorem 3.1, $\mathbb{E}[X] = \mathbb{E}[-X] = 0$ could imply that $\mathbb{E}[\alpha X] = \alpha \mathbb{E}[X]$ for all $\alpha\in \mathbb{R}$, but $\mathbb{E}[X] = \mathbb{E}[-X] = 0$ could not imply that $\mathbb{E}[\alpha X^{\beta}] = \alpha \mathbb{E}[X^{\beta}]$ for all $\alpha\in \mathbb{R}$ and $\beta\neq 1$. By Lemma 2.1, in order to justify $\mathbb{E}[X] = \mathbb{E}[-X] = 0$ in Theorem 3.1, we should have $\mathbb{E}[Z+X] = \mathbb{E}[Z-X]$, for all $Z\in {\mathcal H}$.

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

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

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

A capacity $V$ is called sub-additive if $V(A+B)\le V(A)+V(B)$, $A, B\in {\mathcal F}$.

In this article, given a sub-linear expectation space $(\Omega, {\mathcal H}, \mathbb{E})$, write $\mathbb{V}(A): = \inf\{ \mathbb{E}[\xi]:I_A\le \xi, \xi\in{\mathcal H}\} = \mathbb{E}[I_A]$, $\forall A\in {\mathcal F}$ (see (2.3) and the definitions of $\mathbb{V}$ above (2.3) in Zhang ^[4]). $\mathbb{V}$ is a sub-additive capacity. Define

Suppose that ${\bf{X}} = (X_1, \cdots, X_m)$, $X_i\in{\mathcal H}$ and ${\bf{Y}} = (Y_1, \cdots, Y_n)$, $Y_i\in {\mathcal H}$ are two random vectors on $(\Omega, {\mathcal H}, \mathbb{E})$. ${\bf{Y}}$ is called to be negatively dependent to ${\bf{X}}$, if for each function $\psi_1\in {\mathcal C}_{l, Lip}(\mathbb{R}^m)$, $\psi_2\in {\mathcal C}_{l, Lip}(\mathbb{R}^n)$, we have $\mathbb{E}[\psi_1({\bf{X}})\psi_2({\bf{Y}})]\le \mathbb{E}[\psi_1({\bf{X}})] \mathbb{E}[\psi_2({\bf{Y}})]$ whenever $\psi_1({\bf{X}})\ge 0$, $\mathbb{E}[\psi_2({\bf{Y}})]\ge 0$, $\mathbb{E}[\psi_1({\bf{X}})\psi_2({\bf{Y}})] < \infty$, $\mathbb{E}[|\psi_1({\bf{X}})|] < \infty$, $\mathbb{E}[|\psi_2({\bf{Y}})|] < \infty$, and either $\psi_1$ and $\psi_2$ are coordinatewise nondecreasing or $\psi_1$ and $\psi_2$ are coordinatewise nonincreasing (see Definition 2.3 of Zhang ^[4], Definition 1.5 of Zhang ^[5], Definition 2.5 in Chen ^[14]). $\{X_n\}_{n = 1}^{\infty}$ is named a sequence of negatively dependent random variables, if $X_{n+1}$ is negatively dependent to $(X_1, \cdots, X_n)$ for each $n\ge 1$.

Suppose that ${\bf{X}}_1$ and ${\bf{X}}_2$ are two $n$-dimensional random vectors defined, respectively, in sub-linear expectation spaces $(\Omega_1, {\mathcal H}_1, \mathbb{E}_1)$ and $(\Omega_2, {\mathcal H}_2, \mathbb{E}_2)$. They are named identically distributed if for every Borel-measurable function $\psi$ such that $\psi({\bf{X}}_1)\in {\mathcal H}_1, \psi({\bf{X}}_2)\in {\mathcal H}_2$,

whenever the sub-linear expectations are finite. $\{X_n\}_{n = 1}^{\infty}$ is named to be identically distributed if for each $i\ge 1$, $X_i$ and $X_1$ are identically distributed.

In this sequel we assume that $\mathbb{E}$ is countably sub-additive, i.e., $\mathbb{E}(X)\le \sum_{n = 1}^{\infty} \mathbb{E}(X_n)$, whenever $X\le \sum_{n = 1}^{\infty}X_n$, $X, X_n\in {\mathcal H}$, and $X\ge 0$, $X_n\ge 0$, $n = 1, 2, \ldots$. Let $C$ stand for a positive constant which may differ from place to place.

As discussed in Zhang ^[5], by the definition of negative dependence, 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 following lemmas under sub-linear expectations.

Lemma 2.1. (See Proposition 1.3.7 of Peng ^[2]) Under sub-linear expectation space $(\Omega, {\mathcal H}, \mathbb{E})$, if $X, Y\in {\mathcal H}$, $\mathbb{E}[Y] = \mathbb{E}[-Y] = 0$, then $\mathbb{E}[X+\alpha Y] = \mathbb{E}[X]$, for any $\alpha\in \mathbb{R}.$

Lemma 2.2. (See Lemma 4.5 (iii) of Zhang ^[4]) If $\mathbb{E}$ is countably sub-additive under sub-linear expectation space $(\Omega, {\mathcal H}, \mathbb{E})$, then for $X\in {\mathcal H}$,

Lemma 2.3. (See Theorem 2.1 and its proof of Zhang ^[5]) Assume that $p > 1$ and $\{X_n; n\ge 1\}$ is a sequence of negatively dependent random varables under sub-linear expectation space $(\Omega, {\mathcal H}, \mathbb{E})$. Then for each $n\ge 1$, there exists a positive constant $C = C(p)$ depending on $p$ such that for $1 < p\le 2$,

and for $p > 2,$

Proof. For reader's convenience, here we give the detailed proof. We first prove (2.1). Set $T_k = \max\{X_k, X_k+X_{k-1}, \ldots, X_k+\cdots+X_1\}$, $\breve{T}_n = \max\{|X_n|, |X_n+X_{n-1}|, \ldots, |X_n+\cdots+X_1|\}$. Since $T_{k}^{+}+X_{k+1}+\ldots+X_n\le T_n$, $T_{k}^{+}\le 2\breve{T}_n$. Substituting $x = X_k$ and $y = T_{k-1}^+$, $k = n, \ldots, 2$ to the following elementary inequality

results in

which by the definition of negative dependence and Hölder inequality under sub-linear expectations (see Proposition 1.4.2 of Peng ^[2]), implies that

Similarly,

Therefore

which implies that (2.1) holds.

Next, by (2.4) of Zhang ^[5] and its proof, we see that for $p > 2$

which implies that (2.2) holds.

By Lemma 2.3 and the similar argument as in Theorem 2.3.1 in Stout ^[31], we could obtain the following lemma.

Lemma 2.4. Assume that $q > 1$ and $\{X_n; n\ge 1\}$ is a sequence of negatively dependent random varables under sub-linear expectation space $(\Omega, {\mathcal H}, \mathbb{E})$. Then for each $n\ge 1$, there exists a positive constant $C = C(q)$ depending only on $q$ such that $1 < q\le 2,$

and for $q > 2$,

Proof. For reader's convenience, here we also give detailed proof. We only prove (2.4), since (2.5) is obvious from (2.3). We first prove (2.4) for $n = 2^k$, $k$ being an any positive integer. To avoid confusing the main idea, we just give the proof for $k = 6$. Let $X_{r, s} = \sum_{i = r+1}^{s}X_i$ for $0\le r < s\le 2^6$. We consider the following collections of $X_{r, s}$:

There are $k+1 = 7$ collections. We choose $1\le i\le 2^6$, and expand $S_i$, by using the terms of this expansion from the collections above and using the minimal possible number of terms in the expansion. Clearly at most one term is needed from each collections. As an example,

Hence each expansion has at most $k+1 = 7$ terms in it. Denote the expansion of $S_i$ by

It follows from Hölder inequality that

Now

Hence,

There are $k+1 = 7$ parenthetical expressions inside square brackets. By the $C_r$ inequality, we see that

which implies

By (2.1) and the above discussion,

Using an appropriate notion, the above discussion extended to any $k\ge 1$ implies

Given an $n$ such that $n\neq 2^k$ for any $k\ge 1$, choose $k$ satisfying $2^{k-1} < n < 2^k$ and redefine $X_i = 0$ if $n < i\le 2^k$. By (2.6), we see that

Since $2^{k-1} < n$ implies $(k+1)^q\le [\log(4n)/\log 2]^q$, (2.4) follows.

3.   Main results

Our main results are the following.

Theorem 3.1. Suppose $\alpha > \frac12$, $\alpha p > 1$ and $\{X_n; n\ge 1\}$ is a sequence of negatively dependent random variables, identically distributed as $X$ under sub-linear expectation space $(\Omega, {\mathcal H}, \mathbb{E})$. Assume that $\mathbb{E}(X) = \mathbb{E}(-X) = 0$ while $p > 1$. Suppose that $\{a_{ni}; 1\le i\le n, n\ge 1\}$ is an array of real numbers being all nonnegative or all non-positive such that

Let $C_{ \mathbb{V}}(|X|^p) < \infty$. Then for any $\varepsilon > 0$,

Theorem 3.2. Suppose $p > 1$, $\alpha\ge \frac12$, $\alpha p > 1$ and $\{X_n; n\ge 1\}$ is a sequence of negatively dependent random variables, identically distributed as $X$ under sub-linear expectation space $(\Omega, {\mathcal H}, \mathbb{E})$. Assume that $\mathbb{E}(X) = \mathbb{E}(-X) = 0$. Suppose that $\{a_{ni}; 1\le i\le n, n\ge 1\}$ is an array of real numbers being all nonnegative or all non-positive such that (3.1) holds. Let $C_{ \mathbb{V}}(|X|^p) < \infty$. Then for any $\varepsilon > 0,$

Remark 3.1. Under the assumptions of Theorem 3.2, we see that for all $\varepsilon > 0,$

By (3.4), we can conclude that the complete moment convergence implies the complete convergence.

4.   Proof of major results

4.1.   Proof of Theorem 3.1

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

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

which results in

Firstly, we will establish that

We study the following three cases.

(ⅰ) If $\frac12 < \alpha\le 1,$ then $p > 1$. By $\mathbb{E} X = \mathbb{E}(-X) = 0, \; | \mathbb{E}(X-Y)|\le \mathbb{E}|X-Y|, \; C_{ \mathbb{V}}(|X|^p) < \infty,$ Lemmas 1 and 2, we can see that

(ⅱ) If $\alpha > 1$, $p < 1$, then by $C_{ \mathbb{V}}\left(|X|^p\right) < \infty$, and Lemma 2.2, we see that

(ⅲ) If $\alpha > 1$, $p\ge 1$, then by $\mathbb{E}|X|\le \left(\mathbb{E}|X|^p\right)^{1/p}\le \left(C_{ \mathbb{V}}\left(|X|^p\right)\right)^{1/p} < \infty$, Markov inequality under sub-linear expectations, Hölder inequality, we see that

Combining (4.4)–(4.6) results in (4.3) immediately. Hence, for $n$ sufficiently large,

To prove (3.2), we only need to establish that

and

For $I$, by Markov inequality under sub-linear expectations, and Lemma 2.2, we obtain

As pointed before Lemma 2.2, we see that $\{Y_{ni}- \mathbb{E} Y_{ni}; 1\le i\le n, n\ge 1\}$ is also a sequence of negatively dependent random variables. By Lemma 2.4, Markov inequality under sub-linear expectations, and the $C_r$ inequality, we conclude that for $q > 2$,

Taking $q > \max\{2, p\}$, by the $C_r$ inequality, Markov inequality under sub-linear expectations, and Lemma 2.2, we have

For $II_2$, we study the following cases.

(ⅰ) If $p\ge 2$, observe that $\sum_{i = 1}^{n}a_{ni}^2\le \left(\sum_{i = 1}^{n}|a_{ni}|^p\right)^{2/p}n^{1-2/p}\le n^{1-2(1-\delta)/p}$. Taking $q > \max\left\{2, \frac{2p(\alpha p-1)}{2\alpha p-p+2(1-\delta)}\right\}$, by the $C_r$ inequality, $\mathbb{E} X^2\le \left(\mathbb{E}(|X|^p)\right)^{1/p}\le \left(C_{ \mathbb{V}}\left(|X|^p\right)\right)^{1/p} < \infty$, we see that

(ⅱ) If $p < 2$, we take $q > 2(\alpha p-1)/(\alpha p-\delta)$. By the $C_r$ inequality, Markov inequality under sub-linear expectations, and Lemma 2.2, we see that

For $II_3$, we study the following cases.

(ⅰ) If $\frac12 < \alpha\le 1$, then $p > 1$. Taking $q > \frac{\alpha p-1}{ \alpha p-\delta}$, by $\mathbb{E}(X) = \mathbb{E}(-X) = 0$, Lemmas 1 and 2, we see that

(ⅱ) If $\alpha > 1$, $p < 1$, taking $q > \frac{\alpha p-1}{\alpha p-\delta}$, by Lemma 2.2, we obtain

(ⅲ) If $\alpha > 1$, $p > 1$, then $\mathbb{E}|X|\le(\mathbb{E}|X|^p)^{1/p}\le (C_{ \mathbb{V}}(|X|^p))^{1/p} < \infty$. We take $q > \frac{(\alpha p-1)p}{\alpha p-p+(1-\delta)}$. Hence by $C_r$ inequality, Markov inequality under sub-linear expectations, and Hölder inequality, we see that

Hence, the proof of Theorem 3.1 is finished.

4.2.   Proof of Theorem 3.2

Proof. For all $\varepsilon > 0$ and any $t > 0$, we see that

By Theorem 3.1, we conclude that $III_1 < \infty$. Therefore, it is enough to establish $III_2 < \infty$. Without loss of restriction, assume that $a_{ni}\ge 0$. For all $1\le i\le n$, $n\ge 1$, $t\ge n^{\alpha}$, write

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

which results in

For $III_{21}$, by $p > 1$, and Lemma 2.2, we obtain

For $III_{22}$, we firstly establish that

For $1\le i\le n$, $n\ge 1$ and $p > 1$, by $\mathbb{E} X_n = \mathbb{E} (-X_n) = 0$ and Lemma 2.1, we see that $\mathbb{E} Y_{ni}' = \mathbb{E}(-Z_{ni})$. If $a_{ni}X_i > t$, $0 < Z_{ni} = a_{ni}X_i-t < a_{ni}X_i$. If $a_{ni}X_i < -t$, $a_{ni}X_i < Z_{ni} = a_{ni}X_i+t\le 0$. Hence $|Z_{ni}|\le |a_{ni}X_i|I\left(|a_{ni}X_i| > t\right)$. Then, by Lemma 2.2, we have

Hence, while $n$ is large enough, for $t\ge n^{\alpha}$,

which results in

In the following, we present $III_{22} < \infty$, for $1 < p\le 2$ and $p > 2$.

(ⅰ) If $1 < p\le 2$, by (4.23), Lemma 2.4, Markov inequality under sub-linear expectations, and the $C_r$ inequality, we obtain

For $III_{221}$, by $1 < p\le 2$, we see that

For $III_{222}$, by $1 < p\le 2$, by Markov inequality under sub-linear expectations, and Lemma 2.2, we see that

For $III_{223} < \infty$, by the proof of $III_{21} < \infty$, we can see that $III_{223} < \infty$. For $III_{224}$, by $1 < p\le 2$, $\mathbb{E}(-X) = \mathbb{E}(X) = 0$, and Lemma 2.1, we obtain

Therefore, we conclude that $III_{22} < \infty$ for $1 < p\le 2$.

(ⅱ) If $p > 2$, by (4.23), $\mathbb{E}(X) = \mathbb{E}(-X) = 0$, Markov inequality under sub-linear expectations, the $C_r$ inequality, and Lemma 2.4 (for $q > 2$), we see that

For $IV_1$, we obtain

By the similar proofs of $III_{221} < \infty$ and $III_{222} < \infty$ (with $q$ in place of the exponent $2$), we can see that $IV_{11} < \infty$ and $IV_{12} < \infty$. Similarly, by the proof of $III_{21} < \infty$, we can see that $IV_{13} < \infty$.

For $IV_2$, we obtain

For $IV_{21}$, taking $q > \max\left\{2, \frac{2p(\alpha p-1)}{2\alpha p-p+2(1-\delta)}\right\}$, by the $C_r$ inequality, Jensen inequality under sub-linear expectations, and Lemma 2.2, we obtain

For $IV_{22}$, taking $q > \max\left\{2, \frac{2(\alpha p-1)}{(\alpha p-\delta)}\right\}$, by Jensen inequality under sub-linear expectations, and Lemma 2.2, we have

For $IV_{23}$, by Markov inequality under sub-linear expectations, and Lemma 2.2, we conclude that

Since $t\ge n^{\alpha}$, for all $n$ sufficiently large, we deduce that

By (4.29), we obtain

For $IV_3$, taking $q > \max\left\{\frac{\alpha p-1}{\alpha p-\delta}, 2\right\}$, by $\mathbb{E}(X) = \mathbb{E}(-X) = 0$, Lemma 2.1, and Lemma 2.2, we see that

Hence, the proof of Theorem 3.2 is finished.

5.   Conclusions

We have established the new results of complete convergence and complete moment convergence for weighted sums of negatively dependent random variables under sub-linear expectations. Theorems of this article are the extensions of convergence properties for weighted sums of extended negatively dependent random variables under classical probability space.

Acknowledgments

This research was supported by Doctoral Scientific Research Starting Foundation of Jingdezhen Ceramic University (Nos.102/01003002031), Natural Science Foundation Program of Jiangxi Province 20202BABL211005, and National Natural Science Foundation of China (Nos. 61662037), Jiangxi Province Key S & T Cooperation Project (Nos. 20212BDH80021).

Conflict of interest

All authors declare no conflict of interest in this paper.