Laws of Large Numbers for Uncertain Random Variables in the Framework of U-S Chance Theory

Abstract

The paper introduces U-S chance spaces, a new framework based on uncertainty theory and sub-linear expectation theory, to depict human uncertainty and sub-linear features, simultaneously. These spaces can be used to analyze the characteristics of uncertain random variables and study investments and other related issues in incomplete financial markets. Within the framework, sub-linear expectation theory describes the randomness in financial behaviors, while uncertainty theory describes the uncertainty associated with government macro-control or experts’ opinions. The main achievement of this paper is the derivation of the Kolmogorov law of large numbers for uncertain random variables under U-S chance spaces. Examples are provided, and the theorems can be applied to uncertain random variables that are functions of random variables with symmetric or asymmetric distributions and uncertain variables with symmetric or asymmetric distributions. In some cases, when both random and uncertain variables are symmetric, the limit in the law exhibits the form that is characterized by symmetrical uncertain variables.

1. Introduction

In the 16th century, Cardano introduced a kind of limit theorem within the framework of classical probability theory, which subsequently gained recognition as the “Law of Large Numbers” (abbreviated as LLNs). Since then, LLN has since been investigated by numerous mathematicians, such as Bernoulli, Kolmogorov, and so on. Following a protracted period of study and development, LLN has developed into a nearly flawless theoretical framework and is now frequently applied in practical applications. The additivity of probabilities and the linear property of expectations play significant roles in the proof of such LLNs. Due to the fact that many events in both subjective and objective environments cannot be represented with additive probability or classical linear expectation, the additivity premise is not practical in numerous fields of application. Since Choquet proposed the concept of non-additive probability (capacity) in [1], people have become increasingly interested in non-additive probability theory. Non-additive probabilities are used in objective settings, particularly in quantum mechanics. According to [2], despite the frequentist interpretation commonly applied to quantum phenomena, the probabilities describing them are typically non-additive due to the well-known wave-particle duality. Non-additive probabilities are used in subjective settings because additive ones make it hard to analyze decision-makers’ confidence.

One very significant aspect of the real world is the fuzzy phenomenon. In order to model fuzzy phenomenon, [3] introduced the idea of a fuzzy set, and [4] established possibility theory, which is connected to the theory of fuzzy sets. Following that, further research was performed on the fuzzy measure, Sugeno integral, and Choquet integral (see, e.g., [5,6,7,8]). Furthermore, [6,7] divided the fuzzy measures into eight different categories that are closed under the operations of distortion functions: submeasure, supermeasure, submodular, supermodular, belief, plausibility, possibility, and necessity. Numerous surveys indicated that human uncertainty does not act in the same way as fuzziness. Before [9,10], the argument centered on the idea that the maximum of measurements of individual events is not always the same as the measure of the union of events, a perspective that was eventually summarized in [11].

To manoeuvre around this disadvantage, [9] created uncertainty theory, and [10] also developed it. An uncertain measure fulfills normality, duality, subadditivity, and product axioms. According to uncertainty theory, the degree of belief that an event will occur is indicated by the uncertain measure of the event. In order to represent the quantities under uncertain situations, this theory offers the concept of an uncertain variable. [12] studied the strong law of large numbers and the weak law of large numbers for Bernoulli uncertain sequences. In complex systems, human uncertainty and random factors might coexist at times. To investigate such complex systems, [13,14] developed the notion of an uncertain random variable to describe the quantities under uncertain and random situations, merged the concepts of probability and uncertain measures, and introduced the idea of chance measure. These concepts, which are founded on probability and updated uncertainty theory in [15], are mathematical explanations for uncertain random events. In this chance space, LLNs have produced some excellent study findings. The first demonstration of LLN under chance measure was made by [16]. It shows that the mean of uncertain random variables, which are functions, converges in distribution to an uncertain variable if random variables are independent and identically distributed (IID for short), and uncertain variables are IID and regular. LLNs under chance measures were subsequently developed by [17,18,19,20,21].

In the real world, apart from randomness, fuzziness, and human uncertainty, financial uncertainty is another form of indeterminacy. Inspired by risk measures, super-hedge pricing, and modeling uncertainty in finance, [22] first introduced the concepts of IID random variables, maximum distribution, and G-normal distribution in the framework of sub-linear expectations. Many financial uncertainties are difficult to adequately model using additive probability and traditional linear expectation. Nevertheless, a very adaptable framework for analyzing and explaining them is provided by sub-linear expectation theory. [22] has investigated weak convergences, including weak LLNs and central limit theorems under sub-linear expectation theory. Subsequently, [23,24] discovered three different types of strong LLNs for non-additive probabilities within the same context, resulting in the development of novel strategies for handling uncertain financial issues. A number of researchers have subsequently investigated strong LLNs in the framework of sub-linear expectations theory. For example, [25] made the strong LLNs in [23,24] hold in this case by providing the weaker requirement that random variables be independent, but not necessarily identically distributed. An expansion of the independence of random variables under sub-linear expectation known as negative dependence was first presented by [26]. Strong LLNs in [23,24] were also extended under the weakest moment conditions by [26]. Almost at the same time, the sub-linear expectations theory was also improved in [27,28]. The latest findings on LLNs based on this theory are presented in [29,30].

However, in the intricate and multifaceted real world, there exists a class of highly challenging systems. These systems are characterized not only by randomness with nonlinear features but also by the imprecision introduced by human subjective consciousness and linguistic expression. The stock market stands as a quintessential example of such complex systems, teeming with myriad random factors and human-induced uncertainties. Regrettably, previous theoretical frameworks have struggled to adequately model such systems. Inspired by the method presented in [31], we propose a novel theoretical framework of U-S chance space, aiming to provide a more precise portrayal of the complete picture of these complex systems. The core of the U-S chance space theoretical framework lies in the ingenious integration of sub-linear expectation theory and uncertainty theory, constructing a two-dimensional product space. Within this spatial framework, tools such as chance measures and chance distributions can comprehensively characterize the probabilistic properties and statistical regularities of uncertain random variables. These novel mathematical tools enable a more precise analysis of complex scenarios in financial environments, encompassing both random phenomena with sub-linear expectation characteristics and uncertainties stemming from human subjective judgments. For instance, when studying stock investments in incomplete financial markets, one can utilize sub-linear expectation theory to depict the incomplete nature of the market while leveraging uncertainty theory to address uncertainties arising from government macro-controls or expert opinions, which significantly impact stock returns and prices. Additionally, [32] gave four distinct kinds of expectations and four models of uncertain random programming under U-S chance spaces, and they were used for optimal investment in incomplete financial markets and system reliability design. [33] proposed risk aversion in incomplete markets affected by government regulation in the framework of the U-S chance theory, presented expectations of risk and the risk premium, and proved Pratt’s theorem under U-S chance spaces. Given the significant practical implications of the U-S chance space, the construction of the LLN under this space undoubtedly emerges as a highly valuable research direction. It is noteworthy that the LLNs for uncertain random variables presented in [20] are based on the assumption of linear probability and expectation. To transcend this linear assumption, under a series of reasonable conditional hypotheses, we cleverly employ the results of the LLNs for random variables from [24] and successfully derive the Kolmogorov type LLN for uncertain random variables being functions of IID random variables and IID uncertain variables under U-S chance spaces in Section 3. Here, the assumption of regularity for uncertain variables is necessary. A number of examples are given and explained in order to better illustrate the LLN for uncertain random variables under U-S chance spaces. In certain specific scenarios, if random and uncertain variables are both symmetric, then the limit in the Kolmogorov type LLN for uncertain random variables under U-S chance spaces also exhibits the symmetrical form. The innovations of this paper are primarily manifested in two aspects: firstly, theoretical innovation, as we propose the theoretical framework of the U-S chance space for the first time; secondly, methodological innovation, given the non-additivity of nonlinear expectations, traditional methods and techniques for studying linear expectations and probabilities are no longer applicable in the study of the LLNs within the U-S chance space. Therefore, we construct appropriate inequalities and innovatively improve the proof methods and techniques of the LLNs presented in [20].

This is the structure for the remainder of the paper. Some novel ideas regarding U-S chance spaces are presented in Section 2. In Section 3, we obtain the Kolmogorov type LLN for uncertain random variables that are functions of IID random variables and IID uncertain variables under U-S chance spaces. After that, five examples are given, which can be seen as two applications of the LLN in Section 4, and a short conclusion is presented in Section 5. Appendix A contains some fundamental definitions pertaining to uncertainty theory. Finally, some results of sub-linear expectations that are used in this paper are given in Appendix B.

2. U-S Chance System and Related Concepts

To describe complex systems where human uncertainty and randomness coexist, [13,14] proposed the chance theory. This randomness can be portrayed through classical probability theory. However, with the development of society, many random phenomena cannot be analyzed through additive probabilities and need to be modeled by non-additive probabilities. For complex systems where human uncertainty and randomness with sub-linear characteristics coexist, the chance theory cannot analyze the problems in such systems well. Therefore, a new theoretical framework is needed to be constructed for dealing with the problems in the above systems. We combine uncertainty theory and sub-linear expectation theory to propose U-S chance theory. Chance measures under U-S chance spaces are the most basic concepts, and they can be used to represent the possible degree that an uncertain random event may occur under U-S chance spaces. An uncertain random variable is a measurable function from the U-S chance space to the set of real numbers. It is used to indicate quantities with both uncertainty and randomness with sub-linear characteristics. Chance distributions can be used to characterize uncertain random variables under U-S chance spaces, and they are carriers of incomplete information of uncertain random variables. In some instances, it is more important to understand chance distributions than uncertain random variables themselves. In this section, we develop a novel class of framework for chance spaces called U-S chance spaces. Under the chance spaces, definitions for chance distributions, chance measures, and uncertain random variables are suggested.

Definition 1.

Suppose that $(\Gamma ,\mathcal{L},\mathcal{M})$ is an uncertainty space, $(\Omega ,\mathcal{F})$ is a measurable space, and $(\Omega ,\mathcal{H},\mathbb{E})$ is a sub-linear expectation space. Let $\mathbb{V}$, v be two non-additive probabilities generated by sub-linear expectation $\mathbb{E}$. A pair of chance spaces generated by uncertainty space and sub-linear expectation space (abbreviated U-S chance spaces) are the forms as follows:

$$(\Gamma ,\mathcal{L},\mathcal{M})\times (\Omega ,\mathcal{F},\mathbb{V})=(\Gamma \times \Omega ,\mathcal{L}\times \mathcal{F},\mathcal{M}\times \mathbb{V}),$$

and

$$(\Gamma ,\mathcal{L},\mathcal{M})\times (\Omega ,\mathcal{F},v)=(\Gamma \times \Omega ,\mathcal{L}\times \mathcal{F},\mathcal{M}\times v),$$

where $\Gamma \times \Omega$ represents the universal set, $\mathcal{L}\times \mathcal{F}$ represents the product σ-algebra, $\mathcal{M}\times \mathbb{V}$ and $\mathcal{M}\times v$ represent two product measures. $\Theta$ is called an uncertain random event in U-S chance spaces if $\Theta \in \mathcal{L}\times \mathcal{F}$. The chance measures $\overline{ch}$ and $CH$ of $\Theta$ can be defined by

$$\overline{ch}\{\Theta \}:={\int }_0^{1}v\{\omega \in \Omega \mid \mathcal{M}\{\gamma \in \Gamma \mid (\gamma ,\omega )\in \Theta \}\ge r\}\mathrm{d}r,$$

(1)

and

$$CH\{\Theta \}:={\int }_0^1\mathbb{V}\{\omega \in \Omega \mid \mathcal{M}\{\gamma \in \Gamma \mid (\gamma ,\omega )\in \Theta \}\ge r\}\mathrm{d}r,$$

(2)

respectively.

$(\Omega ,\mathcal{H},\mathbb{E})$, $v,$ and $\mathbb{V}$ are in the framework of sub-linear expectation theory presented by Peng [27,28] and Chen [23,24]. In Appendix B, we provide some fundamental definitions and properties regarding sub-linear expectation theory, including the definitions of sub-linear expectation $\mathbb{E}$, non-additive probabilities v and $\mathbb{V}$, IID random variables, maximal distribution and G-normal distribution and some of their properties.

Remark 1.

It is evident that $\Gamma \times \Omega$ represents the universal set containing all ordered pairs $(\gamma ,\omega )$, where $\gamma \in \Gamma$ and $\omega \in \Omega$, i.e.,

$$\Gamma \times \Omega =\left\{\right(\gamma ,\omega \left)\right|\gamma \in \Gamma ,\omega \in \Omega \}.$$

The product σ-algebra $\mathcal{L}\times \mathcal{F}$ represents the smallest σ-algebra that contains measurable rectangles $\Lambda \times B$, where $\Lambda \in \mathcal{L}$ and $B\in \mathcal{F}$. Under U-S chance spaces, an element is considered an event if it is in $\mathcal{L}\times \mathcal{F}$.

Next, using a similar approach by Liu [15] (pp. 409–410), the product measures $\mathcal{M}\times \mathbb{V}$ and $\mathcal{M}\times v$ are discussed. Assume that $\Theta$ becomes an event in $\mathcal{L}\times \mathcal{F}$. For every $\omega \in \Omega$, we obtain that the set

$${\Theta }_{\omega }=\{\gamma \in \Gamma |(\gamma ,\omega )\in \Theta \}$$

becomes an event in $\mathcal{L}$. Hence, for any $\omega \in \Omega$, uncertain measure $\mathcal{M}\left\{{\Theta }_{\omega }\right\}$ is existing. However, unluckily, there is no guarantee that $\mathcal{M}\left\{{\Theta }_{\omega }\right\}$ is a measurable function about ω. As a result, for any real number x, the set

$${\Theta }_x^{*}=\{\omega \in \Omega |\mathcal{M}\left\{{\Theta }_{\omega }\right\}\ge x\}$$

becomes a subset of Ω but possibly fails to be an event in $\mathcal{F}$. Therefore, the existences of upper probability measure $\mathbb{V}\left\{{\Theta }_x^{*}\right\}$ and lower probability measure $v\left\{{\Theta }_x^{*}\right\}$ are not ensured. For this instance, let

$$\mathbb{V}\left\{{\Theta }_x^{*}\right\}=\left\{\begin{array}{cc}\underset{B\in \mathcal{F},B\supseteq {\Theta }_x^{*}}{inf}\mathbb{V}\left\{B\right\},& \mathrm{if}\underset{B\in \mathcal{F},B\supseteq {\Theta }_x^{*}}{inf}\mathbb{V}\left\{B\right\}<0.5\\ \\ \underset{B\in \mathcal{F},B\subseteq {\Theta }_x^{*}}{sup}\mathbb{V}\left\{B\right\},& \mathrm{if}\underset{B\in \mathcal{F},B\subseteq {\Theta }_x^{*}}{sup}\mathbb{V}\left\{B\right\}>0.5\\ \\ 0.5,& \mathrm{otherwise}\end{array}\right.$$

and

$$v\left\{{\Theta }_x^{*}\right\}=\left\{\begin{array}{cc}\underset{B\in \mathcal{F},B\supseteq {\Theta }_x^{*}}{inf}v\left\{B\right\},& \mathrm{if}\underset{B\in \mathcal{F},B\supseteq {\Theta }_x^{*}}{inf}v\left\{B\right\}<0.5\\ \\ \underset{B\in \mathcal{F},B\subseteq {\Theta }_x^{*}}{sup}v\left\{B\right\},& \mathrm{if}\underset{B\in \mathcal{F},B\subseteq {\Theta }_x^{*}}{sup}v\left\{B\right\}>0.5\\ \\ 0.5,& \mathrm{otherwise}\end{array}\right.$$

based on the principle of maximum uncertainty. For each real number x, this guarantees the existences of upper probability measure $\mathbb{V}\left\{{\Theta }_x^{*}\right\}$ and lower probability measure $v\left\{{\Theta }_x^{*}\right\}$. It is now possible to define $\mathcal{M}\times \mathbb{V}$ and $\mathcal{M}\times v$ of $\Theta$ as the expected values of $\mathcal{M}\left\{{\Theta }_{\omega }\right\}$ with regard to $\omega \in \Omega$, i.e., ${\int }_0^1\mathbb{V}\left\{{\Theta }_r^{*}\right\}\mathrm{d}r$ and ${\int }_0^{1}v\left\{{\Theta }_r^{*}\right\}\mathrm{d}r$. Thus, chance measures $CH$ and $\overline{ch}$ can be well defined.

Proposition 1.

Suppose that $(\Gamma ,\mathcal{L},\mathcal{M})\times (\Omega ,\mathcal{F},\mathbb{V})$ and $(\Gamma ,\mathcal{L},\mathcal{M})\times (\Omega ,\mathcal{F},v)$ are U-S chance spaces. Four properties of the chance measures $\overline{ch}$ and $CH$ can be listed as following:

(i)

$\overline{ch}\{\Lambda \times A\}=\mathcal{M}\left\{\Lambda \right\}\times v\left\{A\right\}$; $CH\{\Lambda \times A\}=\mathcal{M}\left\{\Lambda \right\}\times \mathbb{V}\left\{A\right\}$ for $\forall \Lambda \in \mathcal{L}$ and $A\in \mathcal{F}$. In particular, $\overline{ch}\{\varnothing \}=0$, $\overline{ch}\{\Gamma \times \Omega \}=1$; $CH\{\varnothing \}=0$, $CH\{\Gamma \times \Omega \}=1.$

(ii)

$CH\{\Theta \}+\overline{ch}\left\{{\Theta }^c\right\}=1$, $\Theta \in \mathcal{L}\times \mathcal{F}.$

(iii)

For events ${\Theta }_1$, ${\Theta }_2\in \mathcal{L}\times \mathcal{F},$ if ${\Theta }_1\subseteq {\Theta }_2,$ then $\overline{ch}\left\{{\Theta }_1\right\}\le \overline{ch}\left\{{\Theta }_2\right\}$ and $CH\left\{{\Theta }_1\right\}\le CH\left\{{\Theta }_2\right\}.$

(iv)

$\overline{ch}\{\Theta \}\le CH\{\Theta \}$, $\Theta \in \mathcal{L}\times \mathcal{F}.$

Proof. 

(i) For every $\omega \in A$, it is clear that $\left\{\gamma \in \Gamma \left|\right(\gamma ,\omega )\in \Lambda \times A\right\}=\Lambda$, and for every $\omega \in A^c$, it is easy to find that $\left\{\gamma \in \Gamma \left|\right(\gamma ,\omega )\in \Lambda \times A\right\}=\varnothing$. Then $\mathcal{M}\left\{\gamma \in \Gamma \left|\right(\gamma ,\omega )\in \Lambda \times A\right\}=\mathcal{M}\{\Lambda \}{I}_A\left(\omega \right)$, for all $\omega \in \Omega$. For ∀r, suppose $\mathcal{M}\{\Lambda \}\ge r$, then we have

$$v\left\{\omega \in \Omega \mid \mathcal{M}\{\gamma \in \Gamma \mid (\gamma ,\omega )\in \Lambda \times A\}\ge r\right\}=v\left\{A\right\}.$$

Suppose $\mathcal{M}\{\Lambda \}<r$, then we have

$$v\left\{\omega \in \Omega \mid \mathcal{M}\{\gamma \in \Gamma \mid (\gamma ,\omega )\in \Lambda \times A\}\ge r\right\}=v\{\varnothing \}=0.$$

Thus, it is concluded that

$$\begin{array}{cc}\hfill & \overline{ch}\{\Lambda \times A\}\hfill \\ \hfill & ={\int }_0^{1}v\{\omega \in \Omega \mid \mathcal{M}\{\gamma \in \Gamma \mid (\gamma ,\omega )\in \Lambda \times A\}\ge r\}\mathrm{d}r\hfill \\ \hfill & ={\int }_0^{\mathcal{M}\{\Lambda \}}v\left\{A\right\}\mathrm{d}r+{\int }_{\mathcal{M}\{\Lambda \}}^{1}0\mathrm{d}r=\mathcal{M}\left\{\Lambda \right\}\times v\left\{A\right\}.\hfill \end{array}$$

Moreover, it is easy to obtain that $\overline{ch}\{\varnothing \}=\mathcal{M}\left\{\varnothing \right\}\times v\left\{\varnothing \right\}=0$, $\overline{ch}\{\Gamma \times \Omega \}=\mathcal{M}\left\{\Gamma \right\}\times v\left\{\Omega \right\}=1.$ Using a similar method as above, we have $CH\{\Lambda \times A\}=\mathcal{M}\left\{\Lambda \right\}\times \mathbb{V}\left\{A\right\}$, $CH\{\varnothing \}=0$, $CH\{\Gamma \times \Omega \}=1.$

(ii) Taking $\Theta \in \mathcal{L}\times \mathcal{F}$, since $\mathbb{V}\left\{A\right\}+v\left\{A^c\right\}=1$, $A\in \mathcal{F}$, it follows from the duality of $\mathcal{M}$ that

$$\begin{array}{cc}\hfill & CH\{\Theta \}\hfill \\ \hfill & ={\int }_0^1\mathbb{V}\{\omega \in \Omega \mid \mathcal{M}\{\gamma \in \Gamma \mid (\gamma ,\omega )\in \Theta \}\ge r\}\mathrm{d}r\hfill \\ \hfill & ={\int }_0^1\mathbb{V}\{\omega \in \Omega \mid \mathcal{M}\{\gamma \in \Gamma \mid (\gamma ,\omega )\in {\Theta }^c\}\le 1-r\}\mathrm{d}r\hfill \\ \hfill & ={\int }_0^{1}1-v\{\omega \in \Omega \mid \mathcal{M}\{\gamma \in \Gamma \mid (\gamma ,\omega )\in {\Theta }^c\}>1-r\}\mathrm{d}r\hfill \\ \hfill & =-{\int }_1^{0}1-v\{\omega \in \Omega \mid \mathcal{M}\{\gamma \in \Gamma \mid (\gamma ,\omega )\in {\Theta }^c\}>r\}\mathrm{d}r\hfill \\ \hfill & =1-{\int }_0^{1}v\{\omega \in \Omega \mid \mathcal{M}\{\gamma \in \Gamma \mid (\gamma ,\omega )\in {\Theta }^c\}>r\}\mathrm{d}r\hfill \\ \hfill & =1-\overline{ch}\left\{{\Theta }^c\right\}.\hfill \end{array}$$

So, we obtain $CH\{\Theta \}+\overline{ch}\left\{{\Theta }^c\right\}=1.$

(iii) Suppose that ${\Theta }_1\subseteq {\Theta }_2$, for every $\omega \in \Omega$, we can conclude that $\{\gamma \in \Gamma \mid (\gamma ,\omega )\in {\Theta }_1\}\subseteq \{\gamma \in \Gamma \mid (\gamma ,\omega )\in {\Theta }_2\}.$ It is obvious from the monotonicity of $\mathcal{M}$ that

$$\mathcal{M}\{\gamma \in \Gamma \mid (\gamma ,\omega )\in {\Theta }_1\}\le \mathcal{M}\{\gamma \in \Gamma \mid (\gamma ,\omega )\in {\Theta }_2\}.$$

From the fact that

$$\{\omega \in \Omega \mid \mathcal{M}\{\gamma \in \Gamma \mid (\gamma ,\omega )\in {\Theta }_1\}\ge r\}$$

$$\subseteq \{\omega \in \Omega \mid \mathcal{M}\{\gamma \in \Gamma \mid (\gamma ,\omega )\in {\Theta }_2\}\ge r\}$$

and the monotonicity of v, it follows that

$${\int }_0^{1}v\{\omega \in \Omega \mid \mathcal{M}\{\gamma \in \Gamma \mid (\gamma ,\omega )\in {\Theta }_1\}\ge r\}\mathrm{d}r$$

$$\le {\int }_0^{1}v\{\omega \in \Omega \mid \mathcal{M}\{\gamma \in \Gamma \mid (\gamma ,\omega )\in {\Theta }_2\}\ge r\}\mathrm{d}r.$$

Thus, $\overline{ch}\left\{{\Theta }_1\right\}\le \overline{ch}\left\{{\Theta }_2\right\}$.

Similar to the method above, it is easy to obtain that $CH\left\{{\Theta }_1\right\}\le CH\left\{{\Theta }_2\right\}$ from the monotonicity of $\mathbb{V}$.

(iv) By the fact that $v\left\{A\right\}\le \mathbb{V}\left\{A\right\}$, $A\in \mathcal{F}$, for ∀$\Theta \in \mathcal{L}\times \mathcal{F}$, $0\le r\le 1$, it can be obtained that

$$v\{\omega \in \Omega \mid \mathcal{M}\{\gamma \in \Gamma \mid (\gamma ,\omega )\in \Theta \}\ge r\}$$

$$\le \mathbb{V}\{\omega \in \Omega \mid \mathcal{M}\{\gamma \in \Gamma \mid (\gamma ,\omega )\in \Theta \}\ge r\}.$$

Thus, from (1) and (2), the conclusion $\overline{ch}\{\Theta \}\le CH\{\Theta \}$ can be obtained. □

Remark 2.

In generally, $\overline{ch}$ and $CH$ are both not subadditive because the subadditivity of upper probability $\mathbb{V}$ could not determine the subadditivity of $\overline{ch}$ and $CH$.

Definition 2.

If a function $\xi :\Gamma \times \Omega \mapsto \mathbb{R}$ is measurable, then it is named an uncertain random variable in U-S chance spaces, i.e., for each $B\in \mathcal{B}\left(\mathbb{R}\right)$

$$\{\xi \in B\}=\left\{\right(\gamma ,\omega )\mid \xi (\gamma ,\omega )\in B\}\in \mathcal{L}\times \mathcal{F}.$$

A pair of chance distributions ${\Xi }_1$ and ${\Xi }_2$ for ξ with the functions

$${\Xi }_1\left(x\right)=\overline{ch}\{\xi \le x\},{\Xi }_2\left(x\right)=CH\{\xi \le x\},x\in \mathbb{R},$$

are named the lower distribution and the upper distribution, respectively.

Both here and in the next sections, uncertain random variables are variables under U-S chance spaces $(\Gamma ,\mathcal{L},\mathcal{M})\times (\Omega ,\mathcal{F},\mathbb{V})$ and $(\Gamma ,\mathcal{L},\mathcal{M})\times (\Omega ,\mathcal{F},v)$.

3. LLN for Uncertain Random Variables Under U-S Chance Spaces

In this section, the Kolmogorov type LLN for uncertain random variables under U-S chance spaces is given. We first require a few fundamental notations and assumptions before we provide the LLN.

Let ${\left\{{\eta }_i\right\}}_{i=1}^{\infty }$ be a sequence of random variables, and ${\left\{{\tau }_i\right\}}_{i=1}^{\infty }$ be a sequence of uncertain variables. The class of functions f: $\mathbb{R}\times \mathbb{R}\mapsto \mathbb{R}$ is denoted by $C^I$, which satisfies that $f(x,y)$ is strictly increasing concerning y for every $x\in \mathbb{R}$, in the meantime, $f(x,y)\in C_{l,Lip}\left(\mathbb{R}\right)$ concerning y for every $x\in \mathbb{R}$, and $f(x,y)$ is continuous concerning x for every $y\in \mathbb{R}$. And the class of functions f: $\mathbb{R}\times \mathbb{R}\mapsto \mathbb{R}$ is denoted by $C^D$, which satisfies that $f(x,y)$ is strictly decreasing concerning y for every $x\in \mathbb{R}$, in addition, $f(x,y)\in C_{l,Lip}\left(\mathbb{R}\right)$ concerning y for every $x\in \mathbb{R}$, and $f(x,y)$ is continuous concerning x for every $y\in \mathbb{R}$. Furthermore, we write the class of continuous function $g:\mathbb{R}\mapsto \mathbb{R}$ strictly increasing as ${\overline{C}}^I$, and the class of continuous function $g:\mathbb{R}\mapsto \mathbb{R}$ strictly decreasing as ${\overline{C}}^D$.

Here, $C_{l,Lip}\left(\mathbb{R}\right)$ represents the linear space of (local Lipschitz) functions f that satisfies $|f\left(y_1\right)-f\left(y_2\right)|\le C(1+|y_1{|}^m+|y_2{|}^m\left)\right|y_1-y_2|$, $\forall y_1,y_2\in \mathbb{R}$, for some $C>0$, $m\in \mathbb{N}$ depending on $f.$

Let

$$S_n=\sum _{i=1}^{n}f({\eta }_i,{\tau }_i),n\in \mathbb{N},f\in C^I\cup C^D,$$

$$S_n\left(y\right)=\sum _{i=1}^{n}f({\eta }_i,y),y\in \mathbb{R},n\in \mathbb{N},f\in C^I\cup C^D,$$

$${\overline{S}}_n=\sum _{i=1}^{n}g\left({\tau }_i\right),n\in \mathbb{N},g\in {\overline{C}}^I\cup {\overline{C}}^D,$$

$$F\left(y\right):=\mathbb{E}\left[f({\eta }_1,y)\right],f\in C^I\cup C^D,y\in \mathbb{R},$$

and

$$\overline{F}\left(y\right):=-\mathbb{E}[-f({\eta }_1,y)]=\mathcal{E}\left[f({\eta }_1,y)\right],$$

$$f\in C^I\cup C^D,y\in \mathbb{R}.$$

Remark 3.

Let $(\Omega ,\mathcal{H},\mathbb{E})$ be a sub-linear expectation space. Suppose that ${\eta }_1$ is a random variable within $(\Omega ,\mathcal{H},\mathbb{E})$. For $\forall y\in \mathbb{R}$, if $f\in C^I$ and $\mathbb{E}\left[\right|f({\eta }_1,y)\left|\right]<\infty$, then it is obtained that each of $F\left(y\right)$ and $\overline{F}\left(y\right)$ is continuous and increasing concerning y. In a similar vein, for $\forall y\in \mathbb{R}$, if $f\in C^D$ and $\mathbb{E}\left[\right|f({\eta }_1,y)\left|\right]<\infty$, then it can be obtained that each of $F\left(y\right)$ and $\overline{F}\left(y\right)$ is continuous and decreasing concerning y.

Proof. 

We only need to demonstrate that $F\left(y\right)$ is continuous and monotonic but not strictly monotonic. Due to $\mathcal{E}[·]=-\mathbb{E}[-·]$, the continuity and monotonicity of $\overline{F}\left(y\right)$ can be also obtained.

We first prove the continuity of $F\left(y\right)$. For a random variable ${\eta }_1$, and each $y\in \mathbb{R}$, then $f({\eta }_1,y)\in C_{l,Lip}\left(\mathbb{R}\right)$. So there exist two constants $C>0$, $m\in \mathbb{R}$, for any given $y_1\in \mathbb{R}$, we have

$$|f({\eta }_1,y)-f({\eta }_1,y_1){|\le C(1+\left|y\right|}^m+|y_1{|}^m\left)\right|y-y_1|.$$

Combining Proposition A1 (iii) in Appendix B, and letting $y\to y_1$, we have

$$\begin{array}{cc}\hfill & |F\left(y\right)-F\left(y_1\right)|=|\mathbb{E}\left[f({\eta }_1,y)\right]-\mathbb{E}\left[f({\eta }_1,y_1)\right]|\hfill \\ \hfill & \le \mathbb{E}\left[\right|f({\eta }_1,y)-f({\eta }_1,y_1)\left|\right]\hfill \\ \hfill & \le \mathbb{E}\left[C\right(1+{\left|y\right|}^m+|y_1{|}^m\left)\right|y-y_1\left|\right]\hfill \\ \hfill & ={C(1+|y|}^m+|y_1{|}^m\left)\right|y-y_1|\to 0.\hfill \end{array}$$

Thus, for any given $y_1\in \mathbb{R},\underset{y\to y_1}{lim}F\left(y\right)=F\left(y_1\right)$.

Next, we give the proof of monotonicity of $F\left(y\right)$. With no loss of generality, we only take into account the situation of $f\in C^I$.

Suppose $f(x,y)\in C^I$, then $f(x,y)$ is strictly increasing concerning y for every $x\in \mathbb{R}$. For ∀$y_1,y_2\in \mathbb{R}$, $y_1<y_2$, we have $f({\eta }_1,y_1)<f({\eta }_1,y_2).$ By the monotonicity of $\mathbb{E}$, it follows that $\mathbb{E}\left[f({\eta }_1,y_1)\right]\le \mathbb{E}\left[f({\eta }_1,y_2)\right].$ Hence, it follows that $F\left(y_1\right)\le F\left(y_2\right).$ It is obvious that $F\left(y\right)$ is not strictly increasing since $\mathbb{E}$ is not strictly monotonic. □

Our primary result is given in the following theorem. We establish the Kolmogorov type LLN for uncertain random variables that are functions of IID random variables under $\mathbb{E}$ and IID uncertain variables. And the assumption of regularity for uncertain variables is necessary.

Theorem 1

(The Kolmogorov type LLN for uncertain random variables under U-S chance spaces). Consider a pair of U-S chance spaces $(\Gamma ,\mathcal{L},\mathcal{M})\times (\Omega ,\mathcal{F},\mathbb{V})$ and $(\Gamma ,\mathcal{L},\mathcal{M})\times (\Omega ,\mathcal{F},v)$, and $\overline{ch}$ and $CH$ are the chance measures under U-S chance spaces. Let ${\left\{{\eta }_i\right\}}_{i=1}^{\infty }$ be a sequence of IID random variables under $\mathbb{E}$ (see Definition A9 in Appendix B), ${\left\{{\tau }_i\right\}}_{i=1}^{\infty }$ be a sequence of IID uncertain variables, and $F\left({\tau }_1\right)=\mathbb{E}\left[f({\eta }_1,y)\right]{|}_{y={\tau }_1}$, $\overline{F}\left({\tau }_1\right)=\mathcal{E}\left[f({\eta }_1,y)\right]{|}_{y={\tau }_1}.$ Suppose that $F\left({\tau }_1\right)$ and $\overline{F}\left({\tau }_1\right)$ are regular uncertain variables, $f\in C^I\cup C^D$, and $\mathbb{E}\left[|f({\eta }_1,y){|}^{1+\alpha }\right]<\infty$, for $\forall y\in \mathbb{R}$ and some $\alpha \in (0,1]$. Then, for $\forall z\in (inf\left\{F\left(y\right)\right|y\in \mathbb{R}\},sup\left\{\overline{F}\left(y\right)\right|y\in \mathbb{R}\})$,

$$\begin{array}{cc}\hfill & \mathcal{M}\{F\left({\tau }_1\right)\le z\}\le \underset{n\to \infty }{lim\; inf}\overline{ch}\left\{{\displaystyle \frac{S_n}{n}}\le z\right\}\hfill \\ \hfill & \le \underset{n\to \infty }{lim\; sup}CH\left\{{\displaystyle \frac{S_n}{n}}\le z\right\}\le \mathcal{M}\{\overline{F}\left({\tau }_1\right)\le z\}.\hfill \end{array}$$

(3)

In particularly, if $F\left({\tau }_1\right)=\overline{F}\left({\tau }_1\right)$, then

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}\overline{ch}\left\{{\displaystyle \frac{S_n}{n}}\le z\right\}& =\underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{S_n}{n}}\le z\right\}\hfill \\ \hfill & =\mathcal{M}\{F\left({\tau }_1\right)\le z\}.\hfill \end{array}$$

(4)

Proof. 

For simplification, the uncertainty distributions of $F\left({\tau }_1\right)$ and $\overline{F}\left({\tau }_1\right)$ are denoted by $\Psi$ and $\overline{\Psi }$, respectively. Firstly, we can prove (3) by taking into account the situations $f\in C^I$ and $f\in C^D$, respectively.

Case 1: Let $f\in C^I$. Fix $z\in (inf\{F\left(y\right)|y\in \mathbb{R}\},sup\left\{F\right(y\left)\right|y\in \mathbb{R}\left\}\right)$, and give a $\epsilon >0$ such that $z-\epsilon \in (inf\{F\left(y\right)|y\in \mathbb{R}\},sup\left\{F\right(y\left)\right|y\in \mathbb{R}\left\}\right)$. For ∀$u\in (inf\{F\left(y\right)|y\in \mathbb{R}\},sup\left\{F\right(y\left)\right|y\in \mathbb{R}\left\}\right)$, denote $y_0\left(u\right)=max\left\{y\right|F\left(y\right)=u\}.$ It follows from Theorem A4 (a) in Appendix B, ∃$N_1\in \mathbb{N}$, such that for any $n\ge N_1$,

$$v\left\{\underset{k\ge n}{sup}{\displaystyle \frac{S_k\left(y_0(z-\epsilon )\right)}{k}}\le z\right\}\ge 1-\epsilon .$$

(5)

If $n\ge N_1$,

$$\begin{array}{cc}\hfill & \overline{ch}\left\{{\displaystyle \frac{S_n}{n}}\le z\right\}={\int }_0^{1}v\left\{\mathcal{M}\left\{{\displaystyle \frac{S_n}{n}}\le z\right\}\ge r\right\}\mathrm{d}r\hfill \\ \hfill & \ge {\int }_0^{1}v\{\left\{\underset{k\ge n}{sup}{\displaystyle \frac{S_k\left(y_0(z-\epsilon )\right)}{k}}\le z\right\}\hfill \\ \hfill & \bigcap \left\{\mathcal{M}\left\{{\displaystyle \frac{S_n}{n}}\le z\right\}\ge r\right\}\}\mathrm{d}r\hfill \\ \hfill & \ge {\int }_0^{1}v\{\left\{\underset{k\ge n}{sup}{\displaystyle \frac{S_k\left(y_0(z-\epsilon )\right)}{k}}\le z\right\}\hfill \\ \hfill & \bigcap \left\{\mathcal{M}\left\{{\displaystyle \frac{S_n}{n}}\le \underset{k\ge n}{sup}{\displaystyle \frac{S_k\left(y_0(z-\epsilon )\right)}{k}}\right\}\ge r\right\}\}\mathrm{d}r\hfill \\ \hfill & \ge {\int }_0^{1}v\{\left\{\underset{k\ge n}{sup}{\displaystyle \frac{S_k\left(y_0(z-\epsilon )\right)}{k}}\le z\right\}\hfill \\ \hfill & \bigcap \left\{\mathcal{M}\left\{{\displaystyle \frac{S_n}{n}}\le {\displaystyle \frac{S_n\left(y_0(z-\epsilon )\right)}{n}}\right\}\ge r\right\}\}\mathrm{d}r\hfill \\ \hfill & ={\int }_0^{1}v\{\left\{\underset{k\ge n}{sup}{\displaystyle \frac{S_k\left(y_0(z-\epsilon )\right)}{k}}\le z\right\}\bigcap \hfill \\ \hfill & \left\{\mathcal{M}\left\{\sum _{i=1}^{n}f({\eta }_i,{\tau }_i)\le \sum _{i=1}^{n}f({\eta }_i,y_0(z-\epsilon ))\right\}\ge r\right\}\}\mathrm{d}r\hfill \\ \hfill & \ge {\int }_0^{1}v\{\left\{\underset{k\ge n}{sup}{\displaystyle \frac{S_k\left(y_0(z-\epsilon )\right)}{k}}\le z\right\}\bigcap \hfill \\ \hfill & \left\{\mathcal{M}\left\{\bigcap _{i=1}^n\left\{f({\eta }_i,{\tau }_i)\le f({\eta }_i,y_0(z-\epsilon ))\right\}\right\}\ge r\right\}\}\mathrm{d}r.\hfill \end{array}$$

Notice that ${\left\{{\tau }_i\right\}}_{i=1}^{\infty }$ is an IID uncertain sequence and $f(x,y)$ is a strictly increasing function about y for every x. By (5) and $y_0(z-\epsilon )=max\left\{y\right|F\left(y\right)=z-\epsilon \}$, we obtain

$$\begin{array}{cc}\hfill & \overline{ch}\left\{{\displaystyle \frac{S_n}{n}}\le z\right\}\ge {\int }_0^{1}v\{\left\{\underset{k\ge n}{sup}{\displaystyle \frac{S_k\left(y_0(z-\epsilon )\right)}{k}}\le z\right\}\hfill \\ \hfill & \bigcap \left\{\mathcal{M}\left\{\bigcap _{i=1}^n\left\{{\tau }_i\le y_0(z-\epsilon )\right\}\right\}\ge r\right\}\}\mathrm{d}r\hfill \\ \hfill & ={\int }_0^{1}v\{\left\{\underset{k\ge n}{sup}{\displaystyle \frac{S_k\left(y_0(z-\epsilon )\right)}{k}}\le z\right\}\hfill \\ \hfill & \bigcap \left\{\mathcal{M}\left\{{\tau }_1\le y_0(z-\epsilon )\right\}\ge r\right\}\}\mathrm{d}r\hfill \\ \hfill & ={\int }_0^{1}v\{\left\{\underset{k\ge n}{sup}{\displaystyle \frac{S_k\left(y_0(z-\epsilon )\right)}{k}}\le z\right\}\hfill \\ \hfill & \bigcap \left\{\mathcal{M}\left\{F\left({\tau }_1\right)\le z-\epsilon \right\}\ge r\right\}\}\mathrm{d}r\hfill \\ \hfill & \ge (1-\epsilon )\Psi (z-\epsilon ).\hfill \end{array}$$

Because $F\left({\tau }_1\right)$ is regular, its uncertainty distribution $\Psi$ is continuous. Based on the arguments mentioned above and taking $\epsilon \to 0$, it follows

$$\underset{n\to \infty }{lim\; inf}\overline{ch}\left\{{\displaystyle \frac{S_n}{n}}\le z\right\}\ge \Psi \left(z\right)=\mathcal{M}\{F\left({\tau }_1\right)\le z\}.$$

(6)

Another aspect, given

$$z\in (inf\left\{\overline{F}\left(y\right)\right|y\in \mathbb{R}\},sup\left\{\overline{F}\left(y\right)\right|y\in \mathbb{R}\})$$

and take $\epsilon >0$ such that $z+\epsilon \in (inf\left\{\overline{F}\left(y\right)\right|y\in \mathbb{R}\},sup\left\{\overline{F}\left(y\right)\right|y\in \mathbb{R}\})$. For ∀$u\in (inf\left\{\overline{F}\left(y\right)\right|y\in \mathbb{R}\},sup\left\{\overline{F}\left(y\right)\right|y\in \mathbb{R}\})$, denote $y_1\left(u\right)=max\left\{y\right|\overline{F}\left(y\right)=u\}.$ From Theorem A4 (b) in Appendix B, ∃$N_2\in \mathbb{N}$, such that for any $n\ge N_2$,

$$v\left\{\underset{k\ge n}{inf}{\displaystyle \frac{S_k\left(y_1(z+\epsilon )\right)}{k}}>z\right\}\ge 1-\epsilon .$$

(7)

If $n\ge N_2$,

$$\begin{array}{cc}\hfill & \overline{ch}\left\{{\displaystyle \frac{S_n}{n}}>z\right\}={\int }_0^{1}v\left\{\mathcal{M}\left\{{\displaystyle \frac{S_n}{n}}>z\right\}\ge r\right\}\mathrm{d}r\hfill \\ \hfill & \ge {\int }_0^{1}v\{\left\{\underset{k\ge n}{inf}{\displaystyle \frac{S_k\left(y_1(z+\epsilon )\right)}{k}}>z\right\}\hfill \\ \hfill & \bigcap \left\{\mathcal{M}\left\{{\displaystyle \frac{S_n}{n}}>z\right\}\ge r\right\}\}\mathrm{d}r\hfill \\ \hfill & \ge {\int }_0^{1}v\{\left\{\underset{k\ge n}{inf}{\displaystyle \frac{S_k\left(y_1(z+\epsilon )\right)}{k}}>z\right\}\hfill \\ \hfill & \bigcap \left\{\mathcal{M}\left\{{\displaystyle \frac{S_n}{n}}>\underset{k\ge n}{inf}{\displaystyle \frac{S_k\left(y_1(z+\epsilon )\right)}{k}}\right\}\ge r\right\}\}\mathrm{d}r\hfill \\ \hfill & \ge {\int }_0^{1}v\{\left\{\underset{k\ge n}{inf}{\displaystyle \frac{S_k\left(y_1(z+\epsilon )\right)}{k}}>z\right\}\bigcap \hfill \\ \hfill & \left\{\mathcal{M}\left\{\sum _{i=1}^{n}f({\eta }_i,{\tau }_i)>\sum _{i=1}^{n}f({\eta }_i,y_1(z+\epsilon ))\right\}\ge r\right\}\}\mathrm{d}r\hfill \\ \hfill & \ge {\int }_0^{1}v\{\left\{\underset{k\ge n}{inf}{\displaystyle \frac{S_k\left(y_1(z+\epsilon )\right)}{k}}>z\right\}\bigcap \hfill \\ \hfill & \left\{\mathcal{M}\left\{\bigcap _{i=1}^n\left\{f({\eta }_i,{\tau }_i)>f({\eta }_i,y_1(z+\epsilon ))\right\}\right\}\ge r\right\}\}\mathrm{d}r.\hfill \end{array}$$

Notice that ${\left\{{\tau }_i\right\}}_{i=1}^{\infty }$ is an IID uncertain sequence and $f(x,y)$ is a strictly increasing function about y for every x. By (7) and $y_1(z+\epsilon )=max\left\{y\right|\overline{F}\left(y\right)=z+\epsilon \}$, we have

$$\begin{array}{cc}\hfill & \overline{ch}\left\{{\displaystyle \frac{S_n}{n}}>z\right\}\ge {\int }_0^{1}v\{\left\{\underset{k\ge n}{inf}{\displaystyle \frac{S_k\left(y_1(z+\epsilon )\right)}{k}}>z\right\}\hfill \\ \hfill & \bigcap \left\{\mathcal{M}\left\{\bigcap _{i=1}^n\left\{{\tau }_i>y_1(z+\epsilon )\right\}\right\}\ge r\right\}\}\mathrm{d}r\hfill \\ \hfill & ={\int }_0^{1}v\{\left\{\underset{k\ge n}{inf}{\displaystyle \frac{S_k\left(y_1(z+\epsilon )\right)}{k}}>z\right\}\hfill \\ \hfill & \bigcap \left\{\mathcal{M}\left\{{\tau }_1>y_1(z+\epsilon )\right\}\ge r\right\}\}\mathrm{d}r\hfill \\ \hfill & ={\int }_0^{1}v\{\left\{\underset{k\ge n}{inf}{\displaystyle \frac{S_k\left(y_1(z+\epsilon )\right)}{k}}>z\right\}\hfill \\ \hfill & \bigcap \left\{\mathcal{M}\left\{\overline{F}\left({\tau }_1\right)>z+\epsilon \right\}\ge r\right\}\}\mathrm{d}r\hfill \\ \hfill & \ge (1-\epsilon )(1-\overline{\Psi }(z+\epsilon )).\hfill \end{array}$$

Applying Proposition 1 (ii), it yields

$$CH\left\{{\displaystyle \frac{S_n}{n}}\le z\right\}\le 1-(1-\epsilon )(1-\overline{\Psi }(z+\epsilon )).$$

Because $\overline{F}\left({\tau }_1\right)$ is regular, its uncertainty distribution $\overline{\Psi }$ is continuous. Letting $\epsilon \to 0$, we obtain

$$\underset{n\to \infty }{lim\; sup}CH\left\{{\displaystyle \frac{S_n}{n}}\le z\right\}\le \overline{\Psi }\left(z\right)=\mathcal{M}\{\overline{F}\left({\tau }_1\right)\le z\}.$$

(8)

Since $\mathcal{E}[·]\le \mathbb{E}[·]$, it follows that $inf\left\{\overline{F}\left(y\right)\right|y\in \mathbb{R}\}\le inf\left\{F\left(y\right)\right|y\in \mathbb{R}\},$ and $sup\left\{\overline{F}\left(y\right)\right|y\in \mathbb{R}\}\le sup\left\{F\left(y\right)\right|y\in \mathbb{R}\}.$ Then, for ∀$z\in (inf\left\{F\left(y\right)\right|y\in \mathbb{R}\},sup\left\{\overline{F}\left(y\right)\right|y\in \mathbb{R}\})$, (3) follows, integrating (6) and (8).

Case 2: Let $f\in C^D$, then $-f\in C^I$. To make the notation simpler, we use the notations $F\left({\tau }_1\right)\equiv \mathbb{E}\left[f({\eta }_1,{\tau }_1)\right]$ and $\overline{F}\left({\tau }_1\right)\equiv \mathcal{E}\left[f({\eta }_1,{\tau }_1)\right].$ By taking into account $-f$ and $-z$ instead of f and z in (3), we obtain

$$\begin{array}{cc}\hfill & \mathcal{M}\{\mathbb{E}[-f({\eta }_1,{\tau }_1)]<-z\}\le \underset{n\to \infty }{lim\; inf}\overline{ch}\left\{{\displaystyle \frac{-S_n}{n}}<-z\right\}\hfill \\ \hfill & \le \underset{n\to \infty }{lim\; sup}CH\left\{{\displaystyle \frac{-S_n}{n}}<-z\right\}\le \mathcal{M}\{\mathcal{E}[-f({\eta }_1,{\tau }_1)]<-z\}.\hfill \end{array}$$

Due to the fact $\mathbb{E}[-f({\eta }_1,{\tau }_1)]=-\mathcal{E}\left[f({\eta }_1,{\tau }_1)\right]$,

$$\begin{array}{cc}\hfill & \mathcal{M}\{-\mathcal{E}\left[f({\eta }_1,{\tau }_1)\right]<-z\}\le \underset{n\to \infty }{lim\; inf}\overline{ch}\left\{{\displaystyle \frac{-S_n}{n}}<-z\right\}\hfill \\ \hfill & \le \underset{n\to \infty }{lim\; sup}CH\left\{{\displaystyle \frac{-S_n}{n}}<-z\right\}\le \mathcal{M}\{-\mathbb{E}\left[f({\eta }_1,{\tau }_1)\right]<-z\}.\hfill \end{array}$$

From Proposition 1 (ii) and the duality axiom of $\mathcal{M}$, we obtain

$$\begin{array}{cc}\hfill & 1-\mathcal{M}\{\mathcal{E}\left[f({\eta }_1,{\tau }_1)\right]\le z\}\le 1-\underset{n\to \infty }{lim\; inf}CH\left\{{\displaystyle \frac{S_n}{n}}\le z\right\}\hfill \\ \hfill & \le 1-\underset{n\to \infty }{lim\; sup}\overline{ch}\left\{{\displaystyle \frac{S_n}{n}}\le z\right\}\le 1-\mathcal{M}\{\mathbb{E}\left[f({\eta }_1,{\tau }_1)\right]\le z\}.\hfill \end{array}$$

Moreover,

$$\begin{array}{cc}\hfill & \mathcal{M}\{\mathbb{E}\left[f({\eta }_1,{\tau }_1)\right]\le z\}\le \underset{n\to \infty }{lim\; inf}\overline{ch}\left\{{\displaystyle \frac{S_n}{n}}\le z\right\}\hfill \\ \hfill & \le \underset{n\to \infty }{lim\; sup}CH\left\{{\displaystyle \frac{S_n}{n}}\le z\right\}\le \mathcal{M}\{\mathcal{E}\left[f({\eta }_1,{\tau }_1)\right]\le z\}.\hfill \end{array}$$

Then, for ∀$z\in (inf\left\{F\left(y\right)\right|y\in \mathbb{R}\},sup\left\{\overline{F}\left(y\right)\right|y\in \mathbb{R}\})$, (3) follows.

Finally, we consider (4). If $F\left({\tau }_1\right)=\overline{F}\left({\tau }_1\right)$, we have

$$\mathcal{M}\{F\left({\tau }_1\right)\le z\}=\mathcal{M}\{\overline{F}\left({\tau }_1\right)\le z\}.$$

(9)

By (3), we obtain

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim\; inf}\overline{ch}\left\{{\displaystyle \frac{S_n}{n}}\le z\right\}& =\underset{n\to \infty }{lim\; sup}CH\left\{{\displaystyle \frac{S_n}{n}}\le z\right\}\hfill \\ \hfill & =\mathcal{M}\{F\left({\tau }_1\right)\le z\}.\hfill \end{array}$$

(10)

Applying Proposition 1 (iv), we have

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim\; inf}\overline{ch}\left\{{\displaystyle \frac{S_n}{n}}\le z\right\}\le \underset{n\to \infty }{lim\; sup}\overline{ch}\left\{{\displaystyle \frac{S_n}{n}}\le z\right\}\hfill \\ \hfill & \le \underset{n\to \infty }{lim\; sup}CH\left\{{\displaystyle \frac{S_n}{n}}\le z\right\},\hfill \end{array}$$

(11)

and

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim\; inf}\overline{ch}\left\{{\displaystyle \frac{S_n}{n}}\le z\right\}\le \underset{n\to \infty }{lim\; inf}CH\left\{{\displaystyle \frac{S_n}{n}}\le z\right\}\hfill \\ \hfill & \le \underset{n\to \infty }{lim\; sup}CH\left\{{\displaystyle \frac{S_n}{n}}\le z\right\}.\hfill \end{array}$$

(12)

Combining (10), (11) and (12), we obtain

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}\overline{ch}\left\{{\displaystyle \frac{S_n}{n}}\le z\right\}& =\underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{S_n}{n}}\le z\right\}\hfill \\ \hfill & =\mathcal{M}\{F\left({\tau }_1\right)\le z\}.\hfill \end{array}$$

Hence, (4) is proved. Thus, the proof of Theorem 1 is completed. □

In the degenerate situation for uncertain variables with $f(x,y)=g\left(y\right)$, x, $y\in \mathbb{R}$, Theorem 1 directly leads to the following corollary.

Corollary 1.

Suppose that ${\left\{{\tau }_i\right\}}_{i=1}^{\infty }$ is a sequence of regular and IID uncertain variables, and $g\in {\overline{C}}^I\cup {\overline{C}}^D$. Then for $\forall z\in (inf\{g\left(y\right)|y\in \mathbb{R}\},sup\left\{g\right(y\left)\right|y\in \mathbb{R}\left\}\right)$,

$$\begin{array}{cc}& \underset{n\to \infty }{lim}\overline{ch}\left\{{\displaystyle \frac{{\overline{S}}_n}{n}}\le z\right\}=\underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\overline{S}}_n}{n}}\le z\right\}\hfill \\ \hfill & =\underset{n\to \infty }{lim}\mathcal{M}\left\{{\displaystyle \frac{{\overline{S}}_n}{n}}\le z\right\}=\mathcal{M}\{g\left({\tau }_1\right)\le z\}.\hfill \end{array}$$

Theorem 2 below includes the degenerate LLNs for random variables with $f(x,y)=h\left(x\right)$, x, $y\in \mathbb{R}$. It can be obtained from Theorems A2 and A3 in Appendix B immediately.

Theorem 2.

Assume that ${\left\{{\eta }_i\right\}}_{i=1}^{\infty }$ is a sequence of IID random variables under $\mathbb{E}$, and $h\left(x\right)$, $x\in \mathbb{R}$ is continuous and $\mathbb{E}\left[|h\left({\eta }_1\right){|}^{1+\alpha }\right]<\infty$ for some $\alpha \in (0,1].$ Then for any $\epsilon >0$,

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim}CH\left\{\mathcal{E}\left[h\left({\eta }_1\right)\right]-\epsilon \le {\displaystyle \frac{{\sum }_{i=1}^{n}h\left({\eta }_i\right)}{n}}\le \mathbb{E}\left[h\left({\eta }_1\right)\right]+\epsilon \right\}\hfill \\ \hfill & =\underset{n\to \infty }{lim}\mathbb{V}\left\{\mathcal{E}\left[h\left({\eta }_1\right)\right]-\epsilon \le {\displaystyle \frac{{\sum }_{i=1}^{n}h\left({\eta }_i\right)}{n}}\le \mathbb{E}\left[h\left({\eta }_1\right)\right]+\epsilon \right\}\hfill \\ \hfill & =1,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim}\overline{ch}\left\{\mathcal{E}\left[h\left({\eta }_1\right)\right]-\epsilon \le {\displaystyle \frac{{\sum }_{i=1}^{n}h\left({\eta }_i\right)}{n}}\le \mathbb{E}\left[h\left({\eta }_1\right)\right]+\epsilon \right\}\hfill \\ \hfill & =\underset{n\to \infty }{lim}v\left\{\mathcal{E}\left[h\left({\eta }_1\right)\right]-\epsilon \le {\displaystyle \frac{{\sum }_{i=1}^{n}h\left({\eta }_i\right)}{n}}\le \mathbb{E}\left[h\left({\eta }_1\right)\right]+\epsilon \right\}\hfill \\ \hfill & =1.\hfill \end{array}$$

4. Examples

This section employs the notations in Section 3. Five examples are given, which can be seen as applications of Theorem 1.

Example 1.

Suppose that ${\left\{{\eta }_i\right\}}_{i=1}^{\infty }$ is a sequence of IID random variables with maximal distribution under $\mathbb{E}$ (see Definition A10 in Appendix B), i.e., $\mathbb{E}\left[\phi \left({\eta }_1\right)\right]={sup}_{\underline{\mu }\le y\le \overline{\mu }}\phi \left(y\right)$, for $\phi \in C_{l.lip}\left(\mathbb{R}\right),$ where $\overline{\mu }=\mathbb{E}\left[{\eta }_1\right]$, $\underline{\mu }=\mathcal{E}\left[{\eta }_1\right]$, and ${\left\{{\tau }_i\right\}}_{i=1}^{\infty }$ is a sequence of uncertain variables that are independent, normally distributed, and the uncertainty distribution is

$$\Psi \left(x\right)={\left(1+exp\left({\displaystyle \frac{-\pi x}{\sqrt{3}}}\right)\right)}^{-1},x\in \mathbb{R}.$$

Then, it follows that

(I) 

For every $z\in (-\infty ,\infty )$,

$$\begin{array}{cc}\hfill & \Psi (z-\overline{\mu })\le \underset{n\to \infty }{lim\; inf}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)}{n}}\le z\right\}\hfill \\ \hfill & \le \underset{n\to \infty }{lim\; sup}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)}{n}}\le z\right\}\le \Psi (z-\underline{\mu }),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & 1-\Psi (\overline{\mu }-z)\le \underset{n\to \infty }{lim\; inf}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)}{n}}\le z\right\}\hfill \\ \hfill & \le \underset{n\to \infty }{lim\; sup}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)}{n}}\le z\right\}\le 1-\Psi (\underline{\mu }-z).\hfill \end{array}$$

(II) 

For any $z\in (-\infty ,\infty )$,

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n{\tau }_i^3}{n}}\le z\right\}\hfill \\ \hfill & =\underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n{\tau }_i^3}{n}}\le z\right\}=\Psi \left(\sqrt[3]{z}\right),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n\left(-{\tau }_i^3\right)}{n}}\le z\right\}\hfill \\ \hfill & =\underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n\left(-{\tau }_i^3\right)}{n}}\le z\right\}=1-\Psi (-\sqrt[3]{z}).\hfill \end{array}$$

Example 2.

Suppose that ${\left\{{\eta }_i\right\}}_{i=1}^{\infty }$ is a sequence of IID random variables with maximal distribution under $\mathbb{E}$, i.e., $\mathbb{E}\left[\phi \left({\eta }_1\right)\right]={sup}_{\underline{\mu }\le y\le \overline{\mu }}\phi \left(y\right)$, for $\phi \in C_{l.lip}\left(\mathbb{R}\right),$ where $\overline{\mu }=\mathbb{E}\left[{\eta }_1\right]$, $\underline{\mu }=\mathcal{E}\left[{\eta }_1\right]$, and ${\left\{{\tau }_i\right\}}_{i=1}^{\infty }$ is a sequence of uncertain variables that are independent, linear distributed, and the uncertainty distribution is

$$\Psi \left(x\right)=\left\{\begin{array}{cc}0,& \mathrm{if}x<a\\ \\ {\displaystyle \frac{x-a}{b-a}},& \mathrm{if}a\le x<b\\ \\ 1,& \mathrm{otherwise}\end{array}\right.$$

where a and b are real numbers with $a<b$. Then, it follows that

(I) 

For every $z<a+\underline{\mu }$,

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)}{n}}\le z\right\}=\underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)}{n}}\le z\right\}=0,\end{array}$$

for every $z\ge b+\overline{\mu }$,

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)}{n}}\le z\right\}=\underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)}{n}}\le z\right\}=1,\end{array}$$

for every $z\in [a+\underline{\mu },min\{a+\overline{\mu },b+\underline{\mu }\})$,

$$\begin{array}{c}\hfill 0\le \underset{n\to \infty }{lim\; inf}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)}{n}}\le z\right\}\le \underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)}{n}}\le z\right\}\le {\displaystyle \frac{z-\underline{\mu }-a}{b-a}},\end{array}$$

for every $z\in [max\{a+\overline{\mu },b+\underline{\mu }\},b+\overline{\mu })$,

$$\begin{array}{c}\hfill {\displaystyle \frac{z-\overline{\mu }-a}{b-a}}\le \underset{n\to \infty }{lim\; inf}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)}{n}}\le z\right\}\le \underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)}{n}}\le z\right\}\le 1.\end{array}$$

for every $z\in [a+\overline{\mu },b+\underline{\mu }]$ with $a+\overline{\mu }<b+\underline{\mu }$,

$$\begin{array}{c}\hfill {\displaystyle \frac{z-\overline{\mu }-a}{b-a}}\le \underset{n\to \infty }{lim\; inf}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)}{n}}\le z\right\}\le \underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)}{n}}\le z\right\}\le {\displaystyle \frac{z-\underline{\mu }-a}{b-a}},\end{array}$$

and for every $z\in [b+\underline{\mu },a+\overline{\mu }]$ with $a+\overline{\mu }\ge b+\underline{\mu }$,

$$\begin{array}{c}\hfill 0\le \underset{n\to \infty }{lim\; inf}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)}{n}}\le z\right\}\le \underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)}{n}}\le z\right\}\le 1.\end{array}$$

(II) 

For every $z>\overline{\mu }-a$,

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)}{n}}\le z\right\}=\underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)}{n}}\le z\right\}=1,\end{array}$$

for every $z\le \underline{\mu }-b$,

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)}{n}}\le z\right\}=\underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)}{n}}\le z\right\}=0,\end{array}$$

for every $z\in (\underline{\mu }-b,min\{\underline{\mu }-a,\overline{\mu }-b\}]$,

$$\begin{array}{cc}\hfill & 0\le \underset{n\to \infty }{lim\; inf}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)}{n}}\le z\right\}\hfill \\ \hfill & \le \underset{n\to \infty }{lim\; sup}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)}{n}}\le z\right\}\le 1-{\displaystyle \frac{\underline{\mu }-z-a}{b-a}},\hfill \end{array}$$

for every $z\in (max\{\underline{\mu }-a,\overline{\mu }-b\},\overline{\mu }-a]$,

$$\begin{array}{cc}\hfill & 1-{\displaystyle \frac{\overline{\mu }-z-a}{b-a}}\le \underset{n\to \infty }{lim\; inf}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)}{n}}\le z\right\}\hfill \\ \hfill & \le \underset{n\to \infty }{lim\; sup}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)}{n}}\le z\right\}\le 1,\hfill \end{array}$$

for every $z\in (\overline{\mu }-b,\underline{\mu }-a]$ with $\overline{\mu }-b<\underline{\mu }-a$,

$$\begin{array}{cc}\hfill & 1-{\displaystyle \frac{\overline{\mu }-z-a}{b-a}}\le \underset{n\to \infty }{lim\; inf}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)}{n}}\le z\right\}\hfill \\ \hfill & \le \underset{n\to \infty }{lim\; sup}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)}{n}}\le z\right\}\le 1-{\displaystyle \frac{\underline{\mu }-z-a}{b-a}},\hfill \end{array}$$

and for every $z\in (\underline{\mu }-a,\overline{\mu }-b]$ with $\overline{\mu }-b\ge \underline{\mu }-a$,

$$\begin{array}{c}\hfill 0\le \underset{n\to \infty }{lim\; inf}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)}{n}}\le z\right\}\le \underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)}{n}}\le z\right\}\le 1.\end{array}$$

(III) 

For any $z\in (-\infty ,\infty )$,

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n{\tau }_i^3}{n}}\le z\right\}\hfill \\ \hfill & =\underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n{\tau }_i^3}{n}}\le z\right\}=\Psi \left(\sqrt[3]{z}\right),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n\left(-{\tau }_i^3\right)}{n}}\le z\right\}\hfill \\ \hfill & =\underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n\left(-{\tau }_i^3\right)}{n}}\le z\right\}=1-\Psi (-\sqrt[3]{z}).\hfill \end{array}$$

Example 3.

Suppose that ${\left\{{\eta }_i\right\}}_{i=1}^{\infty }$ is a sequence of IID random variables with G-normal distribution under $\mathbb{E}$ (see Definition A11 in Appendix B), i.e., ${\eta }_1\sim N\{0,[{\underline{\sigma }}^2,{\overline{\sigma }}^2]\},$ where $\mathbb{E}\left[{\eta }_1\right]=\mathcal{E}\left[{\eta }_1\right]=0$, ${\overline{\sigma }}^2=\mathbb{E}\left[{\eta }_1^2\right]$ and ${\underline{\sigma }}^2=\mathcal{E}\left[{\eta }_1^2\right]$, and ${\left\{{\tau }_i\right\}}_{i=1}^{\infty }$ is a sequence of uncertain variables that are independent, normally distributed, and the uncertainty distribution is

$$\Psi \left(x\right)={\left(1+exp\left({\displaystyle \frac{-\pi x}{\sqrt{3}}}\right)\right)}^{-1},x\in \mathbb{R}.$$

Then, we have

(I) 

For any $z\in (-\infty ,\infty )$,

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)}{n}}\le z\right\}\hfill \\ \hfill & =\underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)}{n}}\le z\right\}=\Psi \left(z\right),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)}{n}}\le z\right\}\hfill \\ \hfill & =\underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)}{n}}\le z\right\}=1-\Psi (-z).\hfill \end{array}$$

(II) 

For any $z\in (-\infty ,\infty )$,

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n{\tau }_i^3}{n}}\le z\right\}\hfill \\ \hfill & =\underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n{\tau }_i^3}{n}}\le z\right\}=\Psi \left(\sqrt[3]{z}\right),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n\left(-{\tau }_i^3\right)}{n}}\le z\right\}\hfill \\ \hfill & =\underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n\left(-{\tau }_i^3\right)}{n}}\le z\right\}=1-\Psi (-\sqrt[3]{z}).\hfill \end{array}$$

Example 4.

Suppose that ${\left\{{\eta }_i\right\}}_{i=1}^{\infty }$ is a sequence of IID random variables with G-normal distribution under $\mathbb{E}$, i.e., ${\eta }_1\sim N\{0,[{\underline{\sigma }}^2,{\overline{\sigma }}^2]\},$ where $\mathbb{E}\left[{\eta }_1\right]=\mathcal{E}\left[{\eta }_1\right]=0$, ${\overline{\sigma }}^2=\mathbb{E}\left[{\eta }_1^2\right]$ and ${\underline{\sigma }}^2=\mathcal{E}\left[{\eta }_1^2\right]$, and ${\left\{{\tau }_i\right\}}_{i=1}^{\infty }$ is a sequence of uncertain variables that are independent, linear distributed, and the uncertainty distribution is

$$\Psi \left(x\right)=\left\{\begin{array}{cc}0,& \mathrm{if}x<a\\ \\ {\displaystyle \frac{x-a}{b-a}},& \mathrm{if}a\le x<b\\ \\ 1,& \mathrm{otherwise}\end{array}\right.$$

where a and b are real numbers with $a<b$. Then, it follows that

(I) 

For any $z\in (-\infty ,\infty )$,

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)}{n}}\le z\right\}\hfill \\ \hfill & =\underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)}{n}}\le z\right\}=\Psi \left(z\right),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)}{n}}\le z\right\}\hfill \\ \hfill & =\underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)}{n}}\le z\right\}=1-\Psi (-z).\hfill \end{array}$$

(II) 

For any $z\in (-\infty ,\infty )$,

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n{\tau }_i^3}{n}}\le z\right\}\hfill \\ \hfill & =\underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n{\tau }_i^3}{n}}\le z\right\}=\Psi \left(\sqrt[3]{z}\right),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n\left(-{\tau }_i^3\right)}{n}}\le z\right\}\hfill \\ \hfill & =\underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n\left(-{\tau }_i^3\right)}{n}}\le z\right\}=1-\Psi (-\sqrt[3]{z}).\hfill \end{array}$$

Example 5.

The triplets $(\Omega ,\mathcal{F},\mathbb{V})$ and $(\Omega ,\mathcal{F},v)$ are an upper probability space and a lower probability space, respectively. Assume that ${\left\{{\eta }_i\right\}}_{i=1}^n$ is a sequence of IID random variables. The upper and lower probabilities are defined by $\mathbb{V}\left(A\right)=sup\{P_1\left(A\right),P_2\left(A\right)\}$, $v\left(A\right)=inf\{P_1\left(A\right),P_2\left(A\right)\}$, for any $A\in \mathcal{F}$, and

$$P_1({\eta }_1\le x)=\left\{\begin{array}{cc}1-e^{-{\lambda }_{1}x},& \mathrm{if}x>0\\ \\ 0,& \mathrm{otherwise}\end{array},\right.$$

$$P_2({\eta }_1\le x)=\left\{\begin{array}{cc}1-e^{-{\lambda }_{2}x},& \mathrm{if}x>0\\ \\ 0,& \mathrm{otherwise}\end{array},\right.$$

where $0<{\lambda }_1<{\lambda }_2$, $\mathbb{E}\left[{\eta }_1\right]=sup\{E_{P_1}\left[{\eta }_1\right],E_{P_2}\left[{\eta }_1\right]\}$, and $\mathcal{E}\left[{\eta }_1\right]=inf\{E_{P_1}\left[{\eta }_1\right],E_{P_2}\left[{\eta }_1\right]\}$. And assume that ${\left\{{\tau }_i\right\}}_{i=1}^{\infty }$ is a sequence of uncertain variables that are independent, linear distributed, and the uncertainty distribution is

$$\Psi \left(x\right)=\left\{\begin{array}{cc}0,& \mathrm{if}x<a\\ \\ {\displaystyle \frac{x-a}{b-a}},& \mathrm{if}a\le x<b\\ \\ 1,& \mathrm{otherwise}\end{array}\right.$$

where a and b are real numbers with $a<b$. Then, it follows that

(I) 

For every $z<a+{\displaystyle \frac{1}{{\lambda }_2}}$,

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)}{n}}\le z\right\}=\underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)}{n}}\le z\right\}=0,\end{array}$$

for every $z\ge b+{\displaystyle \frac{1}{{\lambda }_1}}$,

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)}{n}}\le z\right\}=\underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)}{n}}\le z\right\}=1,\end{array}$$

for every $z\in [a+{\displaystyle \frac{1}{{\lambda }_2}},min\{a+{\displaystyle \frac{1}{{\lambda }_1}},b+{\displaystyle \frac{1}{{\lambda }_2}}\})$,

$$\begin{array}{c}\hfill 0\le \underset{n\to \infty }{lim\; inf}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)}{n}}\le z\right\}\le \underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)}{n}}\le z\right\}\le {\displaystyle \frac{z-\frac{1}{{\lambda }_2}-a}{b-a}},\end{array}$$

for every $z\in [max\{a+{\displaystyle \frac{1}{{\lambda }_1}},b+{\displaystyle \frac{1}{{\lambda }_2}}\},b+{\displaystyle \frac{1}{{\lambda }_1}})$,

$$\begin{array}{c}\hfill {\displaystyle \frac{z-\frac{1}{{\lambda }_1}-a}{b-a}}\le \underset{n\to \infty }{lim\; inf}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)}{n}}\le z\right\}\le \underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)}{n}}\le z\right\}\le 1,\end{array}$$

for every $z\in [a+{\displaystyle \frac{1}{{\lambda }_1}},b+{\displaystyle \frac{1}{{\lambda }_2}}]$ with $a+{\displaystyle \frac{1}{{\lambda }_1}}<b+{\displaystyle \frac{1}{{\lambda }_2}}$,

$$\begin{array}{c}\hfill {\displaystyle \frac{z-\frac{1}{{\lambda }_1}-a}{b-a}}\le \underset{n\to \infty }{lim\; inf}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)}{n}}\le z\right\}\le \underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)}{n}}\le z\right\}\le {\displaystyle \frac{z-\frac{1}{{\lambda }_2}-a}{b-a}},\end{array}$$

and for every $z\in [b+{\displaystyle \frac{1}{{\lambda }_2}},a+{\displaystyle \frac{1}{{\lambda }_1}}]$ with $a+{\displaystyle \frac{1}{{\lambda }_1}}\ge b+{\displaystyle \frac{1}{{\lambda }_2}}$,

$$\begin{array}{c}\hfill 0\le \underset{n\to \infty }{lim\; inf}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)}{n}}\le z\right\}\le \underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)}{n}}\le z\right\}\le 1.\end{array}$$

(II) 

For every $z>{\displaystyle \frac{1}{{\lambda }_1}}-a$,

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)}{n}}\le z\right\}=\underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)}{n}}\le z\right\}=1,\end{array}$$

for every $z\le {\displaystyle \frac{1}{{\lambda }_2}}-b$,

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)}{n}}\le z\right\}=\underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)}{n}}\le z\right\}=0,\end{array}$$

for every $z\in ({\displaystyle \frac{1}{{\lambda }_2}}-b,min\{{\displaystyle \frac{1}{{\lambda }_2}}-a,{\displaystyle \frac{1}{{\lambda }_1}}-b\}]$,

$$\begin{array}{cc}\hfill & 0\le \underset{n\to \infty }{lim\; inf}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)}{n}}\le z\right\}\hfill \\ \hfill & \le \underset{n\to \infty }{lim\; sup}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)}{n}}\le z\right\}\le 1-{\displaystyle \frac{\frac{1}{{\lambda }_2}-z-a}{b-a}},\hfill \end{array}$$

for every $z\in (max\{{\displaystyle \frac{1}{{\lambda }_2}}-a,{\displaystyle \frac{1}{{\lambda }_1}}-b\},{\displaystyle \frac{1}{{\lambda }_1}}-a]$,

$$\begin{array}{cc}\hfill & 1-{\displaystyle \frac{\frac{1}{{\lambda }_1}-z-a}{b-a}}\le \underset{n\to \infty }{lim\; inf}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)}{n}}\le z\right\}\hfill \\ \hfill & \le \underset{n\to \infty }{lim\; sup}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)}{n}}\le z\right\}\le 1,\hfill \end{array}$$

for every $z\in ({\displaystyle \frac{1}{{\lambda }_1}}-b,{\displaystyle \frac{1}{{\lambda }_2}}-a]$ with ${\displaystyle \frac{1}{{\lambda }_1}}-b<{\displaystyle \frac{1}{{\lambda }_2}}-a$,

$$\begin{array}{cc}\hfill & 1-{\displaystyle \frac{\frac{1}{{\lambda }_1}-z-a}{b-a}}\le \underset{n\to \infty }{lim\; inf}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)}{n}}\le z\right\}\hfill \\ \hfill & \le \underset{n\to \infty }{lim\; sup}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)}{n}}\le z\right\}\le 1-{\displaystyle \frac{\frac{1}{{\lambda }_2}-z-a}{b-a}},\hfill \end{array}$$

and for every $z\in ({\displaystyle \frac{1}{{\lambda }_2}}-a,{\displaystyle \frac{1}{{\lambda }_1}}-b]$ with ${\displaystyle \frac{1}{{\lambda }_1}}-b\ge {\displaystyle \frac{1}{{\lambda }_2}}-a$,

$$\begin{array}{c}\hfill 0\le \underset{n\to \infty }{lim\; inf}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)}{n}}\le z\right\}\le \underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)}{n}}\le z\right\}\le 1.\end{array}$$

(III) 

For any $z\in (-\infty ,\infty )$,

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n{\tau }_i^3}{n}}\le z\right\}\hfill \\ \hfill & =\underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n{\tau }_i^3}{n}}\le z\right\}=\Psi \left(\sqrt[3]{z}\right),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim}\overline{ch}\left\{{\displaystyle \frac{{\sum }_{i=1}^n\left(-{\tau }_i^3\right)}{n}}\le z\right\}\hfill \\ \hfill & =\underset{n\to \infty }{lim}CH\left\{{\displaystyle \frac{{\sum }_{i=1}^n\left(-{\tau }_i^3\right)}{n}}\le z\right\}=1-\Psi (-\sqrt[3]{z}).\hfill \end{array}$$

Remark 4.

In Theorem 1, if distributions of ${\eta }_i$, $i=1,2,\cdots$ are symmetrical, i.e., the distributions of ${\eta }_i$ and $-{\eta }_i$ are same, ${\tau }_i$, $i=1,2,\cdots$ exhibit symmetry, i.e., $\Psi \left(x\right)+\Psi (-x)=1$, for all $x\in \mathbb{R}$ (see [34]), and $f(x,y)=x+y$ or $f(x,y)=x-y$, then $\underset{n\to \infty }{lim}\overline{ch}\left\{{\displaystyle \frac{S_n}{n}}\le z\right\}$ exhibits the form that is characterized by the symmetrical uncertain variable ${\tau }_1$ or $-{\tau }_1$, respectively. Example 3 illustrates this remark as a special case.

5. Conclusions

To better handle complex systems where human uncertainty and randomness with sub-linear characteristics coexist, a new type of chance spaces named U-S chance spaces is introduced in this paper. The U-S chance spaces are a pair of two-dimensional spaces made up of uncertainty space and sub-linear expectation space. Additionally, under U-S chance spaces, we define uncertain random variables, chance measures and chance distributions. A novel framework for uncertain random variables is also suggested by merging sub-linear expectation theory with uncertainty theory. This paper’s primary contribution is the derivation of the Kolmogorov type LLN for uncertain random variables that are functions of IID random variables and IID uncertain variables under U-S chance spaces. Five examples were given and explained to better illustrate the LLN for uncertain random variables under U-S chance spaces.These results are obtained under the condition that the uncertain variables are regular, and future research may further weaken the regularity assumptions.

The LLNs in the framework of U-S chance theory provides a solid theoretical foundation for statistical analysis within the same theory, enabling the inference of population characteristics through sample analysis. Based on the LLNs under U-S chance spaces, topics such as statistical inference, parameter estimation, and so on within the theory are all worthy of future research endeavors. And the U-S chance theory provides a robust framework for modeling issues such as stock investment, optimal investment, risk aversion, and the design problems of system reliability in incomplete financial markets. And these issues have been studied to a certain extent in related papers.

Based on this theory, [33] studied the risk aversion problem under incomplete markets subject to government regulation, defined the risk premium and the relative risk premium, and proved Pratt’s theorem within the framework of U-S chance theory. [32] proved operational laws for uncertain random variables, gave four expectations and models of uncertain random programming, and used them to tackle optimal investment in incomplete markets and the design problems of system reliability. Other outcomes from classical probability theory, like multi-objective uncertain random programming, uncertain random networks, and so forth, could be generalized under U-S chance spaces in future research.