On Some Laws of Large Numbers for Uncertain Random Variables

Abstract

Baoding Liu created uncertainty theory to describe the information represented by human language. In turn, Yuhan Liu founded chance theory for modelling phenomena where both uncertainty and randomness are present. The first theory involves an uncertain measure and variable, whereas the second one introduces the notions of a chance measure and an uncertain random variable. Laws of large numbers (LLNs) are important theorems within both theories. In this paper, we prove a law of large numbers (LLN) for uncertain random variables being continuous functions of pairwise independent, identically distributed random variables and regular, independent, identically distributed uncertain variables, which is a generalisation of a previously proved version of LLN, where the independence of random variables was assumed. Moreover, we prove the Marcinkiewicz–Zygmund type LLN in the case of uncertain random variables. The proved version of the Marcinkiewicz–Zygmund type theorem reflects the difference between probability and chance theory. Furthermore, we obtain the Chow type LLN for delayed sums of uncertain random variables and formulate counterparts of the last two theorems for uncertain variables. Finally, we provide illustrative examples of applications of the proved theorems. All the proved theorems can be applied for uncertain random variables being functions of symmetrically or asymmetrically distributed random variables, and symmetrical or asymmetrical uncertain variables. Furthermore, in some special cases, under the assumption of symmetry of the random and uncertain variables, the limits in the first and the third theorem have forms of symmetrical uncertain variables.

1. Introduction

The classical probability theory was developed by Kolmogorov. It is a field of mathematics that is used to address practical problems in which randomness naturally occurs. Among other things, the authors would like to highlight its applications in finance, where the notion of a stochastic process, an important part of probability theory, plays a substantial role. Apart from the Kolmogorov theory, other models of randomness have been proposed, including the boolean algebraic probability theory and MV-algebraic probability theory (see [1] and references therein).

Fuzzy sets were introduced in [2] to model non-random imprecision. However, since they can have various interpretations in practice, some generalisations of fuzzy sets have been proposed, such as Atanassov’s intuitionistic fuzzy sets (IFSs) (see [3] and references therein), interval-valued fuzzy sets (IVFs), defined in [4], or grey sets (see [5]).

Fuzzy random variables are used for the description of complex phenomena, which are both random and imprecise. Definitions of fuzzy random variables, e.g., proposed by Kwakernaak [6] or Puri and Ralescu [7], correspond to their different conceptions. For an interesting discussion on this issue, we refer the reader to [8]. The MV-algebraic probability theory, where observables and states are counterparts of random variables and probability measures, has also been applied for intuitionistic fuzzy events and interval-valued fuzzy events, defined as IFSs and IVFs, respectively. MV-algebraic central limit theorems were considered in [9], whereas their counterparts for intuitionistic fuzzy events and interval-valued fuzzy events were proved in [10,11], respectively. In turn, the approach proposed in [12] was devoted to the martingale convergence theorem in the intuitionistic fuzzy framework.

As described in [13], some authors have noted that human uncertainty is not adequately described by fuzziness. This fact was one of the reasons why Baoding Liu created uncertainty theory. Within this theory (see [14]), an uncertain measure of an event indicates the belief degree that this event will occur. The properties of uncertain measures, including normality, self-duality, monotonicity, countable subadditivity, and fulfilling product axiom (see [15]), have made the uncertainty theory a part of mathematics, different from classical probability. Moreover, this theory has provided the notion of an uncertain variable, describing uncertain quantities. Chance theory was proposed by Yuhan Liu in [13], who introduced the notion of a chance measure, integrating both probability and uncertain measures, and the concept of an uncertain random variable to deal with appearance simultaneously uncertain and random phenomena.

The laws of large numbers (LLNs) take an important place among theoretical issues concerning random and fuzzy random variables. The literature devoted to LLNs within probability theory is very rich. For classical versions of these theorems, we refer the reader to [16]. Many papers have been devoted to LLNs for fuzzy random variables too. We only mention a few of them. Basic versions of the law of large numbers were proved in [17] for fuzzy random variables defined by Kwakernaak and in [18] in the case where the definition proposed by Puri and Ralescu was used. Marcinkiewicz type and Chung type LLNs were considered in [19,20], respectively. Finally, the paper [21] was devoted to a weak LLN for linearly negative quadrant dependent fuzzy random variables. LLNs have also been considered for sequences of observables. Basic versions of LLNs in the MV-algebraic framework were discussed in [22], and some of their generalisations were proved in [23]. A weak law of large numbers for intuitionistic fuzzy events was considered in [24]. Moreover, LLNs were obtained in [25,26] for M-observables and IF-observables, respectively. Finally, the paper [27] concerned the interval-valued fuzzy case. In [28], LLN was proved for uncertain random variables being functions of independent, identically distributed random variables and independent, identically distributed regular uncertain variables, in the version corresponding to the classical Kolmogorov Theorem. The LLN mentioned above was further generalised to cases where random and uncertain variables are independent, random variables are identically distributed, and uncertain variables are not identically distributed (see [29]). In turn, the cases where random and uncertain variables are independent, but both are not identically distributed, were considered in [30,31]. Finally, in [32], LLNs were studied for sequences of uncertain random variables described by dependent random variables and independent uncertain variables.

In this paper, we prove an LLN for uncertain random variables being functions of pairwise, independent, identically distributed random variables and independent, identically distributed regular uncertain variables. Thus, the obtained theorem is a generalisation of the LLN from [28]. Moreover, we prove the Marcinkiewicz–Zygmund and the Chow type LLNs for sequences of uncertain random variables. We also present counterparts of the last two theorems for uncertain variables. Finally, we provide illustrative examples of applications of the proved theorems. Although the proving technique in the case of the first and the third theorems is similar to the one used in [28], the mentioned theorems are essential generalisations of the LLN from that paper. As described in Remarks 2 and 5, in some special cases, under the assumption of symmetry of the random and uncertain variables, the limits in the first and the third theorem also have symmetrical forms.

The rest of the paper is organised as follows. In Section 2, we give a brief exposition of basic elements of uncertainty and chance theory, including the notions of uncertain variables and uncertain random variables. Section 3 is dedicated to the generalisation of the LLN from [28]. The Marcinkiewicz–Zygmund type LLN within the theory of uncertain random variables is proved in Section 4. Section 5 is devoted to the Chow LLN for delayed sums of uncertain random variables. A short conclusion is presented in Section 6. Finally, classical probabilistic versions of the proved theorems are presented in Appendix A.

2. Preliminaries

This section is dedicated to basic definitions and theorems concerning uncertain variables and uncertain random variables.

2.1. Uncertainty Space and Uncertain Variable

We denote by $\mathbb{R}$, $\mathcal{B}\left(\mathbb{R}\right)$, and $\mathbb{N}$ the set of real numbers, the $\sigma$-algebra of Borel subsets of $\mathbb{R}$, and the set of positive integers, respectively.

Definition 1

(see [14]). Let Γ be a nonempty set and $\mathcal{L}$ be a σ-algebra of subsets of Γ. We call a set function $\mathcal{M}:\mathcal{L}\to [0,1]$ an uncertain measure and $(\Gamma ,\mathcal{L},\mathcal{M})$ an uncertainty space if $\mathcal{M}$ satisfies the following conditions:

Axiom 1 (normality axiom): $\mathcal{M}\left\{\Gamma \right\}=1$;

Axiom 2 (duality axiom): $\mathcal{M}\left\{\Lambda \right\}+\mathcal{M}\left\{{\Lambda }^c\right\}=1,\Lambda \in \mathcal{L}$;

Axiom 3 (subadditivity axiom): for each sequence of events ${\Lambda }_1$, ${\Lambda }_2$,…, the following inequality holds:

$$\mathcal{M}\left\{\bigcup _{i=1}^{\infty }{\Lambda }_i\right\}\le \sum _{i=1}^{\infty }\mathcal{M}\left\{{\Lambda }_i\right\}.$$

Definition 2

(see [15]). Let $({\Gamma }_k,{\mathcal{L}}_k,{\mathcal{M}}_k),k=1,2,\dots$, be a sequence of uncertainty spaces. We call an uncertain measure $\mathcal{M}$ on the product σ-algebra a product uncertain measure if

$$\mathcal{M}\left\{\prod _{k=1}^{\infty }{\Lambda }_k\right\}=\underset{k=1}{\overset{\infty }{\bigwedge }}{\mathcal{M}}_k\left\{{\Lambda }_k\right\}$$

for arbitrary ${\Lambda }_k\in {\mathcal{L}}_k,k=1,2,\dots$

Definition 3

(see [14]). A function $\tau :\Gamma \to \mathbb{R}$ is called an uncertain variable if it is measurable, i.e., if

$$\left\{\tau \in B\right\}=\left\{\gamma \in \Gamma |\tau (\gamma )\in B\right\}\in \mathcal{L}$$

for each $B\in \mathcal{B}\left(\mathbb{R}\right)$. Its uncertainty distribution is a function given by

$$\Psi \left(x\right)=\mathcal{M}\left\{\tau \le x\right\},x\in \mathbb{R}.$$

If its inverse function ${\Psi }^{-1}(\alpha )$ exists and is unique for each $\alpha \in (0,1)$, then Ψ and τ are called a regular uncertainty distribution and regular uncertain variable, respectively.

Definition 4

(see [15]). Let ${\tau }_1,{\tau }_2,\dots ,{\tau }_n$ be uncertain variables on an uncertainty space $(\Gamma ,\mathcal{L},\mathcal{M})$. They are said to be independent if

$$\mathcal{M}\left\{\bigcap _{i=1}^n({\tau }_i\in B_i)\right\}=\underset{i=1}{\overset{n}{\bigwedge }}\mathcal{M}\left\{{\tau }_i\in B_i\right\}$$

for arbitrary $B_i\in \mathcal{B}\left(\mathbb{R}\right),i=1,2,\dots ,n$.

2.2. Chance Space and Uncertain Random Variable

Definition 5

(see [13]). Let $(\Gamma ,\mathcal{L},\mathcal{M})$ be an uncertainty space and $(\Omega ,\mathcal{A},\mathbb{P})$ be a probability space. A chance space is the space of the form:

$$(\Gamma ,\mathcal{L},\mathcal{M})\times (\Omega ,\mathcal{A},\mathbb{P})=(\Gamma \times \Omega ,\mathcal{L}\times \mathcal{A},\mathcal{M}\times \mathbb{P}).$$

If $\Theta \in \mathcal{L}\times \mathcal{A}$ (i.e., Θ is an uncertain random event), then the chance measure $\mathrm{Ch}$ of Θ is given by

$$\begin{array}{c}\hfill \mathrm{Ch}\left\{\Theta \right\}={\int }_0^1\mathbb{P}\left\{\omega \in \Omega |\mathcal{M}\left\{\gamma \in \Gamma \left|\right(\gamma ,\omega )\in \Theta \right\}\ge r\right\}dr.\end{array}$$

(1)

It was proved in [13] that $\mathrm{Ch}$ satisfies normality, duality, and monotonicity, i.e.,

• $\mathrm{Ch}\left\{\Gamma \times \Omega \right\}=1$,
• $\mathrm{Ch}\left\{\Theta \right\}+\mathrm{Ch}\left\{{\Theta }^c\right\}=1,\Theta \in \mathcal{L}\times \mathcal{A}$,
• $\mathrm{Ch}\left\{{\Theta }_1\right\}\le \mathrm{Ch}\left\{{\Theta }_2\right\}$ for events ${\Theta }_1,{\Theta }_2$, such that ${\Theta }_1\subseteq {\Theta }_2$.

Moreover, the author of [33] proved that $\mathrm{Ch}$ is subadditive.

Definition 6

(see [13]). A function ξ from the chance space $(\Gamma ,\mathcal{L},\mathcal{M})\times (\Omega ,\mathcal{A},\mathbb{P})$ to $\mathbb{R}$ is called an uncertain random variable if it is measurable, i.e., for each $B\in \mathcal{B}\left(\mathbb{R}\right)$

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

Its chance distribution Ξ is defined as follows:

$$\Xi \left(x\right)=\mathrm{Ch}\left\{\xi \le x\right\},x\in \mathbb{R}.$$

Definition 7

(see [34]). An uncertain random sequence ${\left\{{\xi }_n\right\}}_{n=1}^{\infty }$ is said to be convergent to an uncertain random variable ξ in measure if

$$\underset{n\to \infty }{lim}\mathrm{Ch}\left\{(\gamma ,\omega )\in \mathcal{L}\times \mathcal{A}\left|\right|{\xi }_n(\gamma ,\omega )-\xi (\gamma ,\omega )|\ge \epsilon \right\}=0$$

for each $\epsilon >0$.

Definition 8

(see [13]). An uncertain random sequence ${\left\{{\xi }_n\right\}}_{n=1}^{\infty }$ with chance distributions ${\left\{{\Xi }_n\right\}}_{n=1}^{\infty }$ is said to be convergent to an uncertain random variable ξ with chance distribution Ξ in distribution if $\underset{n\to \infty }{lim}{\Xi }_n\left(x\right)=\Xi \left(x\right)$ for each $x\in \mathbb{R}$ at which $\Xi \left(x\right)$ is continuous.

The following theorem from [35] shows the relationship between both types of convergence.

Theorem 1.

If an uncertain random sequence ${\left\{{\xi }_n\right\}}_{n=1}^{\infty }$ converges in measure to an uncertain random variable ξ, then it converges in distribution to ξ.

3. A Generalised Version of LLN

Let $(\Gamma ,\mathcal{L},\mathcal{M})\times (\Omega ,\mathcal{A},\mathbb{P})$ be a chance space.

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. We denote by $C^I$ the class of continuous functions $f:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ such that $f(x,y)$ is strictly increasing with respect to y for each $x\in \mathbb{R}$, by $C^D$ the class of continuous functions $f:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ such that $f(x,y)$ is strictly decreasing with respect to y for each $x\in \mathbb{R}$, by ${\overline{C}}^I$ the class of continuous functions $g:\mathbb{R}\to \mathbb{R}$ strictly increasing, and by ${\overline{C}}^D$ the class of continuous functions $g:\mathbb{R}\to \mathbb{R}$ strictly decreasing. Let

$$\begin{array}{cc}\hfill & S_n=\sum _{i=1}^{n}f({\eta }_i,{\tau }_i),n\in \mathbb{N},f\in C^I\cup C^D,\hfill \\ \hfill & S_n\left(y\right)=\sum _{i=1}^{n}f({\eta }_i,y),y\in \mathbb{R},n\in \mathbb{N},f\in C^I\cup C^D,\hfill \\ \hfill & {\overline{S}}_n=\sum _{i=1}^{n}g\left({\tau }_i\right),n\in \mathbb{N},g\in {\overline{C}}^I\cup {\overline{C}}^D.\hfill \end{array}$$

We use the symbol $\mathbb{E}$ to denote the expected value with respect to probability measure $\mathbb{P}$.

Definition 9.

Random variables ${\left\{{\eta }_i\right\}}_{i=1}^{\infty }$ are said to be pairwise independent if for each $i,j\in \mathbb{N}$, $i\ne j$, the random variables ${\eta }_i$ and ${\eta }_j$ are independent.

Clearly, independence of random variables implies their pairwise independence.

Remark 1.

Let Φ be a cumulative distribution function, τ be a regular uncertain variable with uncertainty distribution Ψ, and $F\left(y\right)={\int }_{-\infty }^{\infty }f(x,y)d\Phi \left(x\right),y\in \mathbb{R}$. Then from considerations in [28], it follows that for $z\in \left(inf\left\{F\right(y\left)\right|y\in \mathbb{R}\},sup\{F\left(y\right)|y\in \mathbb{R}\}\right)$ the function

$$\begin{array}{c}\hfill z\mapsto \mathcal{M}\left\{{\int }_{-\infty }^{\infty }f(x,\tau )d\Phi \left(x\right)\le z\right\}=\left\{\begin{array}{ccc}\Psi \left(F^{-1}\left(z\right)\right)& if& f\in C^I,\\ 1-\Psi \left(F^{-1}\left(z\right)\right)& if& f\in C^D\end{array}\right.\end{array}$$

(2)

is continuous and strictly increasing.

Theorem 2.

Let random variables ${\left\{{\eta }_i\right\}}_{i=1}^{\infty }$ be pairwise independent, identically distributed with cumulative distribution function Φ and uncertain variables ${\left\{{\tau }_i\right\}}_{i=1}^{\infty }$ be regular, independent, identically distributed with uncertainty distribution Ψ. Assume that $f\in C^I\cup C^D$ and $\mathbb{E}\left|f\right({\eta }_1,y\left)\right|<\infty ,y\in \mathbb{R}$. Then

$$\begin{array}{c}\hfill {\displaystyle \underset{n\to \infty }{lim}\frac{S_n}{n}={\int }_{-\infty }^{\infty }f(x,{\tau }_1)d\Phi \left(x\right)}\end{array}$$

(3)

in the sense of convergence in distribution.

Proof. 

In the proof, we use the approach from [28].

Assume that $f\in C^I$. Fix $z\in \mathbb{R}$. Let $\epsilon >0$. By Theorem A1 in Appendix A, there exists $N_1\in \mathbb{N}$ such that for each $n\ge N_1$

$$\begin{array}{cc}\hfill & \mathbb{P}\left\{\frac{S_n\left(F^{-1}(z-\epsilon )\right)}{n}\le z\right\}\ge 1-\epsilon ,\hfill \end{array}$$

(4)

where F is defined as in Remark 1 for the cumulative distribution function $\Phi$ of the random variables ${\left\{{\eta }_i\right\}}_{i=1}^{\infty }$.

Let $n\ge N_1$.

$$\begin{array}{cc}\hfill \mathrm{Ch}& \left\{\frac{S_n}{n}\le z\right\}={\int }_0^1\mathbb{P}\left\{\mathcal{M}\left\{\frac{S_n}{n}\le z\right\}\ge r\right\}dr\hfill \\ \hfill & \ge {\int }_0^1\mathbb{P}\left\{\left\{\frac{S_n\left(F^{-1}(z-\epsilon )\right)}{n}\le z\right\}\right.\left.\cap \left\{\mathcal{M}\left\{\frac{S_n}{n}\le z\right\}\ge r\right\}\right\}dr\hfill \\ \hfill & \ge {\int }_0^1\mathbb{P}\left\{\left\{\frac{S_n\left(F^{-1}(z-\epsilon )\right)}{n}\le z\right\}\right.\left.\cap \left\{\mathcal{M}\left\{\frac{S_n}{n}\le \frac{S_n\left(F^{-1}(z-\epsilon )\right)}{n}\right\}\ge r\right\}\right\}dr\hfill \\ \hfill & ={\int }_0^1\mathbb{P}\left\{\left\{\frac{S_n\left(F^{-1}(z-\epsilon )\right)}{n}\le z\right\}\right.\hfill \\ \hfill & \left.\cap \left\{\mathcal{M}\left\{\sum _{i=1}^{n}f({\eta }_i,{\tau }_i)\le \sum _{i=1}^{n}f({\eta }_i,F^{-1}(z-\epsilon ))\right\}\ge r\right\}\right\}dr\hfill \\ \hfill & \ge {\int }_0^1\mathbb{P}\left\{\left\{\frac{S_n\left(F^{-1}(z-\epsilon )\right)}{n}\le z\right\}\right.\hfill \\ \hfill & \left.\cap \left\{\mathcal{M}\left\{\bigcap _{i=1}^n\left\{f({\eta }_i,{\tau }_i)\le f({\eta }_i,F^{-1}(z-\epsilon ))\right\}\right\}\ge r\right\}\right\}dr.\hfill \end{array}$$

(5)

${\left\{{\tau }_i\right\}}_{i=1}^{\infty }$ are independent and $f(x,y)$ is a strictly increasing function of y for each x. Thus, applying (4) and (5) we obtain

$$\begin{array}{cc}\hfill \mathrm{Ch}& \left\{\frac{S_n}{n}\le z\right\}\ge {\int }_0^1\mathbb{P}\left\{\left\{\frac{S_n\left(F^{-1}(z-\epsilon )\right)}{n}\le z\right\}\right.\left.\cap \left\{\underset{i=1}{\overset{n}{\bigwedge }}\mathcal{M}\left\{{\tau }_i\le F^{-1}(z-\epsilon )\right\}\ge r\right\}\right\}dr\hfill \\ \hfill & ={\int }_0^1\mathbb{P}\left\{\left\{\frac{S_n\left(F^{-1}(z-\epsilon )\right)}{n}\le z\right\}\right.\left.\cap \left\{\underset{i=1}{\overset{n}{\bigwedge }}\Psi \left(F^{-1}(z-\epsilon )\right)\ge r\right\}\right\}dr\hfill \\ \hfill & \ge (1-\epsilon )\Psi \left(F^{-1}(z-\epsilon )\right).\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim}\mathrm{Ch}\left\{\frac{S_n}{n}\le z\right\}\ge \Psi \left(F^{-1}\left(z\right)\right).\hfill \end{array}$$

(6)

In a similar way, let $\epsilon >0$. Theorem A1 in Appendix A implies that there exists $N_2\in \mathbb{N}$ such that for each $n\ge N_2$

$$\begin{array}{cc}\hfill & \mathbb{P}\left\{\frac{S_n\left(F^{-1}(z+\epsilon )\right)}{n}>z\right\}\ge 1-\epsilon .\hfill \end{array}$$

(7)

Let $n\ge N_2.$

$$\begin{array}{cc}\hfill \mathrm{Ch}& \left\{\frac{S_n}{n}>z\right\}={\int }_0^1\mathbb{P}\left\{\mathcal{M}\left\{\frac{S_n}{n}>z\right\}\ge r\right\}dr\hfill \\ \hfill & \ge {\int }_0^1\mathbb{P}\left\{\left\{\frac{S_n\left(F^{-1}(z+\epsilon )\right)}{n}>z\right\}\cap \left\{\mathcal{M}\left\{\frac{S_n}{n}>z\right\}\ge r\right\}\right\}dr\hfill \\ \hfill & \ge {\int }_0^1\mathbb{P}\left\{\left\{\frac{S_n\left(F^{-1}(z+\epsilon )\right)}{n}>z\right\}\right.\left.\cap \left\{\mathcal{M}\left\{\frac{S_n}{n}>\frac{S_n\left(F^{-1}(z+\epsilon )\right)}{n}\right\}\ge r\right\}\right\}dr\hfill \\ \hfill & ={\int }_0^1\mathbb{P}\left\{\left\{\frac{S_n\left(F^{-1}(z+\epsilon )\right)}{n}>z\right\}\right.\hfill \\ \hfill & \left.\cap \left\{\mathcal{M}\left\{\sum _{i=1}^{n}f({\eta }_i,{\tau }_i)>\sum _{i=1}^{n}f({\eta }_i,F^{-1}(z+\epsilon ))\right\}\ge r\right\}\right\}dr\hfill \\ \hfill & \ge {\int }_0^1\mathbb{P}\left\{\left\{\frac{S_n\left(F^{-1}(z+\epsilon )\right)}{n}>z\right\}\right.\hfill \\ \hfill & \left.\cap \left\{\mathcal{M}\left\{\bigcap _{i=1}^n\left\{f({\eta }_i,{\tau }_i)>f({\eta }_i,F^{-1}(z+\epsilon ))\right\}\right\}\ge r\right\}\right\}dr.\hfill \end{array}$$

(8)

From independence of ${\left\{{\tau }_i\right\}}_{i=1}^{\infty }$, the assumption that $f(x,y)$ is a strictly increasing function of y for each x, (7), and (8) it follows that

$$\begin{array}{cc}\hfill \mathrm{Ch}& \left\{\frac{S_n}{n}>z\right\}\ge {\int }_0^1\mathbb{P}\left\{\left\{\frac{S_n\left(F^{-1}(z+\epsilon )\right)}{n}>z\right\}\right.\left.\cap \left\{\underset{i=1}{\overset{n}{\bigwedge }}\mathcal{M}\left\{{\tau }_i>F^{-1}(z+\epsilon )\right\}\ge r\right\}\right\}dr\hfill \\ \hfill & ={\int }_0^1\mathbb{P}\left\{\left\{\frac{S_n\left(F^{-1}(z+\epsilon )\right)}{n}>z\right\}\right.\left.\cap \left\{\underset{i=1}{\overset{n}{\bigwedge }}(1-\Psi \left(F^{-1}(z+\epsilon )\right))\ge r\right\}\right\}dr\hfill \\ \hfill & \ge (1-\epsilon )(1-\Psi \left(F^{-1}(z+\epsilon )\right)).\hfill \end{array}$$

(9)

By (9) and the duality property of $\mathrm{Ch}$,

$$\begin{array}{cc}\hfill \mathrm{Ch}& \left\{\frac{S_n}{n}\le z\right\}\le 1-(1-\epsilon )(1-\Psi \left(F^{-1}(z+\epsilon )\right))\hfill \end{array}$$

(10)

for each $n\ge N_2$.

(10) implies

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim}\mathrm{Ch}\left\{\frac{S_n}{n}\le z\right\}\le \Psi \left(F^{-1}\left(z\right)\right).\hfill \end{array}$$

(11)

Thus, by (6) and (11),

$$\underset{n\to \infty }{lim}\mathrm{Ch}\left\{\frac{S_n}{n}\le z\right\}=\Psi \left(F^{-1}\left(z\right)\right)=\mathcal{M}\left\{{\int }_{-\infty }^{\infty }f(x,{\tau }_1)d\Phi \left(x\right)\le z\right\}.$$

If $f\in C^D$, then $-f\in C^I$. Proceeding like in the previous part of the proof, we get

$$\underset{n\to \infty }{lim}\mathrm{Ch}\left\{-\frac{S_n}{n}<-z\right\}=\mathcal{M}\left\{-{\int }_{-\infty }^{\infty }f(x,{\tau }_1)d\Phi \left(x\right)<-z\right\},$$

being equivalent to

$$\underset{n\to \infty }{lim}\mathrm{Ch}\left\{\frac{S_n}{n}>z\right\}=\mathcal{M}\left\{{\int }_{\infty }^{\infty }f(x,{\tau }_1)d\Phi \left(x\right)>z\right\}.$$

From the duality properties of $\mathrm{Ch}$ and $\mathcal{M}$ we obtain

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}\mathrm{Ch}& \left\{\frac{S_n}{n}\le z\right\}=\mathcal{M}\left\{{\int }_{-\infty }^{\infty }f(x,{\tau }_1)d\Phi \left(x\right)\le z\right\}.\hfill \end{array}$$

 □

The LLN proposed in [28] follows from Theorem 2, since independent random variables are pairwise independent.

Example 1.

We consider a sequence of independent Rademacher random variables ${\left\{{ϵ}_i\right\}}_{i=-1}^{\infty }$, i.e., random variables with distribution $\mathbb{P}\left\{{ϵ}_i=-1\right\}=\mathbb{P}\left\{{ϵ}_i=1\right\}=\frac{1}{2}$, $i\in \mathbb{N}\cup \left\{-1,0\right\}$, and a sequence of independent, identically distributed indicator functions ${\mathbb{1}}_{A_i}$, $i\in \mathbb{N}$, such that $\mathbb{P}\left\{A_i\right\}=\alpha$, $0<\alpha <1$, $i\in \mathbb{N}$. We assume that both sequences are independent. Let ${\left\{{\eta }_i\right\}}_{i=1}^{\infty }$ be also Rademacher distributed random variables, given by

$${\eta }_i={\mathbb{1}}_{A_i}{ϵ}_i+{\mathbb{1}}_{A_i^c}{ϵ}_{i-1}{ϵ}_{i-2},i\in \mathbb{N}.$$

Moreover, let uncertain variables ${\left\{{\tau }_i\right\}}_{i=1}^{\infty }$ be independent, normally distributed with uncertainty distribution

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

Then ${\left\{{\eta }_i\right\}}_{i=1}^{\infty }$ are pairwise independent, but dependent (see [36] for the proof). Therefore, the LLN proposed in [28] can not be applied to the sequence of uncertain random variables ${\left\{{\eta }_i+{\tau }_i\right\}}_{i=1}^{\infty }$. However, from Theorem 2 it follows that

$$\begin{array}{cc}\hfill & {\displaystyle \underset{n\to \infty }{lim}\frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)}{n}={\tau }_1}\hfill \end{array}$$

in the sense of convergence in distribution.

The above example is an illustration of the following remark.

Remark 2.

If random variables ${\left\{{\eta }_i\right\}}_{i=1}^{\infty }$ and uncertain variables ${\left\{{\tau }_i\right\}}_{i=1}^{\infty }$ satisfy the assumptions of Theorem 2, ${\left\{{\eta }_i\right\}}_{i=1}^{\infty }$ are symmetrically distributed, i.e., ${\eta }_i$ and $-{\eta }_i$ have the same distribution for $i\in \mathbb{N}$, ${\left\{{\tau }_i\right\}}_{i=1}^{\infty }$ are symmetrical, i.e., $\Psi \left(x\right)+\Psi (-x)=1$, $x\in \mathbb{R}$ (see, e.g., [37]), and $f(x,y)=x+y$ or $f(x,y)=x-y$, then ${\displaystyle \underset{n\to \infty }{lim}\frac{S_n}{n}}$ has the form of the symmetrical uncertain variable ${\tau }_1$ or $-{\tau }_1$, respectively.

Remark 3.

Remark 1 and Formula (3) imply that for $z\in \left(inf\left\{F\right(y\left)\right|y\in \mathbb{R}\},sup\{F\left(y\right)|y\in \mathbb{R}\}\right)$

$$\begin{array}{cc}\hfill \psi \left(z\right)& :=\underset{n\to \infty }{lim}\mathrm{Ch}\left\{\frac{S_n}{n}\le z\right\}=\mathcal{M}\left\{{\int }_{-\infty }^{\infty }f(x,{\tau }_1)d\Phi \left(x\right)\le z\right\}\hfill \\ \hfill & =\left\{\begin{array}{ccc}\Psi \left(F^{-1}\left(z\right)\right)& if& f\in C^I,\\ 1-\Psi \left(F^{-1}\left(z\right)\right)& if& f\in C^D\end{array}\right.\hfill \end{array}$$

(12)

is a continuous, strictly increasing function.

4. The Marcinkiewicz–Zygmund Type LLN for Independent Uncertain Random Variables

We will use the notation from the previous section. This section is devoted to the problem of convergence of the sequence

$$\frac{S_n-nc}{n^{1/p}}\mathrm{as}n\to \infty ,\mathrm{where}p>0\mathrm{and}c\in \mathbb{R}.$$

(13)

At the beginning, we consider convergence of the sequence given in formula (13) in measure for $p\in (0,1)$.

Theorem 3.

Let random variables ${\left\{{\eta }_i\right\}}_{i=1}^{\infty }$ be independent, identically distributed with cumulative distribution function Φ and uncertain variables ${\left\{{\tau }_i\right\}}_{i=1}^{\infty }$ be independent, regular with uncertainty distributions ${\left\{{\Psi }_i\right\}}_{i=1}^{\infty }$ such that

$$\begin{array}{cc}\hfill & \underset{y\to \infty }{lim}\overline{\Psi }\left(y\right)=1,\underset{y\to -\infty }{lim}\overline{\overline{\Psi }}\left(y\right)=1,\hfill \end{array}$$

(14)

where

$$\overline{\Psi }\left(y\right)=\underset{n\to \infty }{lim}\underset{i=1}{\overset{n}{\bigwedge }}{\Psi }_i\left(y\right),\overline{\overline{\Psi }}\left(y\right)=\underset{n\to \infty }{lim}\underset{i=1}{\overset{n}{\bigwedge }}(1-{\Psi }_i\left(y\right)),$$

$y\in \mathbb{R}$. Let $p\in (0,1)$. If $f\in C^I\cup C^D$ and $\mathbb{E}|f({\eta }_1,y){|}^p<\infty$,$y\in \mathbb{R}$, then for arbitrary $c\in \mathbb{R}$

$${\displaystyle \underset{n\to \infty }{lim}\frac{S_n-nc}{n^{1/p}}=0}$$

in the sense of convergence in measure.

Proof. 

Fix $\epsilon >0$. Let $0<{\epsilon }_1,{\epsilon }_2,{\epsilon }_1^{\prime },{\epsilon }_1^{″},{\epsilon }_2^{\prime },{\epsilon }_2^{″}<1$ be such that

$$(1-{\epsilon }_1)<(1-{\epsilon }_1^{\prime })(1-{\epsilon }_1^{″}),(1-{\epsilon }_2)<(1-{\epsilon }_2^{\prime })(1-{\epsilon }_2^{″}).$$

By Theorem A2 in Appendix A, there exist $N_1,N_2\in \mathbb{N},y_1,y_2\in \mathbb{R}$ such that

$$\begin{array}{cc}\hfill & \mathbb{P}\left\{\frac{S_n\left(y_1\right)-nc}{n^{1/p}}<\epsilon \right\}>1-{\epsilon }_1^{\prime },\underset{i=1}{\overset{n}{\bigwedge }}{\Psi }_i\left(y_1\right)\ge \overline{\Psi }\left(y_1\right)>1-{\epsilon }_1^{″},n\ge N_1,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & \mathbb{P}\left\{\frac{S_n\left(y_2\right)-nc}{n^{1/p}}>-\epsilon \right\}>1-{\epsilon }_2^{\prime },\underset{i=1}{\overset{n}{\bigwedge }}(1-{\Psi }_i\left(y_2\right))\ge \overline{\overline{\Psi }}\left(y_2\right)>1-{\epsilon }_2^{″},n\ge {\mathbb{N}}_2.\hfill \end{array}$$

(16)

Assume that $f\in C^I$. Let $n\ge N_1$.

$$\begin{array}{cc}\hfill \mathrm{Ch}& \left\{\frac{S_n-nc}{n^{1/p}}<\epsilon \right\}={\int }_0^1\mathbb{P}\left\{\mathcal{M}\left\{\frac{S_n-nc}{n^{1/p}}<\epsilon \right\}\ge r\right\}dr\hfill \\ \hfill & \ge {\int }_0^1\mathbb{P}\left\{\left\{\frac{S_n\left(y_1\right)-nc}{n^{1/p}}<\epsilon \right\}\right.\left.\cap \left\{\mathcal{M}\left\{\frac{S_n-nc}{n^{1/p}}<\epsilon \right\}\ge r\right\}\right\}dr\hfill \\ \hfill & \ge {\int }_0^1\mathbb{P}\left\{\left\{\frac{S_n\left(y_1\right)-nc}{n^{1/p}}<\epsilon \right\}\right.\left.\cap \left\{\mathcal{M}\left\{\frac{S_n}{n^{1/p}}<\frac{S_n\left(y_1\right)}{n^{1/p}}\right\}\ge r\right\}\right\}dr\hfill \\ \hfill & ={\int }_0^1\mathbb{P}\left\{\left\{\frac{S_n\left(y_1\right)-nc}{n^{1/p}}<\epsilon \right\}\right.\hfill \\ \hfill & \left.\cap \left\{\mathcal{M}\left\{\sum _{i=1}^{n}f({\eta }_i,{\tau }_i)<\sum _{i=1}^{n}f({\eta }_i,y_1)\right\}\ge r\right\}\right\}dr\hfill \\ \hfill & \ge {\int }_0^1\mathbb{P}\left\{\left\{\frac{S_n\left(y_1\right)-nc}{n^{1/p}}<\epsilon \right\}\right.\hfill \\ \hfill & \left.\cap \left\{\mathcal{M}\left\{\bigcap _{i=1}^n\left\{f({\eta }_i,{\tau }_i)<f({\eta }_i,y_1)\right\}\right\}\ge r\right\}\right\}dr.\hfill \end{array}$$

(17)

By independence and regularity of ${\left\{{\tau }_i\right\}}_{i=1}^{\infty }$, the assumption that $f(x,y)$ is a strictly increasing function of y for each x, (15), and (17), it follows that

$$\begin{array}{cc}\hfill \mathrm{Ch}& \left\{\frac{S_n-nc}{n^{1/p}}<\epsilon \right\}\hfill \\ \hfill & \ge {\int }_0^1\mathbb{P}\left\{\left\{\frac{S_n\left(y_1\right)-nc}{n^{1/p}}<\epsilon \right\}\right.\left.\cap \left\{\underset{i=1}{\overset{n}{\bigwedge }}\mathcal{M}\left\{{\tau }_i<y_1\right\}\ge r\right\}\right\}dr\hfill \\ \hfill & \ge {\int }_0^1\mathbb{P}\left\{\left\{\frac{S_n\left(y_1\right)-nc}{n^{1/p}}<\epsilon \right\}\right.\left.\cap \left\{\underset{i=1}{\overset{n}{\bigwedge }}{\Psi }_i\left(y_1\right)\ge r\right\}\right\}dr\hfill \\ \hfill & =\mathbb{P}\left\{\frac{S_n\left(y_1\right)-nc}{n^{1/p}}<\epsilon \right\}\underset{i=1}{\overset{n}{\bigwedge }}{\Psi }_i\left(y_1\right)>(1-{\epsilon }_1^{\prime })(1-{\epsilon }_1^{″})>1-{\epsilon }_1.\hfill \end{array}$$

(18)

Thus,

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim}\mathrm{Ch}\left\{\frac{S_n-nc}{n^{1/p}}<\epsilon \right\}=1.\hfill \end{array}$$

(19)

Similarly, for $n\ge N_2$

$$\begin{array}{cc}\hfill \mathrm{Ch}& \left\{\frac{S_n-nc}{n^{1/p}}>-\epsilon \right\}={\int }_0^1\mathbb{P}\left\{\mathcal{M}\left\{\frac{S_n-nc}{n^{1/p}}>-\epsilon \right\}\ge r\right\}dr\hfill \\ \hfill & \ge {\int }_0^1\mathbb{P}\left\{\left\{\frac{S_n\left(y_2\right)-nc}{n^{1/p}}>-\epsilon \right\}\cap \left\{\mathcal{M}\left\{\frac{S_n-nc}{n^{1/p}}>-\epsilon \right\}\ge r\right\}\right\}dr\hfill \\ \hfill & \ge {\int }_0^1\mathbb{P}\left\{\left\{\frac{S_n\left(y_2\right)-nc}{n^{1/p}}>-\epsilon \right\}\cap \left\{\mathcal{M}\left\{\frac{S_n}{n^{1/p}}>\frac{S_n\left(y_2\right)}{n^{1/p}}\right\}\ge r\right\}\right\}dr\hfill \\ \hfill & ={\int }_0^1\mathbb{P}\left\{\left\{\frac{S_n\left(y_2\right)-nc}{n^{1/p}}>-\epsilon \right\}\right.\hfill \\ \hfill & \left.\cap \left\{\mathcal{M}\left\{\sum _{i=1}^{n}f({\eta }_i,{\tau }_i)>\sum _{i=1}^{n}f({\eta }_i,y_2)\right\}\ge r\right\}\right\}dr\hfill \\ \hfill & \ge {\int }_0^1\mathbb{P}\left\{\left\{\frac{S_n\left(y_2\right)-nc}{n^{1/p}}>-\epsilon \right\}\right.\hfill \\ \hfill & \left.\cap \left\{\mathcal{M}\left\{\bigcap _{i=1}^n\left\{f({\eta }_i,{\tau }_i)>f({\eta }_i,y_2)\right\}\right\}\ge r\right\}\right\}dr.\hfill \end{array}$$

(20)

${\left\{{\tau }_i\right\}}_{i=1}^{\infty }$ are independent and regular, $f(x,y)$ is a strictly increasing function of y for each x. Therefore, (16) and (20) imply

$$\begin{array}{cc}\hfill \mathrm{Ch}& \left(\frac{S_n-nc}{n^{1/p}}>-\epsilon \right)\hfill \\ \hfill & \ge {\int }_0^1\mathbb{P}\left\{\left\{\frac{S_n\left(y_2\right)-nc}{n^{1/p}}>-\epsilon \right\}\cap \left\{\underset{i=1}{\overset{n}{\bigwedge }}\mathcal{M}\left\{{\tau }_i>y_2\right\}\ge r\right\}\right\}dr\hfill \\ \hfill & ={\int }_0^1\mathbb{P}\left\{\left\{\frac{S_n\left(y_2\right)-nc}{n^{1/p}}>-\epsilon \right\}\cap \left\{\underset{i=1}{\overset{n}{\bigwedge }}(1-{\Psi }_i\left(y_2\right))\ge r\right\}\right\}dr\hfill \\ \hfill & =\mathbb{P}\left\{\frac{S_n\left(y_2\right)-nc}{n^{1/p}}>-\epsilon \right\}\underset{i=1}{\overset{n}{\bigwedge }}(1-{\Psi }_i\left(y_2\right))\hfill \\ \hfill & >(1-{\epsilon }_2^{\prime })(1-{\epsilon }_2^{″})>1-{\epsilon }_2.\hfill \end{array}$$

(21)

Thus,

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim}\mathrm{Ch}\left\{\frac{S_n-nc}{n^{1/p}}>-\epsilon \right\}=1.\hfill \end{array}$$

(22)

From (19), (22), and properties of $\mathrm{Ch}$ it follows that

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}& \mathrm{Ch}\left\{\left|\frac{S_n-nc}{n^{1/p}}\right|\ge \epsilon \right\}\hfill \\ \hfill =& \underset{n\to \infty }{lim}\mathrm{Ch}\left\{\left\{\frac{S_n-nc}{n^{1/p}}\le -\epsilon \right\}\cup \left\{\frac{S_n-nc}{n^{1/p}}\ge \epsilon \right\}\right\}\hfill \\ \hfill \le & \underset{n\to \infty }{lim}\mathrm{Ch}\left\{\frac{S_n-nc}{n^{1/p}}\le -\epsilon \right\}+\underset{n\to \infty }{lim}\mathrm{Ch}\left\{\frac{S_n-nc}{n^{1/p}}\ge \epsilon \right\}\hfill \\ \hfill =& 1-\underset{n\to \infty }{lim}\mathrm{Ch}\left\{\frac{S_n-nc}{n^{1/p}}>-\epsilon \right\}+1-\underset{n\to \infty }{lim}\mathrm{Ch}\left\{\frac{S_n-nc}{n^{1/p}}<\epsilon \right\}=0.\hfill \end{array}$$

(23)

If $f\in C^D$, then $-f\in C^I$. Therefore, we can apply (23) for $-f$ and $-c$. Thus,

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}& \mathrm{Ch}\left\{\left|\frac{{\sum }_{i=1}^n-f({\eta }_i,{\tau }_i)+nc}{n^{1/p}}\right|\ge \epsilon \right\}\hfill \\ \hfill =& \underset{n\to \infty }{lim}\mathrm{Ch}\left\{\left|\frac{{\sum }_{i=1}^{n}f({\eta }_i,{\tau }_i)-nc}{n^{1/p}}\right|\ge \epsilon \right\}=0.\hfill \end{array}$$

(24)

 □

The following corollary concerns the case of strictly monotone functions of uncertain variables.

Corollary 1.

Let uncertain variables ${\left\{{\tau }_i\right\}}_{i=1}^{\infty }$ be independent, regular with uncertainty distributions ${\left\{{\Psi }_i\right\}}_{i=1}^{\infty }$ satisfying (14), and $p\in (0,1)$. If $g\in {\overline{C}}^I\cup {\overline{C}}^D$, then for arbitrary $c\in \mathbb{R}$ ${\displaystyle \underset{n\to \infty }{lim}\frac{{\overline{S}}_n-nc}{n^{1/p}}=0}$ in the sense of convergence in measure.

The convergence (13) in distribution for $p>0$ is established by our next theorem.

Theorem 4.

Let random variables ${\left\{{\eta }_i\right\}}_{i=1}^{\infty }$ be identically distributed with cumulative distribution function Φ and uncertain variables ${\left\{{\tau }_i\right\}}_{i=1}^{\infty }$ be regular, independent, identically distributed with uncertainty distribution Ψ. Let $f\in C^I\cup C^D$.

(i) If ${\left\{{\eta }_i\right\}}_{i=1}^{\infty }$ are independent, $p\in (0,1)$, $\mathbb{E}|f({\eta }_1,y){|}^p<\infty ,y\in \mathbb{R}$, and $c\in \mathbb{R}$, then

$${\displaystyle \underset{n\to \infty }{lim}\frac{S_n-nc}{n^{1/p}}=0}$$

in the sense of convergence in distribution.

(ii) If ${\left\{{\eta }_i\right\}}_{i=1}^{\infty }$ are pairwise independent, $p=1$, $\mathbb{E}\left|f\right({\eta }_1,y\left)\right|<\infty ,y\in \mathbb{R}$, and $c\in \mathbb{R}$, then

$${\displaystyle \underset{n\to \infty }{lim}\frac{S_n-nc}{n^{1/p}}=\underset{n\to \infty }{lim}\frac{S_n}{n}-c={\int }_{-\infty }^{\infty }f(x,{\tau }_1)d\Phi \left(x\right)-c}$$

in the sense of convergence in distribution.

(iii) If ${\left\{{\eta }_i\right\}}_{i=1}^{\infty }$ are pairwise independent, $p>1$, $\mathbb{E}\left|f\right({\eta }_1,y\left)\right|<\infty ,y\in \mathbb{R}$, and $c\in \{u\in \mathbb{R}|\mathcal{M}\left\{{\int }_{-\infty }^{\infty }f(x,{\tau }_1)d\Phi \left(x\right)\le u\right\}\in (0,1)\}$, then

$${\displaystyle \underset{n\to \infty }{lim}\frac{S_n-nc}{n^{1/p}}=m_{f,c}}$$

in the sense of convergence in distribution, where $m_{f,c}$ is an uncertain variable with constant uncertainty distribution equal to

$$\mathcal{M}\left\{{\int }_{-\infty }^{\infty }f(x,{\tau }_1)d\Phi \left(x\right)\le c\right\}.$$

Proof. 

Case (i) We use Theorem 3. Clearly, $\overline{\Psi }=\Psi$, $\overline{\overline{\Psi }}=1-\Psi .$ Therefore, ${\displaystyle \underset{y\to \infty }{lim}\overline{\Psi }\left(y\right)=1}$ and ${\displaystyle \underset{y\to -\infty }{lim}\overline{\overline{\Psi }}\left(y\right)=1}$. Thus, the assumptions of Theorem 3 are satisfied and application of Theorem 1 finishes the proof.

Case (ii) Let $z\in \mathbb{R}$. By Theorem 2,

$$\begin{array}{cc}\hfill {\displaystyle \underset{n\to \infty }{lim}\mathrm{Ch}}& {\displaystyle \left\{\frac{S_n-nc}{n^{1/p}}\le z\right\}=\underset{n\to \infty }{lim}\mathrm{Ch}\left\{\frac{S_n}{n}\le z+c\right\}}\hfill \\ & =\mathcal{M}\left\{\mathbb{E}f({\eta }_1,{\tau }_1)\le z+c\right\}=\mathcal{M}\left\{\mathbb{E}f({\eta }_1,{\tau }_1)-c\le z\right\}.\hfill \end{array}$$

(25)

This finishes the proof.

Case (iii) Fix $z\in \mathbb{R}$. For each $\delta >0$ there exists $N\in \mathbb{N}$ such that $\frac{\left|z\right|}{n^{1-1/p}}<\delta$ for each $n\ge N$. If $z\ge 0$, then

$$\begin{array}{cc}\hfill \mathrm{Ch}& \left\{\frac{S_n}{n}\le c\right\}\le \mathrm{Ch}\left\{\frac{S_n-nc}{n^{1/p}}\le z\right\}=\mathrm{Ch}\left\{\frac{S_n}{n}\le c+\frac{z}{n^{1-1/p}}\right\}\hfill \\ & \le \mathrm{Ch}\left\{\frac{S_n}{n}\le c+\delta \right\},n\ge N.\hfill \end{array}$$

By Theorem 2,

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}& \mathrm{Ch}\left\{\frac{S_n}{n}\le c\right\}=\mathcal{M}\left\{{\int }_{-\infty }^{\infty }f(x,{\tau }_1)d\Phi \left(x\right)\le c\right\}=\psi \left(c\right),\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}& \mathrm{Ch}\left\{\frac{S_n}{n}\le c+\delta \right\}=\mathcal{M}\left\{{\int }_{-\infty }^{\infty }f(x,{\tau }_1)d\Phi \left(x\right)\le c+\delta \right\}=\psi (c+\delta ).\hfill \end{array}$$

(27)

From (26) and (27) it follows that

$$\begin{array}{cc}\hfill \psi \left(c\right)& \le \underset{n\to \infty }{lim\; inf}\mathrm{Ch}\left\{\frac{S_n-nc}{n^{1/p}}\le z\right\}\le \underset{n\to \infty }{lim\; sup}\mathrm{Ch}\left\{\frac{S_n-nc}{n^{1/p}}\le z\right\}\le \psi (c+\delta ).\hfill \end{array}$$

(28)

Since $\delta >0$ is arbitrary, (28) and continuity of $\psi$ imply

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim\; inf}\mathrm{Ch}& \left\{\frac{S_n-nc}{n^{1/p}}\le z\right\}=\underset{n\to \infty }{lim\; sup}\mathrm{Ch}\left\{\frac{S_n-nc}{n^{1/p}}\le z\right\}\hfill \\ \hfill =\underset{n\to \infty }{lim}& \mathrm{Ch}\left\{\frac{S_n-nc}{n^{1/p}}\le z\right\}=\psi \left(c\right)=\mathcal{M}\left\{{\int }_{-\infty }^{\infty }f(x,{\tau }_1)d\Phi \left(x\right)\le c\right\}.\hfill \end{array}$$

(29)

If $z<0$, then

$$\begin{array}{cc}\hfill \mathrm{Ch}& \left\{\frac{S_n}{n}\le c-\delta \right\}\le \mathrm{Ch}\left\{\frac{S_n-nc}{n^{1/p}}\le z\right\}=\mathrm{Ch}\left\{\frac{S_n}{n}\le c+\frac{z}{n^{1-1/p}}\right\}\hfill \\ & \le \mathrm{Ch}\left\{\frac{S_n}{n}\le c\right\},n\ge N.\hfill \end{array}$$

By Theorem 2,

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}& \mathrm{Ch}\left\{\frac{S_n}{n}\le c-\delta \right\}=\mathcal{M}\left\{{\int }_{-\infty }^{\infty }f(x,{\tau }_1)d\Phi \left(x\right)\le c-\delta \right\}=\psi (c-\delta ).\hfill \end{array}$$

(30)

From (26) and (30)

$$\begin{array}{cc}\hfill \psi (c-\delta )& \le \underset{n\to \infty }{lim\; inf}\mathrm{Ch}\left\{\frac{S_n-nc}{n^{1/p}}\le z\right\}\hfill \end{array}$$

$$\begin{array}{c}\hfill \le \underset{n\to \infty }{lim\; sup}\mathrm{Ch}\left\{\frac{S_n-nc}{n^{1/p}}\le z\right\}\le \psi \left(c\right).\end{array}$$

(31)

$\delta >0$ is arbitrary. Therefore, by (31) and continuity of $\psi$

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim\; inf}& \mathrm{Ch}\left\{\frac{S_n-nc}{n^{1/p}}\le z\right\}=\underset{n\to \infty }{lim\; sup}\mathrm{Ch}\left\{\frac{S_n-nc}{n^{1/p}}\le z\right\}\hfill \\ \hfill =\underset{n\to \infty }{lim}& \mathrm{Ch}\left\{\frac{S_n-nc}{n^{1/p}}\le z\right\}=\psi \left(c\right)=\mathcal{M}\left\{{\int }_{-\infty }^{\infty }f(x,{\tau }_1)d\Phi \left(x\right)\le c\right\}.\hfill \end{array}$$

(32)

 □

Remark 4.

Case (iii) of the above theorem reflects the difference between chance theory and probability theory, where such convergence for random variables generally does not hold for $p>2$.

Example 2.

Let random variables ${\left\{{\eta }_i\right\}}_{i=1}^{\infty }$ be independent, identically distributed with cumulative distribution function Φ. Let uncertain variables ${\left\{{\tau }_i\right\}}_{i=1}^{\infty }$ be regular, independent, identically distributed with uncertainty distribution Ψ. Then from Theorem 4 it follows that

(i) If $p\in (0,1)$, $\mathbb{E}|{\eta }_1{|}^p<\infty$, and $c\in \mathbb{R}$, then

$$\begin{array}{cc}\hfill & {\displaystyle \underset{n\to \infty }{lim}\frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)-nc}{n^{1/p}}=0,\underset{n\to \infty }{lim}\frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)-nc}{n^{1/p}}=0}\hfill \end{array}$$

in the sense of convergence in distribution.

(ii) If $p=1$, $\mathbb{E}|{\eta }_1|<\infty$, and $c\in \mathbb{R}$, then

$$\begin{array}{cc}\hfill & {\displaystyle \underset{n\to \infty }{lim}\frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)-nc}{n^{1/p}}=\underset{n\to \infty }{lim}\frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)}{n}-c}\hfill \\ & ={\tau }_1+\mathbb{E}{\eta }_1-c,\hfill \\ \hfill & {\displaystyle \underset{n\to \infty }{lim}\frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)-nc}{n^{1/p}}=\underset{n\to \infty }{lim}\frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)}{n}-c}\hfill \\ & =-{\tau }_1+\mathbb{E}{\eta }_1-c\hfill \end{array}$$

in the sense of convergence in distribution.

(iii) If $p>1$ and $\mathbb{E}|{\eta }_1|<\infty$, then

$$\begin{array}{cc}\hfill & {\displaystyle \underset{n\to \infty }{lim}\frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)-nc}{n^{1/p}}=m_1,c\in \{u\in \mathbb{R}|\Psi \left(u-\mathbb{E}{\eta }_1\right)\in (0,1)\},}\hfill \\ \hfill & {\displaystyle \underset{n\to \infty }{lim}\frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)-nc}{n^{1/p}}=m_2,c\in \{u\in \mathbb{R}|\Psi \left(\mathbb{E}{\eta }_1-u\right)\in (0,1)\},}\hfill \end{array}$$

in the sense of convergence in distribution, where $m_1$ and $m_2$ are uncertain variables with constant uncertainty distributions equal to $\Psi \left(c-\mathbb{E}{\eta }_1\right)$ and $1-\Psi \left(\mathbb{E}{\eta }_1-c\right)$, respectively.

Example 3.

Let the assumptions from Example 2 be fulfilled, ${\eta }_1$ have the discrete distribution

$$\mathbb{P}\left\{{\eta }_1=2^{5i/8}\right\}=1/2^i,i\in \mathbb{N},$$

and ${\tau }_1$ have the linear distribution

$$\Psi \left(x\right)=\left\{\begin{array}{ccc}0& \mathrm{if}& x\le 0;\\ x& \mathrm{if}& 0<x\le 1;\\ 1& \mathrm{if}& x>1.\end{array}\right.$$

$\mathbb{E}|{\eta }_1{|}^p<\infty$ if and only if $p<8/5$. Moreover, $\mathbb{E}{\eta }_1=\frac{2^{-3/8}}{1-2^{-3/8}}=\frac{1}{2^{3/8}-1}$. Therefore,

$$\begin{array}{cc}\hfill & {\displaystyle \underset{n\to \infty }{lim}\frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)-nc}{n^{1/p}}=0,\underset{n\to \infty }{lim}\frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)-nc}{n^{1/p}}=0,p\in (0,1),c\in \mathbb{R},}\hfill \\ \hfill & {\displaystyle \underset{n\to \infty }{lim}\frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)-nc}{n}={\tau }_1+\frac{1}{2^{3/8}-1}-c,}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\displaystyle \underset{n\to \infty }{lim}\frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)}{n}-c=-{\tau }_1+\frac{1}{2^{3/8}-1}-c,c\in \mathbb{R},}\hfill \\ \hfill & {\displaystyle \underset{n\to \infty }{lim}\frac{{\sum }_{i=1}^n({\eta }_i+{\tau }_i)-nc}{n^{1/p}}=m_1,}\hfill \\ \hfill & p>1,c\in (\frac{1}{2^{3/8}-1},\frac{1}{2^{3/8}-1}+1),\hfill \\ \hfill & {\displaystyle \underset{n\to \infty }{lim}\frac{{\sum }_{i=1}^n({\eta }_i-{\tau }_i)-nc}{n^{1/p}}=m_2,}\hfill \\ \hfill & p>1,c\in (\frac{1}{2^{3/8}-1}-1,\frac{1}{2^{3/8}-1}),\hfill \end{array}$$

in the sense of convergence in distribution, where $m_1$ and $m_2$ are the uncertain variables with constant uncertainty distributions equal to $c-\frac{1}{2^{3/8}-1}$ and $c-\frac{1}{2^{3/8}-1}+1$, respectively.

In the case of strictly monotone functions of uncertain variables, we have the following corollary.

Corollary 2.

Let uncertain variables ${\left\{{\tau }_i\right\}}_{i=1}^{\infty }$ be regular, independent, identically distributed with uncertainty distribution Ψ. Let $g\in {\overline{C}}^I\cup {\overline{C}}^D$.

(i) If $p\in (0,1)$ and $c\in \mathbb{R}$, then

$${\displaystyle \underset{n\to \infty }{lim}\frac{{\overline{S}}_n-nc}{n^{1/p}}=0}$$

in the sense of convergence in distribution.

(ii) If $p=1$ and $c\in \mathbb{R}$, then

$${\displaystyle \underset{n\to \infty }{lim}\frac{{\overline{S}}_n-nc}{n^{1/p}}=\underset{n\to \infty }{lim}\frac{{\overline{S}}_n}{n}-c=g\left({\tau }_1\right)-c}$$

in the sense of convergence in distribution.

(iii) If $p>1$ and $c\in \{u\in \mathbb{R}|\mathcal{M}\left\{g\left({\tau }_1\right)\le u\right\}\in (0,1)\}$, then

$${\displaystyle \underset{n\to \infty }{lim}\frac{{\overline{S}}_n-nc}{n^{1/p}}=m_{g,c}}$$

in the sense of convergence in distribution, where $m_{g,c}$ is an uncertain variable with constant uncertainty distribution equal to $\mathcal{M}\left\{g\left({\tau }_1\right)\le c\right\}$.

Example 4.

Let uncertain variables ${\left\{{\tau }_i\right\}}_{i=1}^{\infty }$ be regular, independent, identically distributed with uncertainty distribution Ψ from Example 3. We apply Corollary 2.

(i) If $p\in (0,1)$ and $c\in \mathbb{R}$, then

$${\displaystyle \underset{n\to \infty }{lim}\frac{{\sum }_{i=1}^n{\tau }_i^3-nc}{n^{1/p}}=0\mathit{and}\underset{n\to \infty }{lim}\frac{{\sum }_{i=1}^n(-{\tau }_i^3)-nc}{n^{1/p}}=0}$$

in the sense of convergence in distribution.

(ii) If $p=1$ and $c\in \mathbb{R}$, then

$$\begin{array}{cc}\hfill & {\displaystyle \underset{n\to \infty }{lim}\frac{{\sum }_{i=1}^n{\tau }_i^3-nc}{n^{1/p}}=\underset{n\to \infty }{lim}\frac{{\sum }_{i=1}^n{\tau }_i^3}{n}-c={\tau }_1^3-c}\hfill \\ \hfill \mathit{and}& {\displaystyle \underset{n\to \infty }{lim}\frac{{\sum }_{i=1}^n(-{\tau }_i^3)-nc}{n^{1/p}}=\underset{n\to \infty }{lim}\frac{{\sum }_{i=1}^n(-{\tau }_i^3)}{n}-c=-{\tau }_1^3-c}\hfill \end{array}$$

in the sense of convergence in distribution.

(iii) If $p>1$, then

$$\begin{array}{cc}\hfill & {\displaystyle \underset{n\to \infty }{lim}\frac{{\sum }_{i=1}^n{\tau }_i^3-nc}{n^{1/p}}=m_1,c\in (0,1),}\hfill \\ \hfill \mathit{and}& {\displaystyle \underset{n\to \infty }{lim}\frac{{\sum }_{i=1}^n(-{\tau }_i^3)-nc}{n^{1/p}}=m_2,c\in (-1,0),}\hfill \end{array}$$

in the sense of convergence in distribution, where $m_1$ and $m_2$ are uncertain variables with constant uncertainty distributions equal to $c^{1/3}$ and $c^{1/3}+1$, respectively.

5. The Chow LLN for Delayed Sums of Uncertain Random Variables

We will use the notation from the previous sections. Let additionally,

$$\begin{array}{cc}\hfill & T_{n,n+k}=\sum _{i=n+1}^{n+k}f({\eta }_i,{\tau }_i),n\in \mathbb{N}\cup \left\{0\right\},k\in \mathbb{N},f\in C^I\cup C^D,\hfill \\ \hfill & T_{n,n+k}\left(y\right)=\sum _{i=n+1}^{n+k}f({\eta }_i,y),y\in \mathbb{R},n\in \mathbb{N}\cup \left\{0\right\},k\in \mathbb{N},f\in C^I\cup C^D,\hfill \\ \hfill & {\overline{T}}_{n,n+k}=\sum _{i=n+1}^{n+k}g\left({\tau }_i\right),n\in \mathbb{N}\cup \left\{0\right\},k\in \mathbb{N},g\in {\overline{C}}^I\cup {\overline{C}}^D.\hfill \end{array}$$

We will prove the LLN for uncertain random variables, which corresponds to the Chow law for delayed sums of random variables. In contradistinction to Theorem 2 and the LLN proved in [28], where for each $n\in \mathbb{N}$ the scaled sums $S_n={\sum }_{i=1}^{n}f({\eta }_i,{\tau }_i)$ were considered, the following Theorem 5 concerns the scaled sums of the form

$$T_{n,n+n^{\alpha }}=\sum _{i=n+1}^{n+n^{\alpha }}f({\eta }_i,{\tau }_i)$$

for a fixed number $\alpha \in (0,1)$.

Theorem 5.

Let $\alpha \in (0,1)$. Let random variables ${\left\{{\eta }_i\right\}}_{i=1}^{\infty }$ be independent, identically distributed with cumulative distribution function Φ. Let uncertain variables ${\left\{{\tau }_i\right\}}_{i=1}^{\infty }$ be regular, independent, identically distributed with uncertainty distribution Ψ. Assume that $f\in C^I\cup C^D$ and $\mathbb{E}|f({\eta }_1,y){|}^{1/\alpha }<\infty ,y\in \mathbb{R}$. Then

$$\begin{array}{c}\hfill {\displaystyle \underset{n\to \infty }{lim}\frac{T_{n,n+n^{\alpha }}}{n^{\alpha }}={\int }_{-\infty }^{\infty }f(x,{\tau }_1)d\Phi \left(x\right)}\end{array}$$

(33)

in the sense of convergence in distribution.

Proof. 

In the following proof of this theorem, we use the general approach proposed in [28] with several important modifications.

Assume that $f\in C^I$. Fix $z\in \mathbb{R}$. Let $\epsilon >0$. By Corollary A1 in Appendix A, there exists $N_1\in \mathbb{N}$ such that for each $n\ge N_1$

$$\begin{array}{cc}\hfill & \mathbb{P}\left\{\frac{T_{n,n+n^{\alpha }}\left(F^{-1}(z-\epsilon )\right)}{n^{\alpha }}\le z\right\}\ge 1-\epsilon .\hfill \end{array}$$

(34)

Let $n\ge N_1$.

$$\begin{array}{cc}\hfill \mathrm{Ch}& \left\{\frac{T_{n,n+n^{\alpha }}}{n^{\alpha }}\le z\right\}={\int }_0^1\mathbb{P}\left\{\mathcal{M}\left\{\frac{T_{n,n+n^{\alpha }}}{n^{\alpha }}\le z\right\}\ge r\right\}dr\hfill \\ \hfill & \ge {\int }_0^1\mathbb{P}\left\{\left\{\frac{T_{n,n+n^{\alpha }}\left(F^{-1}(z-\epsilon )\right)}{n^{\alpha }}\le z\right\}\right.\hfill \\ \hfill & \left.\cap \left\{\mathcal{M}\left\{\frac{T_{n,n+n^{\alpha }}}{n^{\alpha }}\le z\right\}\ge r\right\}\right\}dr\hfill \\ \hfill & \ge {\int }_0^1\mathbb{P}\left\{\left\{\frac{T_{n,n+n^{\alpha }}\left(F^{-1}(z-\epsilon )\right)}{n^{\alpha }}\le z\right\}\right.\hfill \\ \hfill & \left.\cap \left\{\mathcal{M}\left\{\frac{T_{n,n+n^{\alpha }}}{n^{\alpha }}\le \frac{T_{n,n+n^{\alpha }}\left(F^{-1}(z-\epsilon )\right)}{n^{\alpha }}\right\}\ge r\right\}\right\}dr\hfill \\ \hfill & ={\int }_0^1\mathbb{P}\left\{\left\{\frac{T_{n,n+n^{\alpha }}\left(F^{-1}(z-\epsilon )\right)}{n^{\alpha }}\le z\right\}\right.\hfill \\ \hfill & \left.\cap \left\{\mathcal{M}\left\{\sum _{i=n+1}^{n+n^{\alpha }}f({\eta }_i,{\tau }_i)\le \sum _{i=n+1}^{n+n^{\alpha }}f({\eta }_i,F^{-1}(z-\epsilon ))\right\}\ge r\right\}\right\}dr\hfill \\ \hfill & \ge {\int }_0^1\mathbb{P}\left\{\left\{\frac{T_{n,n+n^{\alpha }}\left(F^{-1}(z-\epsilon )\right)}{n^{\alpha }}\le z\right\}\right.\hfill \\ \hfill & \left.\cap \left\{\mathcal{M}\left\{\bigcap _{i=n+1}^{n+n^{\alpha }}\left\{f({\eta }_i,{\tau }_i)\le f({\eta }_i,F^{-1}(z-\epsilon ))\right\}\right\}\ge r\right\}\right\}dr.\hfill \end{array}$$

(35)

Independence of ${\left\{{\tau }_i\right\}}_{i=1}^{\infty }$, the assumption that $f(x,y)$ is a strictly increasing function of y for each x, (34), and (35) imply that

$$\begin{array}{cc}\hfill \mathrm{Ch}& \left\{\frac{T_{n,n+n^{\alpha }}}{n^{\alpha }}\le z\right\}\hfill \\ \hfill & \ge {\int }_0^1\mathbb{P}\left\{\left\{\frac{T_{n,n+n^{\alpha }}\left(F^{-1}(z-\epsilon )\right)}{n^{\alpha }}\le z\right\}\right.\hfill \\ \hfill & \left.\cap \left\{\underset{i=n+1}{\overset{n+n^{\alpha }}{\bigwedge }}\mathcal{M}\left\{{\tau }_i\le F^{-1}(z-\epsilon )\right\}\ge r\right\}\right\}dr\hfill \\ \hfill & ={\int }_0^1\mathbb{P}\left\{\left\{\frac{T_{n,n+n^{\alpha }}\left(F^{-1}(z-\epsilon )\right)}{n^{\alpha }}\le z\right\}\right.\hfill \\ \hfill & \left.\cap \left\{\underset{i=n+1}{\overset{n+n^{\alpha }}{\bigwedge }}\Psi \left(F^{-1}(z-\epsilon )\right)\ge r\right\}\right\}dr\ge (1-\epsilon )\Psi \left(F^{-1}(z-\epsilon )\right).\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim}\mathrm{Ch}\left\{\frac{T_{n,n+n^{\alpha }}}{n^{\alpha }}\le z\right\}\ge \Psi \left(F^{-1}\left(z\right)\right).\hfill \end{array}$$

(36)

Similarly, let $\epsilon >0$. By Corollary A1 in Appendix A, there exists $N_2\in \mathbb{N}$ such that for each $n\ge N_2$

$$\begin{array}{cc}\hfill & \mathbb{P}\left\{\frac{T_{n,n+n^{\alpha }}\left(F^{-1}(z+\epsilon )\right)}{n^{\alpha }}>z\right\}\ge 1-\epsilon .\hfill \end{array}$$

(37)

Let $n\ge N_2$.

$$\begin{array}{cc}\hfill \mathrm{Ch}& \left\{\frac{T_{n,n+n^{\alpha }}}{n^{\alpha }}>z\right\}={\int }_0^1\mathbb{P}\left\{\mathcal{M}\left\{\frac{T_{n,n+n^{\alpha }}}{n^{\alpha }}>z\right\}\ge r\right\}dr\hfill \\ \hfill & \ge {\int }_0^1\mathbb{P}\left\{\left\{\frac{T_{n,n+n^{\alpha }}\left(F^{-1}(z+\epsilon )\right)}{n^{\alpha }}>z\right\}\cap \left\{\mathcal{M}\left\{\frac{T_{n,n+n^{\alpha }}}{n^{\alpha }}>z\right\}\ge r\right\}\right\}dr\hfill \\ \hfill & \ge {\int }_0^1\mathbb{P}\left\{\left\{\frac{T_{n,n+n^{\alpha }}\left(F^{-1}(z+\epsilon )\right)}{n^{\alpha }}>z\right\}\right.\hfill \\ \hfill & \left.\cap \left\{\mathcal{M}\left\{\frac{T_{n,n+n^{\alpha }}}{n^{\alpha }}>\frac{T_{n,n+n^{\alpha }}\left(F^{-1}(z+\epsilon )\right)}{n^{\alpha }}\right\}\ge r\right\}\right\}dr\hfill \\ \hfill & ={\int }_0^1\mathbb{P}\left\{\left\{\frac{T_{n,n+n^{\alpha }}\left(F^{-1}(z+\epsilon )\right)}{n^{\alpha }}>z\right\}\right.\hfill \\ \hfill & \left.\cap \left\{\mathcal{M}\left\{\sum _{i=n+1}^{n+n^{\alpha }}f({\eta }_i,{\tau }_i)>\sum _{i=n+1}^{n+n^{\alpha }}f({\eta }_i,F^{-1}(z+\epsilon ))\right\}\ge r\right\}\right\}dr\hfill \\ \hfill & \ge {\int }_0^1\mathbb{P}\left\{\left\{\frac{T_{n,n+n^{\alpha }}\left(F^{-1}(z+\epsilon )\right)}{n^{\alpha }}>z\right\}\right.\hfill \\ \hfill & \left.\cap \left\{\mathcal{M}\left\{\bigcap _{i=n+1}^{n+n^{\alpha }}\left\{f({\eta }_i,{\tau }_i)>f({\eta }_i,F^{-1}(z+\epsilon ))\right\}\right\}\ge r\right\}\right\}dr.\hfill \end{array}$$

(38)

By independence of ${\left\{{\tau }_i\right\}}_{i=1}^{\infty }$, the assumption that $f(x,y)$ is a strictly increasing function of y for each x, (37), and (38), we obtain

$$\begin{array}{cc}\hfill \mathrm{Ch}& \left\{\frac{T_{n,n+n^{\alpha }}}{n^{\alpha }}>z\right\}\hfill \\ \hfill & \ge {\int }_0^1\mathbb{P}\left\{\left\{\frac{T_{n,n+n^{\alpha }}\left(F^{-1}(z+\epsilon )\right)}{n^{\alpha }}>z\right\}\right.\hfill \\ \hfill & \left.\cap \left\{\underset{i=n+1}{\overset{n+n^{\alpha }}{\bigwedge }}\mathcal{M}\left\{{\tau }_i>F^{-1}(z+\epsilon )\right\}\ge r\right\}\right\}dr\hfill \\ \hfill & ={\int }_0^1\mathbb{P}\left\{\left\{\frac{T_{n,n+n^{\alpha }}\left(F^{-1}(z+\epsilon )\right)}{n^{\alpha }}>z\right\}\right.\hfill \\ \hfill & \left.\cap \left\{\underset{i=n+1}{\overset{n+n^{\alpha }}{\bigwedge }}(1-\Psi \left(F^{-1}(z+\epsilon )\right))\ge r\right\}\right\}dr\hfill \\ \hfill & \ge (1-\epsilon )(1-\Psi \left(F^{-1}(z+\epsilon )\right)).\hfill \end{array}$$

(39)

By (39) and the duality property of $\mathrm{Ch}$,

$$\begin{array}{cc}\hfill \mathrm{Ch}& \left\{\frac{T_{n,n+n^{\alpha }}}{n^{\alpha }}\le z\right\}\le 1-(1-\epsilon )(1-\Psi \left(F^{-1}(z+\epsilon )\right))\hfill \end{array}$$

(40)

for each $n\ge N_2$. From (40) it follows that

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim}\mathrm{Ch}\left\{\frac{T_{n,n+n^{\alpha }}}{n^{\alpha }}\le z\right\}\le \Psi \left(F^{-1}\left(z\right)\right).\hfill \end{array}$$

(41)

Therefore, by (36) and (41)

$$\underset{n\to \infty }{lim}\mathrm{Ch}\left\{\frac{T_{n,n+n^{\alpha }}}{n^{\alpha }}\le z\right\}=\Psi \left(F^{-1}\left(z\right)\right)=\mathcal{M}\left\{{\int }_{-\infty }^{\infty }f(x,{\tau }_1)d\Phi \left(x\right)\le z\right\}.$$

If $f\in C^D$, then $-f\in C^I$. Thus, proceeding similarly as in the previous part of the proof, we obtain

$$\underset{n\to \infty }{lim}\mathrm{Ch}\left\{-\frac{T_{n,n+n^{\alpha }}}{n^{\alpha }}<-z\right\}=\mathcal{M}\left\{-{\int }_{-\infty }^{\infty }f(x,{\tau }_1)d\Phi \left(x\right)<-z\right\},$$

which is equivalent to

$$\underset{n\to \infty }{lim}\mathrm{Ch}\left\{\frac{T_{n,n+n^{\alpha }}}{n^{\alpha }}>z\right\}=\mathcal{M}\left\{{\int }_{\infty }^{\infty }f(x,{\tau }_1)d\Phi \left(x\right)>z\right\}.$$

The duality properties of $\mathrm{Ch}$ and $\mathcal{M}$ imply

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}\mathrm{Ch}& \left\{\frac{T_{n,n+n^{\alpha }}}{n^{\alpha }}\le z\right\}=\mathcal{M}\left\{{\int }_{-\infty }^{\infty }f(x,{\tau }_1)d\Phi \left(x\right)\le z\right\}.\hfill \end{array}$$

 □

Remark 5.

If random variables ${\left\{{\eta }_i\right\}}_{i=1}^{\infty }$ and uncertain variables ${\left\{{\tau }_i\right\}}_{i=1}^{\infty }$ satisfy the assumptions of Theorem 5, ${\left\{{\eta }_i\right\}}_{i=1}^{\infty }$ are symmetrically distributed, ${\left\{{\tau }_i\right\}}_{i=1}^{\infty }$ are symmetrical, and $f(x,y)=x+y$ or $f(x,y)=x-y$, then ${\displaystyle \underset{n\to \infty }{lim}\frac{T_{n,n+n^{\alpha }}}{n^{\alpha }}}$ has the form of the symmetrical uncertain variable ${\tau }_1$ or $-{\tau }_1$, respectively.

In the special case of strictly monotone functions of independent uncertain variables, we obtain.

Corollary 3.

Let $\alpha \in (0,1)$. Let uncertain variables ${\left\{{\tau }_i\right\}}_{i=1}^{\infty }$ be regular, independent, identically distributed with uncertainty distribution Ψ. Assume that $g\in {\overline{C}}^I\cup {\overline{C}}^D$. Then

$${\displaystyle \underset{n\to \infty }{lim}\frac{{\overline{T}}_{n,n+n^{\alpha }}}{n^{\alpha }}=g\left({\tau }_1\right)}$$

in the sense of convergence in distribution.

Example 5.

Let $\alpha =2/3$. Let random variables ${\left\{{\eta }_i\right\}}_{i=1}^{\infty }$ and uncertain variables ${\left\{{\tau }_i\right\}}_{i=1}^{\infty }$ satisfy the assumptions from Example 3. Then

$$\mathbb{E}({\eta }_1+y)=y+\frac{1}{2^{3/8}-1}\mathit{and}\mathbb{E}{|{\eta }_1+y|}^{3/2}<\infty ,y\in \mathbb{R}.$$

Therefore, by Theorem 5 and Corollary 3

$${\displaystyle \underset{n\to \infty }{lim}\frac{{\sum }_{i=n+1}^{n+n^{2/3}}({\eta }_i+{\tau }_i)}{n^{2/3}}={\tau }_1+\frac{1}{2^{3/8}-1}and\underset{n\to \infty }{lim}\frac{{\sum }_{i=n+1}^{n+n^{2/3}}{\tau }_i}{n^{2/3}}={\tau }_1}$$

in the sense of convergence in distribution.

6. Conclusions

This paper is devoted to the development of the theory of uncertain random variables and uncertain variables. We have considered uncertain random variables in the form of continuous functions of a random variable and an uncertain variable. It has been additionally assumed that the functions are strictly monotone with respect to their second arguments. The sequences of uncertain random variables of this form have been defined for a fixed function satisfying the above conditions and given sequences of pairwise independent or independent, identically distributed random variables and regular, independent, identically distributed uncertain variables. This paper’s main contribution is the formulation and proof of the generalisation of LLN from [28], the Marcinkiewicz–Zygmund type LNN, and the Chow type LLN for such defined sequences of uncertain random variables. The first theorem has the form corresponding to the classical Etemadi Theorem. Within the second theorem, under the assumption that for each fixed value of the second argument, the p-th moment of the considered function of random variables exists, $p\in (0,1]$, the convergence in distribution of scaled sums of the uncertain random variables has been proved. Moreover, their convergence in measure has been obtained in the case of $p\in (0,1)$, where the assumption of the same distribution of the regular uncertain variables has been weakened. In turn, the third theorem has established the convergence in distribution of delayed, scaled sums of uncertain random variables. Furthermore, we have formulated counterparts of the last two theorems for regular uncertain variables. Finally, we have provided illustrative examples of applications of the proved theorems. Our future research plans concern the formulation and proof of other versions of LLN, in particular, for dependent uncertain random variables. One possible direction for further research is a generalisation of Theorem 5, corresponding to the Chow law for delayed sums of random variables, to the case where uncertain random variables are functions of pairwise independent, identically distributed random variables and regular, independent, identically distributed uncertain variables. This generalisation requires having first proved a version of the Chow law for random variables that are pairwise independent.