On the convergence of complex uncertain random sequences

Abstract

This paper aims at putting forward several types of convergence concepts of complex uncertain random sequences. The relations among convergence concepts are derived by some limit theorems. In addition, for illustrating of convergence theorems, lots of examples are derived. Finally, a lot of counterexamples about relationships between convergence concepts are stated.

Keywords

Chance theory complex uncertain random variable chance distribution convergence in distribution convergence in mean convergence in measure

1 Introduction

When no samples are available to estimate a probability distribution, the researcher has to invite some domain experts to evaluate the belief degree that each event will happen. Perhaps some people think that the belief degree should be modeled by subjective probability or fuzzy set theory. Since, the complex number set allows us to solve many problems that traditionally cannot be solved by invoking the real number set, such as improper integrals that represent electrical resistance in the field of engineering, applying soft, fuzzy and hybrid sets to complex numbers is an essential step to incorporate the advantages of complex numbers into the notion of soft sets, fuzzy sets, and generalizations. Thus, Romat et al. [22] introduced the concept of complex fuzzy sets which extend the notion of fuzzy sets, are made possible by adding a phase term that describes the periodicity of the elements with respect to time. After that, Selvachandran et al. [23] discussed about relations between vague soft sets. As an application of soft sets and complex soft sets in decision making, several authors devoted their works to this topic, such as [12 , 25– 27].

However, it is usually inappropriate because both of them may lead to counterintuitive results in this case. In order to rationally deal with personal belief degrees, uncertainty theory was founded in 2007 and consequently studied by many researchers. Nowadays, uncertainty theory has become a branch of mathematics. In fact, the world is neither random nor uncertain, but sometimes it can be analyzed by probability theory, and sometimes by uncertainty theory. In order to derive our main results, we recall several works in this topic.

After the introducing of uncertain measure, as a fundamental device in uncertainty theory, Liu [13] presented the concept of uncertain variable. For describing an uncertain variable, the concept of uncertainty distribution is introduced by Liu [15]. After introducing uncertainty distribution, Liu [16] proposed the concept of inverse uncertainty distribution, and a sufficient and necessary condition for it is verified by Liu [17]. As major applications of uncertainty theory on uncertain system, the concept of reliability index is presented by Liu [16]. Gao and Yao [9] studied the reliability index of uncertain lifetime in the case of k-out-of-n systems. Since, complex quantities are applicable in two-dimensional potential flow, alternating current and so on, Peng [21] introduced the concept of complex uncertain variables for modeling complex quantities in uncertainty theory. Chen et al. [4] established several formulas for calculating the variance and pseudo-variance of complex uncertain variables. Furthermore, convergence theorems of complex uncertain sequences are studied by Chen et al. [3].

As a mixture of uncertainty theory and probability theory, the concept of uncertain random variable is introduced by Liu [18]. Also, Liu [18] studied the concepts of expected value and variance of uncertain random variables. Furthermore, as major device in chance theory, Liu [19] presented operational law of uncertain random variables. After that, several scholars devoted their studies to applications of uncertain random variables, for instance Guo and Wang [10], Gao et al. [6], Ahmadzade et al. [1, 2], Gao and Sheng [7], Gao and Ralescu [5], Liu and Ralescu [20], Wang et al. [24] and Gao and Yao [9]. For modeling complex quantities in chance theory, Gao et al. [8] introduced the concept of complex uncertain random variables. Also, Gao et al. [8] presented the concepts of complex chance distribution and expected value for describing and ranking a complex uncertain variable, respectively. Furthermore, the variance of a complex uncertain random variable was studied.

The rest of this paper is organized as follows. Some basic concepts and theorems about uncertainty theory and chance theory are presented in Section 2 including some concepts of uncertain random variables, expected value, variance. Several convergence concepts of complex uncertain random variables are introduced in Section 3. Also, some limit theorems for complex uncertain random sequences are established. Furthermore, for better illustrations of limit theorems, several examples are stated in this section. Finally, some brief conclusions are given in Section 4.

2 Preliminaries

In this section, we review some concepts concerning uncertainty theory and chance theory, including uncertain variable, complex uncertain variable, chance distribution, operational law, expected value, variance.

2.1 Uncertain Variables

In this subsection, we provide several concepts of uncertainty theory that will be used throughout the paper. For more details, see [13, 14].

Let ℒ be a σ-algebra on a nonempty set Γ. A set function M : ℒ → [0, 1] is called an uncertain measure if it satisfies the following axioms: (i) (Normality Axiom) M {Γ} =1 for the universal set Γ. (ii) (Duality Axiom) M {Λ} + M {Λ^c} =1 for any event Λ. (iii) (Subadditivity Axiom) For every countable sequence of events Λ₁, Λ₂, ⋯, we have $ℒ M {⋃_{i = 1}^{\infty} Λ_{i}} \leq \sum_{i = 1}^{\infty} ℒ M {Λ_{i}} .$ The triplet (Γ, ℒ, M) is called an uncertainty space. Furthermore, the product uncertain measure on the product σ-algebra ℒ is defined by Liu [14] as follows: (iv) (Product Axiom) Let (Γ_k, ℒ_k, M_k) be uncertainty spaces for k = 1, 2, ⋯ the product uncertain measure M is an uncertain measure satisfying $ℒ M {\prod_{k = 1}^{\infty} Λ_{k}} = ⋀_{k = 1}^{\infty} {ℒ M}_{k} {Λ_{k}}$ where Λ_k are arbitrarily chosen events from ℒ_k for k = 1, 2, ⋯, respectively.

Definition 1. (Liu [13]) An uncertain variable ξ is a function from an uncertainty space (Γ, ℒ, M) to the set of real numbers such that {ξ ∈ B} is an event for any Borel set B.

Definition 2. (Liu [13]) The uncertain variables ξ₁, ξ₂, ⋯, ξ_n are said to be independent if $M {⋂_{i = 1}^{n} {ξ_{i} \in B_{i}}} = ⋀_{i = 1}^{n} ℒ M {ξ_{i} \in B_{i}}$ for any Borel sets B₁, B₂, ⋯, B_n of real numbers.

Theorem 1. (Liu [13]) Suppose that ξ₁, ξ₂, ⋯, ξ_n are independent uncertain variables, and f₁, f₂, ⋯, f_n be measurable functions. Then f₁ (ξ₁), f₂ (ξ₂), ⋯, f_n (ξ_n) are independent uncertain variables.

Definition 3. (Liu [14]) The uncertainty distribution Φ of an uncertain variable ξ is defined by $Φ (x) = ℒ M {ξ \leq x}$ for any real number x.

Definition 4. (Liu [14]) Let ξ be an uncertain variable with regular uncertainty distribution Φ (x). Then the inverse function Φ^-1 (α) is called the inverse uncertainty distribution of ξ.

Theorem 2. (Liu [14]) Let ξ₁, ⋯, ξ_n be independent uncertain variables with regular uncertainty distributions Φ₁, Φ₂, ⋯, Φ_n, respectively. If f is a strictly increasing function, then ξ = f (ξ₁, ξ₂, ⋯, ξ_n) is an uncertain variable with inverse uncertainty distribution $Ψ^{- 1} (α) = f (Φ_{1}^{- 1} (α), \dots, Φ_{n}^{- 1} (α)) .$

Definition 5. (Peng [21]) A complex uncertain variable is a measurable function τ from an uncertainty space (Γ, ℒ, M) to the set of complex numbers, i.e., for any Borel set B of complex numbers, the set $\begin{matrix} {τ \in B} = {γ \in Γ | τ (γ) \in B}, \end{matrix}$ is an event.

Definition 6. (Peng [21]) The complex uncertainty distribution Φ (x) of a complex uncertain variable ξ is a function from $ℂ$ to [0, 1] defined by $\begin{matrix} Φ (z) = ℒ M {Re (ξ) \leq Re (z), Im (ξ) \leq Im (z)} \end{matrix}$ for any complex number z.

Definition 7. (Chen et al. [4]) The complex uncertain random sequence {ξ_n, n ≥ 1} is said to be convergent in measure to ξ if $\begin{matrix} lim_{n \to \infty} ℒ M {γ \in Γ | | | ξ_{n} (γ) - ξ (γ) | | > ∊} = 0, \forall ∊ > 0 . \end{matrix}$

Definition 8. (Chen et al. [4]) The complex uncertain sequence {ξ_n, n ≥ 1} is said to be convergent in mean to ξ if $\begin{matrix} lim_{n \to \infty} E | | ξ_{n} - ξ | | = 0 . \end{matrix}$

Definition 9. (Chen et al. [4]) Let Φ, Φ₁, Φ₂, ⋯ be the complex uncertainty distributions of complex uncertain variables ξ, ξ₁, ξ₂, ⋯, respectively.We say the complex uncertain random sequence {ξ_n, n ≥ 1} converges in distribution to ξ if $\begin{matrix} lim_{n \to \infty} Φ_{n} (z) = Φ (z) \end{matrix}$ for any continuous point z of Φ.

2.2 Uncertain Random Variable

The chance space refers to the product (Γ, ℒ, M) × (Ω, A, Pr), in which (Γ, ℒ, M) is an uncertainty space and (Ω, A, Pr) is a probability space, respectively.

Definition 10. (Liu [18]) Let (Γ, ℒ, M) × (Ω, A, Pr) be a chance space, and Θ ∈ ℒ × A be an uncertain random event. Then the chance measure of Θ is defined as $\begin{matrix} Ch {Θ} \\ = \int_{0}^{1} \Pr {ω \in Ω ∣ ℒ M {γ \in Γ | (γ, ω) \in Θ} \geq x} \\ d x . \end{matrix}$

It is mentioned that a chance measure satisfies normality, duality, and monotonicity properties, that is (i) Ch {Γ × Ω} =1; (ii) Ch {Θ} + Ch {Θ^c} =1 for any event Θ; (iii) Ch {Θ₁} ≤ Ch {Θ₂} for any real number set Θ₁ ⊂ Θ₂, for more details, see Liu [18]. Besides, the subadditivity of chance measure is proved by Hou [11], that is, $Ch {⋃_{i = 1}^{\infty} Θ_{i}} \leq \sum_{i = 1}^{\infty} Ch {Θ_{i}}$ for a sequence of events Θ₁, Θ₂, ⋯.

Definition 11. (Liu [18]) An uncertain random variable is a measurable function ξ from a chance space (Γ, ℒ, M) × (Ω, A, Pr) to the set of real numbers, i. e., {ξ ∈ B} is an event for any Borel set B.

Theorem 3. (Liu [18]) Let ξ₁, ξ₂, …, ξ_n be uncertain random variables on the chance space (Γ, ℒ, M) × (Ω, A, Pr) and let f be a measurable function. then $ξ = f (ξ_{1}, ξ_{2}, . . ., ξ_{n})$ is an uncertain random variable determined by $ξ (γ, ω) = f (ξ_{1} (γ, ω), ξ_{2} (γ, ω), . . ., ξ_{n} (γ, ω))$ for all (γ, ω) ∈ Γ × Ω.

To describe an uncertain random variable, Liu [19] presented a definition of chance distribution.

Definition 12. (Liu [19]) Let ξ be an uncertain random variable. Then its chance distribution is defined by $Φ (x) = Ch {ξ \leq x}$ for any x ∈ eulerR.

The chance distribution of a random variable is just its probability distribution, and the chance distribution of an uncertain variable is just its uncertainty distribution.

Theorem 4. (Liu [19]) Let η₁, η₂, ⋯, η_m be independent random variables with probability distributions Ψ₁, Ψ₂, ⋯, Ψ_m, respectively, and let τ₁, τ₂, ⋯, τ_n be uncertain variables. Then the uncertain random variable ξ = f (η₁, η₂, ⋯, η_m, τ₁, τ₂, ⋯, τ_n) has a chance distribution $Φ (x) = \int_{ℒ R e^{m}} F (x, y_{1}, \dots, y_{m}) d Ψ_{1} (y_{1}) \dots d Ψ_{m} (y_{m})$ where F (x, y₁, ⋯, y_m) is the uncertainty distribution of uncertain variable f (η₁, η₂, ⋯, η_m, τ₁, τ₂, ⋯, τ_n) for any real numbers y₁, y₂, ⋯, y_m.

Definition 13. (Liu [19]) Let ξ be an uncertain random variable. Then its expected value is defined by $E [ξ] = \int_{0}^{+ \infty} Ch {ξ \geq x} d x - \int_{- \infty}^{0} Ch {ξ \leq x} d x$ provided that at least one of the two integrals is finite.

Let Φ denote the chance distribution of ξ. Liu [19] gave a formula to calculate the expected value of uncertain random variable with chance distribution, that is, $E [ξ] = \int_{0}^{+ \infty} (1 - Φ (x)) d x - \int_{- \infty}^{0} Φ (x) d x .$

Theorem 5. (Liu [18]) Let η₁, η₂, ⋯, η_m be independent random variables with probability distributions Ψ₁, Ψ₂, ⋯, Ψ_m, respectively, and τ₁, τ₂, ⋯, τ_n be independent uncertain variables (not necessarily independent). Then the uncertain random variable ξ = f (η₁, ⋯, η_m, τ₁, ⋯, τ_n) has an expected value $\begin{matrix} E [ξ] & = \int_{ℝ^{m}} E [f (y_{1}, \dots, y_{m}, τ_{1}, \dots, τ_{n})] \\ d Ψ_{1} (y_{1}) \dots d Ψ_{m} (y_{m}) \end{matrix}$ where E [f (y₁, ⋯, y_m, τ₁, ⋯, τ_n)] is the expected value of the uncertain variable f (y₁, ⋯, y_m, τ₁, ⋯, τ_n) for any real numbers y₁, ⋯, y_m.

In order to study the convergence theorems of uncertain random sequences, Ahmadzade et al. [2] introduced the following concepts of convergence:

Definition 14. (Ahmadzade et al. [2]) An uncertain random sequence {ξ_n, n ≥ 1} is said to be almost surely convergent (a. s.) to an uncertain random variable ξ if there exists an event Θ with Ch {Θ} =1 such that $lim_{n \to \infty} | ξ_{n} (γ, ω) - ξ (γ, ω) | = 0$ for every (γ, ω) ∈ Θ.

Definition 15. (Ahmadzade et al. [2]) An uncertain random sequence {ξ_n, n ≥ 1} is said to be convergent in measure to an uncertain random variable ξ if $lim_{n \to \infty} Ch {(γ, ω) \in Θ | | ξ_{n} (γ, ω) - ξ (γ, ω) | \geq ∊} = 0$ for any ∊ > 0.

Definition 16. (Gao et al. [8]) A complex uncertain random variable is a function ξ from a chance space (Γ × Ω, ℒ × A, M × Pr) to the set of complex numbers such that {ξ ∈ B} = {(γ, ω) ∈ Γ × Ω|ξ (γ, ω) ∈ B}is an event in Γ × Ω for any Borel Set B of complex numbers.

Definition 17. (Gao et al. [8]) Let ξ be a complex uncertain random variable. Then the complex chance distribution of ξ is defined by $F (z) = Ch {Re (ξ) \leq x, Im (ξ) \leq y},$ for any complex number z = x + iy.

Now, in order to study properties of complex uncertain random sequence, we introduce the following convergence concepts:

Definition 18. The complex uncertain random sequence {ξ_n, n ≥ 1} is said to be convergent in measure to ξ if $\begin{matrix} lim_{n \to \infty} Ch {(γ, ω) \in Θ | | | ξ_{n} (γ, ω) - ξ (γ, ω) | | > ∊} = 0 \end{matrix}$ for any ∊ > 0.

Definition 19. The complex uncertain random sequence {ξ_n, n ≥ 1} is said to be convergent in mean to ξ if $\begin{matrix} lim_{n \to \infty} E | | ξ_{n} - ξ | | = 0 . \end{matrix}$

It is mentioned that for ξ = τ_n + iη_n and ξ = τ + iη, ||ξ_n - ξ|| is computed as follows: $\begin{matrix} | | ξ_{n} - ξ | | = \sqrt{| τ_{n} - τ |^{2} + | η_{n} - η |^{2}}, \end{matrix}$ where, τ, τ₁, ⋯ are uncertain variables and η, η₁, ⋯ are random variables. Furthermore, it is obvious that {||ξ_n - ξ||, n ≥ 1} is a sequence of uncertain random variables.

Definition 20. Let Φ, Φ₁, Φ₂, ⋯ be the complex chance distributions of complex uncertain random variables ξ, ξ₁, ξ₂, ⋯, respectively.We say the complex uncertain random sequence {ξ_n, n ≥ 1} converges in distribution to ξ if $\begin{matrix} lim_{n \to \infty} Φ_{n} (z) = Φ (z) \end{matrix}$ for all continuous point z of Φ.

3 Convergence Theorems of Complex Uncertain Random Sequences

In this section, we establish several convergence theorems for complex uncertain random sequences. Furthermore, for illustration of convergence concepts, we derive several examples and counterexamples.

Theorem 6. If the complex uncertain random sequence {ξ_n, n ≥ 1} converges in mean to ξ, then {ξ_n, n ≥ 1} converges in measure to ξ.

Proof. By using Markov’s inequality, we have $\begin{matrix} Ch {| | ξ_{n} - ξ | | > ∊} \leq \frac{E | | ξ_{n} - ξ | |}{∊} \to 0, as n \to \infty, \end{matrix}$ i.e. the sequence {ξ_n, n ≥ 1} converges in measure.

Theorem 7. Suppose that {τ_n, n ≥ 1} is a sequence of uncertain variables and {η_n, n ≥ 1} is a sequence of random variables. If {τ_n, n ≥ 1} converges in measure to τ and {η_n, n ≥ 1} converges in probability η, then the sequences {ξ_n = τ_n + iη_n, n ≥ 1} and ${ξ_{n}^{'} = η_{n} + i τ_{n}, n \geq 1}$ converge in measure to ξ = τ + iη and ξ′ = η + iτ, respectively.

Proof. By using definition of convergence in measure, we should prove $\begin{matrix} ℒ M {| | ξ_{n} - ξ | | \geq ∊} \\ = ℒ M {\sqrt{| τ_{n} - τ |^{2} + | η_{n} - η |^{2}} \geq ∊} \to 0, \\ as n \to \infty . \end{matrix}$ By using definition of ||. || and triangular inequality, we have $\begin{matrix} {| | ξ_{n} - ξ | | > ∊} & = {| | ξ_{n} - ξ | |^{2} > ∊^{2}} \\ = {| τ_{n} - τ |^{2} + | η_{n} - η |^{2} > ∊^{2}} \\ \subseteq {| τ_{n} - τ |^{2} > \frac{∊^{2}}{2}} \\ ⋃ {| η_{n} - η |^{2} > \frac{∊^{2}}{2}} . \end{matrix}$

On the other hand, subadditivity axiom of chance measure implies that

$\begin{matrix} M {| | ξ_{n} - ξ | | \geq ∊} & \leq ℒ M {| τ_{n} - τ |^{2} \geq \frac{∊^{2}}{2}} \\ + ℒ M {| η_{n} - η |^{2} \geq \frac{∊^{2}}{2}} . \end{matrix}$ (1) Since, the sequence {τ_n, n ≥ 1} converges in measure to τ and {η_n, n ≥ 1} converges in probability to η, relation (1) implies that $\begin{matrix} M {| | ξ_{n} - ξ | | \geq ∊} \to 0, as n \to \infty . \end{matrix}$ By using the similar method, we can prove that ${ξ_{n}^{'}, n \geq 1}$ converges in measure to ξ′.

The following example satisfies the conditions of Theorem 6 and 7.

Example 1. Let {η_n, n ≥ 1} be a sequence of random variables defined by $\Pr (η_{n} = 0) = 1 - \frac{1}{n^{α}}, \Pr (η_{n} = n) = \frac{1}{n^{α}}, α > 1 .$ Consider an uncertainty space (Γ, ℒ, M) to be {γ₁, γ₂, ⋯} with $M {Λ} = {\begin{matrix} sup_{γ_{n} \in Λ} \frac{1}{n}, & if sup_{γ_{n} \in Λ} \frac{1}{n} < 0.5, \\ 1 - sup_{γ_{n} \notin Λ} \frac{1}{n}, & if sup_{γ_{n} \notin Λ} \frac{1}{n} < 0.5, \\ 0.5, & otherwise . \end{matrix}$ And the uncertain variables are defined by $τ_{n} (γ_{m}) = {\begin{matrix} \frac{1}{n}, & if n = m, \\ 0, & otherwise, \end{matrix}$ for i = 1, 2, ⋯ and consider ξ_n = η_n + iτ_n. For some small number ∊ > 0 and ξ ≡ 0, we have $\begin{matrix} Ch {| | ξ_{n} - ξ | | > ∊} \\ = E_{\Pr} [ℒ M {η_{n}^{2} + τ_{n}^{2} > ∊}] \\ = M {τ_{n} = 1 / n} [\Pr {η_{n} = n} \\ + \Pr {η_{n} = 0}] + ℒ M {τ_{n} = 0} \Pr {η_{n} = n} \\ = \frac{1}{n} + (1 - \frac{1}{n}) \frac{1}{n^{α}} \to 0, \end{matrix}$ as n→ ∞. i.e. {ξ_n, i ≥ 1} converges in measure to ξ. Also, it is easy to that {η_n, n ≥ 1} converges in probability to η and {τ_n, n ≥ 1} converges in measure to τ. Since, $E_{Un} [\sqrt{τ_{n}^{2} + η_{n}^{2}}]$ is a random variable, we consider $η_{n}^{*} = E_{Un} [\sqrt{τ_{n}^{2} + η_{n}^{2}}],$ and we have $\begin{matrix} E [| | ξ_{n} - ξ | |] & = E [\sqrt{τ_{n}^{2} + η_{n}^{2}}] \\ = E_{\Pr} E_{Un} [\sqrt{τ_{n}^{2} + η_{n}^{2}}] \\ = E_{\Pr} [η_{n}^{*}], \end{matrix}$ where $η_{n}^{*} = {\begin{matrix} \frac{1}{n^{2}}, & if η_{n} = 0, \\ \frac{\sqrt{n^{4} + 1}}{n^{2}} + n (1 - \frac{1}{n}), & if η_{n} = n, \\ 0, & otherwise . \end{matrix}$ Therefore, $\begin{matrix} E [\sqrt{τ_{n}^{2} + η_{n}^{2}}] & = \frac{1}{n^{2}} (1 - \frac{1}{n^{α}}) \\ + \frac{\sqrt{n^{4} + 1}}{n^{2}} \times \frac{1}{n^{α}} \\ + \frac{1}{n^{α - 1}} (1 - \frac{1}{n}) \to 0 . \end{matrix}$ That is, the sequence {ξ_n, n ≥ 1} converges in mean to ξ.

Following example is a counterexample showing that the converse of the Theorem 6 is not true in general.

Example 2. Suppose that (Ω, A, Pr) is a probability space on the interval [0, 1] with Borel algebra and Lebesgue measure. The random variables are considered as follows: $η_{n} (ω) = {\begin{matrix} 0, & if \frac{1}{n} < ω < 1, \\ 2^{n}, & if 0 < ω < \frac{1}{n} . \end{matrix}$ Take an uncertainty space (Γ, ℒ, M) to be {γ₁, γ₂, ⋯} with $M {Λ} = {\begin{matrix} sup_{γ_{n} \in Λ} \frac{1}{n + 1}, & if sup_{γ_{n} \in Λ} \frac{1}{n + 1} < 0.5, \\ 1 - sup_{γ_{n} \notin Λ} \frac{1}{n + 1}, & if sup_{γ_{n} \notin Λ} \frac{1}{n + 1} < 0.5, \\ 0.5, & otherwise . \end{matrix}$ The uncertain variables are defined by $τ_{n} (γ_{m}) = {\begin{matrix} n, & if n = m, \\ 0, & otherwise . \end{matrix}$ Suppose that ξ_n = τ_n + iη_n and ξ ≡ 0. Then we have $\begin{matrix} Ch {| | ξ_{i} - ξ | | > ∊} \\ = E_{\Pr} [ℒ M {| | ξ_{i} - ξ | | > ∊}] \\ = M {τ_{n} = n} [P {η_{n} = 0} + P {η_{n} = 2^{n}}] \\ + M {τ_{n} = 0} P {τ_{n} = 2^{n}} \\ = \frac{1}{n + 1} \times 1 + (1 - \frac{1}{n + 1}) \times \frac{1}{n} \to 0, \end{matrix}$ as n→ + ∞. This means the sequence {ξ_n, n ≥ 1} converges in measure to ξ. By considering $η_{n}^{*} = E_{Un} [\sqrt{τ_{n}^{2} + η_{n}^{2}}],$ we have $\begin{matrix} E [| | ξ_{n} - ξ | |] & = E [\sqrt{τ_{n}^{2} + η_{n}^{2}}] \\ = E_{\Pr} E_{Un} [\sqrt{τ_{n}^{2} + η_{n}^{2}}] \\ = E [η_{n}^{*}], \end{matrix}$ $η_{n}^{*} = {\begin{matrix} \frac{n}{n + 1}, & if η_{n} = 0, \\ 2^{n} (1 - \frac{1}{n + 1}) + \frac{\sqrt{4^{n} + n^{2}}}{n + 1}, & if η_{n} = 2^{n}, \\ 0, & otherwise . \end{matrix}$ Therefore, $\begin{matrix} E [\sqrt{τ_{n}^{2} + η_{n}^{2}}] & = \frac{n}{n + 1} (1 - \frac{1}{n}) \\ + \frac{2^{n}}{n} (1 - \frac{1}{n + 1}) \\ + \frac{\sqrt{4^{n} + n^{2}}}{n (n + 1)} \to \infty, \end{matrix}$ as n→ ∞, i.e. the sequence {ξ_n, n ≥ 1} does not converge in mean to ξ.

Theorem 8. Suppose that {τ_n, n ≥ 1} is a sequence of uncertain variables and {η_n, n ≥ 1} is a sequence of random variables. If {τ_n, n ≥ 1} converges in measure to τ and {η_n, n ≥ 1} converges in probability η, then the sequences {ξ_n = τ_n + iη_n, n ≥ 1} and ${ξ_{n}^{'} = η_{n} + i τ_{n}, n \geq 1}$ converge in distribution to ξ = τ + iη and ξ′ = η + iτ, respectively.

Proof. For a continuity point of the complex chance distribution Φ such as z = x + iy, we obtain $\begin{matrix} {τ_{n} \leq x, η_{n} \leq y} \\ = {τ_{n} \leq x, η_{n} \leq y, τ > u, η > v} \\ \cup {τ_{n} \leq x, η_{n} \leq y, τ \leq u, η > v} \\ \cup {τ_{n} \leq x, η_{n} \leq y, τ > u, η \leq v} \\ \cup {τ_{n} \leq x, η_{n} \leq y, τ \leq u, η > v} \\ \subseteq {τ \leq u, η \leq v} \cup {| τ_{n} - τ | > u - x} \\ \cup {| η_{n} - η | > v - y} \end{matrix}$ for any u > x, v > y. The subadditivity axiom of chance measure implies that $\begin{matrix} Φ_{n} (z) & = Ch {τ_{n} \leq x, η_{n} \leq y} \\ \leq Ch {τ \leq u, η \leq v} \\ + Ch {| τ_{n} - τ | > u - x} \\ + Ch {| η_{n} - η | > v - y} \\ = Φ (u + iv) + Ch {| τ_{n} - τ | > u - x} \\ + Ch {| η_{n} - η | > v - y} \end{matrix}$ Since the sequence {τ_n, n ≥ 1} converges in measure to τ and {τ_n, n ≥ 1} converges in probability to η, respectively, we obtain $\begin{matrix} \underset{n \to \infty}{lim sup} Φ_{n} (z) \leq Φ (u + iy) \end{matrix}$ for any u > x, v > y. Since z is a continuity point of Φ, we have

$\underset{n \to \infty}{lim sup} Φ_{n} (z) \leq Φ (z) .$ (2) by letting u + iv → x + iy. Now, by considering u < x and m < y, we have $\begin{matrix} {τ \leq w, η \leq m} \\ = {τ_{n} \leq x, η_{n} \leq y, τ \leq w, η \leq m} \\ \cup {τ_{n} > x, η_{n} \leq y, τ \leq w, η \leq m} \\ \cup {τ_{n} \leq x, η_{n} > y, τ \leq w, η \leq m} \\ \cup {τ_{n} > x, η_{n} > y, τ \leq w, η \leq m} \\ \subseteq {τ_{n} \leq x, η_{n} \leq y} \\ \cup {| τ_{n} - τ | > x - w} \\ \cup {| η_{n} - η | > y - m} . \end{matrix}$ Then the subadditivity axiom of chance measure implies that $\begin{matrix} Φ (w + im) & = Ch {τ \leq w, η \leq m} \\ \leq Ch {τ_{n} \leq x, η_{n} \leq y} \\ + Ch {| τ_{n} - τ | > x - w} \\ + Ch {| η_{n} - η | > y - m} \\ = Φ (x + iy) \\ + Ch {| τ_{n} - τ | > x - w} \\ + Ch {| η_{n} - η | > y - m} . \end{matrix}$ Since the sequence {τ_n, n ≥ 1} converges in measure to τ and {η_n, n ≥ 1} converges in probability to η, respectively, we obtain $\begin{matrix} Φ (w + im) \leq \underset{n \to \infty}{lim inf} Φ_{n} (x + iy) = \underset{n \to \infty}{lim inf} Φ_{n} (z) . \end{matrix}$ Since z is a continuous point of Φ, we have

$Φ (z) \leq \underset{n \to \infty}{lim inf} Φ_{n} (z)$ (3) by letting w + im → x + iy. The relations (1) and (2) imply that $\begin{matrix} lim_{n \to \infty} Φ_{n} (z) \to Φ (z), \end{matrix}$ i.e. the sequence {ξ_n, n ≥ 1} converges in distribution to ξ. The similar proof can be written for the sequence ${ξ_{n}^{'}, n \geq 1}$ . Thus the proof is finished.

The following example is an evidence of Theorem 8.

Example 3. Let {η_n, n ≥ 1} be a sequence of random variables defined by $\Pr (η_{n} = 0) = 1 - \frac{1}{n}, \Pr (η_{n} = 1) = \frac{1}{n} .$ Take an uncertainty space (Γ, ℒ, M) to be {γ₁, γ₂, ⋯} with $M {Λ} = {\begin{matrix} sup_{γ_{n} \in Λ} \frac{1}{n}, & if sup_{γ_{n} \in Λ} \frac{1}{n} < 0.5, \\ 1 - sup_{γ_{n} \notin Λ} \frac{1}{n}, & if sup_{γ_{n} \notin Λ} \frac{1}{n} < 0.5, \\ 0.5, & otherwise . \end{matrix}$ The uncertain variables are defined by $τ_{n} (γ_{m}) = {\begin{matrix} 1, & if n = m, \\ 0, & otherwise . \end{matrix}$ Considering ξ_n = τ_n + iη_n, we have $Φ_{n} (z) = {\begin{matrix} 0, & if x < 0, y < \infty, \\ 0, & if x \geq 0, y < 0, \\ {(1 - \frac{1}{n})}^{2}, & if 0 \leq x < 1, 0 \leq y < 1, \\ 1 - \frac{1}{n}, & if 0 \leq x < 1, y \geq 1, \\ 1 - \frac{1}{n}, & if x \geq 1, 0 \leq y < 1, \\ 1, & if x \geq 1, y \geq 1 . \end{matrix}$ On the other hand, the complex uncertainty distribution of ξ is $Φ (z) = {\begin{matrix} 0, & if x < 0, y < \infty, \\ 0, & if x \geq 0, y < 0, \\ 1, & if x \geq 0, y \geq 0 . \end{matrix}$ Therefore, the sequence {ξ_n, n ≥ 1} converges to ξ in distribution. For some small number ∊ > 0 and ξ ≡ 0, we have $\begin{matrix} Ch {| | ξ_{n} - ξ | | > ∊} \\ = E_{\Pr} [M {η_{n}^{2} + τ_{n}^{2} > ∊}] \\ = M {τ_{n} = 1} [P {η_{n} = 1} + \Pr {η_{n} = 0}] \\ + M {τ_{n} = 0} \Pr {η_{n} = 1} \\ = \frac{1}{n} + (1 - \frac{1}{n}) \frac{1}{n} \to 0, as n \to \infty . \end{matrix}$ i.e. {ξ_n, n ≥ 1} converges in measure to ξ. Following example is a counterexample showing that the converse of the Theorem 8 is not true in general.

Example 4. Consider a symmetric coin with η = 1 for heads and η = 0 for tails, and let η_n = 1 - η, n ≥ 1. Take an uncertainty space (Γ, ℒ, M) to be {γ₁, γ₂} with $M {γ_{1}} = \frac{1}{2}$ and $M {γ_{2}} = \frac{1}{2}$ . We define an uncertain variable as $τ (γ) = {\begin{matrix} - 1, & if γ = γ_{1}, \\ 1, & if γ = γ_{2} . \end{matrix}$ We also define τ_n = - τ, ξ_n = τ_n + iη_n and ξ = τ + iη. Then ξ_n and ξ have the same chance distribution. Thus {ξ_n, n ≥ 1} converges in distribution to ξ. However, for some small number ∊ > 0, we have $\begin{matrix} Ch {| | ξ_{n} - ξ | | > ∊} \\ = Ch {| η_{n} - η |^{2} + | τ_{n} - τ |^{2} > ∊^{2}} \\ = E_{\Pr} [ℒ M {| η_{n} - η |^{2} + | τ_{n} - τ |^{2} > ∊^{2}}] \\ = [\Pr {η = 1} + \Pr {η = 0}] ℒ M {τ_{n} = 1} \\ + [\Pr {η = 1} + \Pr {η = 0}] ℒ M {τ_{n} = - 1} \\ = 1 \times \frac{1}{2} + 1 \times \frac{1}{2} = 1 . \end{matrix}$ That is, the sequence {ξ_n, n ≥ 1} does not converge in measure to ξ.

4 Conclusions

Several types of convergence concepts of complex uncertain random sequences were introduced in this paper. For illustrating relations among convergence concepts, several examples, counterexamples and theorems were stated and proved. It is mentioned that, if an complex uncertain random sequence reduced to the case of complex uncertain variable, all results are right. As different types of indeterminacy, soft sets, fuzzy sets and their extensions can be applied to complex numbers. Thus, studying the convergence properties of complex soft sets maybe a potential work for future research.

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