Convergence of complex uncertain sequences

Abstract

Complex uncertain variables are measurable functions from an uncertainty space to the set of complex numbers and are used to model complex uncertain quantities. This paper introduces the convergence concepts of complex uncertain sequences: convergence almost surely (a.s.), convergence in measure, convergence in mean, convergence in distribution and convergence uniformly almost surely. In addition, relationships among them are discussed.

Keywords

Uncertainty theory complex uncertain variable convergence

1 Introduction

Sequence convergence, as a fundamental theory of mathematics, plays a very important role in probability theory. There are many convergence concepts of random sequences such as convergence in probability, convergence in mean, convergence almost surely, convergence in distribution and so on. Based on these concepts of convergence, “limits theorem” for random sequences were derived, which had been an important issue in probability theory. A notion called “statistical convergence” in sequences of real or complex terms has similarity with convergence in the notion of probability theory. In this case the distribution of the terms of the sequences is taken into account. The density of subscripts relative to the set of natural numbers has been considered. It is found in the works published by Rath and Tripathy [17], Tripathy and Sen [18], Tripathy et al. [19].

However, in our daily life, we often encounter the case that there are lack of or no observed data about the events, not only for economic reasons or technical difficulties, but also for influence of unexpected events. Due to not enough data to get probability distribution of events, we have to consult with some domain experts to give belief degree that each event would happen while making decisions. Kahneman and Tversky [4] showed that human beings usually overweight unlikely events. From another side, Liu [14] showed that human beings usually estimate a much wider range of values than the object actually takes. Thus the belief degree may have much larger variance than the real frequency. In this case, if we insist on dealing with the belief degree by using probability theory, counterintuitive results may occur. Interested readers may refer to Liu [12] for an example.

In order to deal with belief degree, an uncertainty theory was founded by Liu [5] in 2007, and refined by Liu [10] in 2010 which based on an uncertain measure which satisfies normality, duality, subadditivity, and product axioms. Thereafter, a concept of uncertain variable was proposed to represent the uncertain quantity and a concept of uncertainty distribution to describe uncertain variables. Up to now, uncertainty theory has successfully been applied to uncertain programming (Liu [7], Liu and Chen [15]), uncertain risk analysis and uncertain reliability analysis (Liu [9]), uncertain logic (Liu [11]), uncertain differential equation (Liu [6], Yao and Chen [20]), uncertain graphs (Gao and Gao [2], Zhang and Peng [22]), uncertain calculus (Liu [8]) and uncertain finance (Chen [1], Liu [8], Liu [13]), etc.

In real life, uncertainty not only appears in real quantities but also in complex quantities. In order to model complex uncertain quantities, Peng [16] presented the concepts of complex uncertain variable and complex uncertainty distribution, and also the expected value was proposed to measure a complex uncertain variable in 2012.

Since sequence convergence plays an important role in the fundamental theory of mathematics, there are also many convergence concepts in uncertainty theory. In 2007, Liu [5] first introduced convergence in measure, convergence in mean, convergence almost surely (a.s.) and convergence in distribution and their relationships were also discussed. You [21] introduced another type of convergence named convergence uniformly almost surely and showed the relationships among those convergence concepts. Zhang [23] proved some theorems on the convergence of uncertain sequence. After that, Guo and Xu [3] gave the concept of convergence in mean square for uncertain sequence and showed that an uncertain sequence converged in mean square if and only if it was a Cauchy sequence. Inspired by these, we study the convergence concepts of complex uncertain sequence and discuss the relationships among them in this paper.

2 Preliminaries

In this section, some foundational concepts and theorems in uncertainty theory are introduced, which are used throughout this paper.

Definition 2.1. Liu [5] Let ℒ be a σ-algebra on a nonempty set Γ. A set function ℳ is called an uncertain measure if it satisfies the following axioms: Axiom 1. (Normality Axiom) ℳ {Γ} =1; Axiom 2. (Duality Axiom) ℳ {Λ} + ℳ {Λ^c} =1 for any Λ ∈ ℒ; Axiom 3. (Subadditivity Axiom) For every countable sequence of {Λ_j} ∈ ℒ, we have $ℳ {⋃_{j = 1}^{\infty} Λ_{j}} \leq \sum_{j = 1}^{\infty} ℳ {Λ_{j}} .$

The triplet (Γ, ℒ, ℳ) is called an uncertainty space, and each element Λ in ℒ is called an event. In order to obtain an uncertain measure of compound event, a product uncertain measure is defined by Liu [8] as follows:

Axiom 4. (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 $ℳ {\prod_{k = 1}^{\infty} Λ_{k}} = ⋀_{k = 1}^{\infty} ℳ_{k} {Λ_{k}}$ where Λ_k are arbitrarily chosen events from ℒ_k for k = 1, 2, ⋯, respectively.

Definition 2.2. Liu [5] An uncertain variableξ is a measurable function from an uncertainty space (Γ, ℒ, ℳ) to the set of real numbers, i.e., for any Borel set B of real numbers, the set ${ξ \in B} = {γ \in Γ | ξ (γ) \in B}$ is an event.

Definition 2.3. Liu [5] The uncertainty distributionΦ of an uncertain variable ξ is defined by $Φ (x) = ℳ {ξ \leq x}, \forall x \in R .$

An uncertain variable ξ is said to be normal if it has a normal uncertainty distribution $Φ (x) = {(1 + \exp (\frac{π (e - x)}{\sqrt{3} σ}))}^{- 1}, x \in R$ denoted by ξ ∼ 𝒩 (e, σ) where e and σ are real numbers with σ > 0.

Definition 2.4. Liu [8] The uncertain variables ξ₁, ξ₂, ⋯ , ξ_n are said to be independent if $ℳ {⋂_{j = 1}^{n} (ξ_{j} \in B_{j})} = ⋀_{j = 1}^{n} ℳ {ξ_{j} \in B_{j}}$ for any Borel sets B₁, B₂, ⋯ , B_n of real numbers.

Definition 2.5. Liu [5] Let ξ be an uncertain variable. The expected value of ξ is defined by $E [ξ] = \int_{0}^{+ \infty} ℳ {ξ \geq r} d r - \int_{- \infty}^{0} ℳ {ξ \leq r} d r$ provided that at least one of the above two integrals is finite.

Considering the importance role of sequence convergence in mathematics, some concepts of convergence for uncertain sequences were introduced in [5,21, 5,21] as follows:

Definition 2.6. Liu [5] The uncertain sequence {ξ_n} is said to be convergent almost surely (a.s.) to ξ if there exists an event Λ with ℳ {Λ} =1 such that $lim_{n \to \infty} | ξ_{n} (γ) - ξ (γ) | = 0$ for every γ ∈ Λ. In that case we write ξ_n → ξ, a . s .

Definition 2.7. Liu [5] The uncertain sequence {ξ_n} is said to be convergent in measure toξif $lim_{n \to \infty} ℳ {| ξ_{n} - ξ | \geq ɛ} = 0$ for every ɛ > 0.

Definition 2.8. Liu [5] The uncertain sequence {ξ_n} is said to be convergent in mean toξ if $lim_{n \to \infty} E [| ξ_{n} - ξ |] = 0 .$

Definition 2.9. Liu [5] Let Φ, Φ₁, Φ₂, ⋯ be the uncertainty distributions of uncertain variables ξ, ξ₁, ξ₂, ⋯, respectively. We say the uncertain sequence {ξ_n} converges in distribution to ξif $lim_{n \to \infty} Φ_{n} (x) = Φ (x)$ for all x at which Φ (x) is continuous.

Definition 2.10. You [21] The sequence {ξ_n} is said to be convergent uniformly almost surely toξ if there exists {E_k}, ℳ {E_k} →0 such that {ξ_n} converges uniformly to ξ in Γ - E_k, for any fixed k ∈ N.

3 Complex uncertain variable

In this section, we introduce some concepts and theorems of complex uncertain variable which were first proposed by Peng [16] in 2012.

As a complex function on uncertainty space, complex uncertain variable is mainly used to model a complex uncertain quantity.

Definition 3.1. Peng [16] A complex uncertain variable is a measurable function ζ 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 ${ζ \in B} = {γ \in Γ | ζ (γ) \in B}$ is an event.

Theorem 3.1. Peng [16] A variable ζ from an uncertainty space (Γ, ℒ, ℳ) to the set of complex numbers is a complex uncertain variable if and only if Reζ and Imζ are uncertain variables where Reζ and Imζ represent the real and the imaginary part of ζ, respectively.

Definition 3.2. Peng [16] Let ξ and η be independent normal uncertain variables. Then ζ = ξ + iη is called complex normal uncertain variable.

Definition 3.3. Peng [16] The complex uncertainty distributionΦ (x) of a complex uncertain variable ζ is a function from $ℂ$ to [0, 1] defined by $Φ (c) = ℳ {Re (ζ) \leq Re (c), Im (ζ) \leq Im (c)}$ for any complex c.

Theorem 3.2.Peng [16] A function Φ (c) → [0, 1] is a complex uncertainty distribution if and only if it is increasing with respect to the real part Re (c) and the imaginary part Im (c) such that $\begin{matrix} \begin{matrix} i) & lim_{x \to - \infty} Φ (x + ib) \neq 1, \\ lim_{y \to - \infty} Φ (a + iy) \neq 1, \end{matrix} \end{matrix}$ for any $a, b \in R$ ; $\begin{matrix} ii) & lim_{x \to + \infty, y \to + \infty} Φ (x + iy) \neq 0 \end{matrix}$ where $i = \sqrt{- 1}$ is the imaginary unit.

4 Convergence concepts of complex uncertain sequence

Complex uncertain sequence is a sequence of complex uncertain variables indexed by integers. In this section, we introduce five convergence concepts of complex uncertain sequence: convergence almost surely (a.s.), convergence in measure, convergence in mean, convergence in distribution and convergence uniformly almost surely.

Definition 4.1. The complex uncertain sequence {ζ_n} is said to be convergent almost surely (a.s.) toζ if there exists an event Λ with ℳ {Λ} =1 such that $lim_{n \to \infty} ∥ ζ_{n} (γ) - ζ (γ) ∥ = 0$ for every γ ∈ Λ.

Definition 4.2. The complex uncertain sequence {ζ_n} is said to be convergent in measure toζ if $lim_{n \to \infty} ℳ {∥ ζ_{n} - ζ ∥ \geq ɛ} = 0$ for every ɛ > 0.

Definition 4.3. Let ζ, ζ₁, ζ₂, ⋯ be complex uncertain variables with finite expected values. Then the complex uncertain sequence {ζ_n} is said to be convergent in mean toζ if $lim_{n \to \infty} E [∥ ζ_{n} - ζ ∥] = 0 .$ Definition 4.4. Let Φ, Φ₁, Φ₂, ⋯ be the complex uncertainty distributions of complex uncertain variables ζ, ζ₁, ζ₂, ⋯, respectively. We say the complex uncertain sequence {ζ_n} converges in distribution toζ if $lim_{n \to \infty} Φ_{n} (c) = Φ (c)$ for all c at which Φ (c) is continuous.

Definition 4.5. The sequence {ζ_n} is said to be convergent uniformly almost surely toζ if there exists ${E_{k}^{'}}$ , $ℳ {E_{k}^{'}} \to 0$ such that {ζ_n} converges uniformly to ζ in $Γ - E_{k}^{'}$ , for any fixed k ∈ N.

5 Relationships among convergence concepts

In this section, relationships among the five convergence concepts of complex uncertain sequences are studied.

5.1 Convergence in mean and convergence in measure

Theorem 5.1. If the complex uncertain sequence {ζ_n} converges in mean to ζ, then {ζ_n} converges in measure to ζ.

Proof: It follows from the Markov inequality that for any given number ɛ > 0, we have $ℳ {∥ ζ_{n} - ζ ∥ \geq ɛ} \leq \frac{E [∥ ζ_{n} - ζ ∥]}{ɛ} \to 0 .$ as n→ ∞. Thus {ζ_n} converges in measure to ζ and the theorem is thus proved. □

Remark 5.1. Assuming ζ_n = ξ_n + iη_n for n = 1, 2, ⋯ and ζ = ξ + iη, we have $∥ ζ_{n} - ζ ∥ = \sqrt{(ξ_{n} - ξ)^{2} + (η_{n} - η)^{2}}$ is just an uncertain variable.

Example 5.1. Take an uncertainty space (Γ, ℒ, ℳ) to be {γ₁, γ₂} with $ℳ {γ_{1}} = ℳ {γ_{2}} = \frac{1}{2}$ . We define complex uncertain variables by $ζ_{n} (γ) = {\begin{matrix} i, & if γ = γ_{1} \\ 1 + \frac{1}{n}, & if γ = γ_{2} \end{matrix}$ for n = 1, 2, ⋯. We also define ζ by $ζ (γ) = {\begin{matrix} i, & if γ = γ_{1} \\ 1, & if γ = γ_{2} . \end{matrix}$

Then for any given number ɛ > 0 we have $E [∥ ζ_{n} - ζ ∥] \to 0$ as n→ ∞ which implies {ζ_n} converges in mean to ζ. It follows from Theorem 5.1 that {ζ_n} converges in measure. In fact, we can easily obtain $ℳ {∥ ζ_{n} - ζ ∥ \geq ɛ} \to 0$ as n→ ∞ which implies {ζ_n} converges in measure to ζ .

Example 5.2. Convergence in measure does not imply convergence in mean. Take an uncertainty space (Γ, ℒ, ℳ) to be {γ₁, γ₂, ⋯} with $ℳ {Λ} = {\begin{matrix} \sup_{γ_{n} \in Λ} \frac{1}{n + 1}, & if \sup_{γ_{n} \in Λ} \frac{1}{n + 1} < \frac{1}{2} \\ 1 - \sup_{γ_{n} \in Λ^{c}} \frac{1}{n + 1}, & if \sup_{γ_{n} \in Λ^{c}} \frac{1}{n + 1} < \frac{1}{2} \\ \frac{1}{2}, & otherwise . \end{matrix}$

And the complex uncertain variables are defined by $ζ_{n} (γ) = {\begin{matrix} (n + 1) i, & if γ = γ_{n} \\ 0, & otherwise \end{matrix}$

for n = 1, 2, ⋯ and ζ ≡ 0. For some small number ɛ > 0 and n ≥ 2, we have $\begin{matrix} ℳ {∥ ζ_{n} - ζ ∥ \geq ɛ} & = & ℳ {γ | ∥ ζ_{n} (γ) - ζ (γ) ∥ \geq ɛ} \\ = & ℳ {γ_{n}} \\ = & \frac{1}{n + 1} \to 0 \end{matrix}$ as n→ ∞. So the sequence {ζ_n} converges in measure to ζ. However, for each n ≥ 2, we have the uncertainty distribution of uncertain variable ∥ζ_n - ζ∥ = ∥ ζ_n ∥ is $Φ_{n} (x) = {\begin{matrix} 0, & if x < 0 \\ 1 - \frac{1}{n + 1}, & if 0 \leq x < n + 1 \\ 1, & if x \geq n + 1 . \end{matrix}$

So for each n ≥ 2, we have $E [∥ ζ_{n} - ζ ∥] = \int_{0}^{n + 1} 1 - (1 - \frac{1}{n + 1}) d x = 1 .$

That is, the sequence {ζ_n} does not converge in mean to ζ.

5.2 Convergence in measure and convergence in distribution

Theorem 5.2. Assume complex uncertain sequence {ζ_n} with real part {ξ_n} and imaginary part {η_n}, respectively, for n = 1, 2, ⋯. If uncertain sequences {ξ_n} and {η_n} converge in measure to ξ and η, respectively, then complex uncertain sequence {ζ_n} converges in measure to ζ = ξ + iη.

Proof: It follows from the definition of convergence in measure of uncertain sequence that for any small number ɛ > 0, $lim_{n \to \infty} ℳ {| ξ_{n} - ξ | \geq \frac{ɛ}{\sqrt{2}}} = 0$ and $lim_{n \to \infty} ℳ {| η_{n} - η | \geq \frac{ɛ}{\sqrt{2}}} = 0 .$

Note that $∥ ζ_{n} - ζ ∥ = \sqrt{| ξ_{n} - ξ |^{2} + | η_{n} - η |^{2}} .$

Thus we have $\begin{matrix} {∥ ζ_{n} - ζ ∥ \geq ɛ} \subset {| ξ_{n} - ξ | \geq \frac{ɛ}{\sqrt{2}}} \\ \cup {| η_{n} - η | \geq \frac{ɛ}{\sqrt{2}}} . \end{matrix}$

Using subadditivity axiom of uncertain measure, we obtain $\begin{matrix} \begin{matrix} ℳ {∥ ζ_{n} - ζ ∥ \geq ɛ} \leq \\ ℳ {| ξ_{n} - ξ | \geq \frac{ɛ}{\sqrt{2}}} + ℳ {| η_{n} - η | \geq \frac{ɛ}{\sqrt{2}}} . \end{matrix} \end{matrix}$

Hence, we have $\begin{matrix} \begin{matrix} 0 & \leq lim_{n \to \infty} ℳ {∥ ζ_{n} - ζ ∥ \geq ɛ} \\ \leq lim_{n \to \infty} ℳ {| ξ_{n} - ξ | \geq \frac{ɛ}{\sqrt{2}}} \\ + lim_{n \to \infty} ℳ {| η_{n} - η | \geq \frac{ɛ}{\sqrt{2}}} \\ = 0 . \end{matrix} \end{matrix}$

So $lim_{n \to \infty} ℳ {∥ ζ_{n} - ζ ∥ \geq ɛ} = 0$ . That is, {ζ_n} converges in measure to ζ. □

In Example 5.1, assume $ξ_{n} (γ) = {\begin{matrix} 0, & if γ = γ_{1} \\ 1 + \frac{1}{n}, & if γ = γ_{2}, \end{matrix}$ $η_{n} (γ) = η (γ) = {\begin{matrix} 1, & if γ = γ_{1} \\ 0, & if γ = γ_{2}, \end{matrix}$ and $ξ (γ) = {\begin{matrix} 0, & if γ = γ_{1} \\ 1, & if γ = γ_{2} . \end{matrix}$

By the expressions of ζ_n, ξ_n and η_n we have $ζ_{n} = ξ_{n} + i η_{n} .$

Then for any given number ɛ > 0 we have $ℳ {| ξ_{n} - ξ | \geq ɛ} \to 0$ as n→ ∞ and $ℳ {| η_{n} - η ∥ \geq ɛ} = 0$ for n = 1, 2, ⋯. Hence, {ξ_n} and {η_n} converge in measure to ξ and η, respectively. By Theorem 5.2, we also obtain {ζ_n} converges to ζ = ξ + iη in measure.

Theorem 5.3. Assume complex uncertain sequence {ζ_n} with real part {ξ_n} and imaginary part {η_n}, respectively, for n = 1, 2, ⋯. If uncertain sequences {ξ_n} and {η_n} converge in measure to ξ and η, respectively, then complex uncertain sequence {ζ_n} converges in distribution to ζ = ξ + iη.

Proof: Let c = a + ib be a given continuity point of the complex uncertainty distribution Φ. On the one hand, for any α > a, β > b, we have $\begin{matrix} {ξ_{n} \leq a, η_{n} \leq b} \\ = {ξ_{n} \leq a, η_{n} \leq b, ξ \leq α, η \leq β} \\ \cup {ξ_{n} \leq a, η_{n} \leq b, ξ > α, η > β} \\ \cup {ξ_{n} \leq a, η_{n} \leq b, ξ \leq α, η > β} \\ \cup {ξ_{n} \leq a, η_{n} \leq b, ξ > α, η \leq β} \\ \subset {ξ \leq α, η \leq β} \cup {| ξ_{n} - ξ | \geq α - a} \\ \cup {| η_{n} - η | \geq β - b} . \end{matrix}$

It follows from the subadditivity axiom that $\begin{matrix} \begin{matrix} Φ_{n} (c) & = Φ_{n} (a + ib) \\ \leq Φ (α + i β) + ℳ {| ξ_{n} - ξ | \geq α - a} \\ + ℳ {| η_{n} - η | \geq β - b} . \end{matrix} \end{matrix}$

Since {ξ_n} and {η_n} converge in measure to ξ and η, respectively, we have $lim_{n \to \infty} ℳ {| ξ_{n} - ξ | \geq α - a} = 0$ and $lim_{n \to \infty} ℳ {| η_{n} - η | \geq β - b} = 0 .$

Thus we obtain $lim_{n \to \infty} sup Φ_{n} (c) \leq Φ (α + i β)$ for any α > a, β > b. Letting α + iβ → a + ib, we get $lim_{n \to \infty} sup Φ_{n} (c) \leq Φ (c) .$ (1)

On the other hand, for any x < a, y < b, we have $\begin{matrix} {ξ \leq x, η \leq y} \\ = {ξ_{n} \leq a, η_{n} \leq b, ξ \leq x, η \leq y} \\ \cup {ξ_{n} \leq a, η_{n} > b, ξ \leq x, η \leq y} \\ \cup {ξ_{n} > a, η_{n} \leq b, ξ \leq x, η \leq y} \\ \cup {ξ_{n} > a, η_{n} > b, ξ \leq x, η \leq y} \\ \subset {ξ_{n} \leq a, η_{n} \leq b} \cup {| ξ_{n} - ξ | \geq a - x} \\ \cup {| η_{n} - η | \geq b - y} \end{matrix}$ which implies $\begin{matrix} Φ (x + iy) & \leq Φ_{n} (a + ib) + ℳ {| ξ_{n} - ξ | \geq a - x} \\ + ℳ {| η_{n} - η | \geq b - y} . \end{matrix}$

Since $lim_{n \to \infty} ℳ {| ξ_{n} - ξ | \geq a - x} = 0$ and $lim_{n \to \infty} ℳ {| η_{n} - η | \geq b - y} = 0,$ we obtain $Φ (x + iy) \leq lim_{n \to \infty} inf Φ_{n} (a + ib)$ for any x < a, y < b. On x + iy → a + ib, we get $Φ (c) \leq lim_{n \to \infty} inf Φ_{n} (c) .$ (2)

It follows from (1) and (2) that Φ_n (c) → Φ (c) as n→ ∞. That is, complex uncertain sequence {ζ_n} converges in distribution to ζ = ξ + iη. The theorem is verified. □

Example 5.3. Consider complex normal uncertain variables ζ_n = ξ_n + iη_n where $ξ_{n} = 𝒩 (0, 1 + \frac{1}{n})$ and $η_{n} = 𝒩 (0, 2 + \frac{1}{n})$ for n = 1, 2, ⋯, respectively. Assume ζ = ξ + iη where ξ = 𝒩 (0, 1) and η = 𝒩 (0, 2), respectively. In the following, we will show that {ξ_n} and {η_n} converge in measure to ξ and η, respectively, and {ζ_n} converges in distributionto ζ.

Since $E [| ξ_{n} - ξ |] = 0 and E [| η_{n} - η |] = 0$ as n→ ∞, {ξ_n} and {η_n} converge in mean to ξ and η, respectively. It follows from Theorem 5.1 that {ξ_n} and {η_n} converge in measure to ξ and η, respectively. And thus by Theorem 5.3 {ζ_n} converges in distribution to ζ.

In fact, the complex uncertainty distribution of ζ is $\begin{matrix} \begin{matrix} Φ (c) & = Φ (x + iy) = ℳ {ξ \leq x, η \leq y} \\ = ℳ {ξ \leq x} \land ℳ {η \leq y} \\ = Φ (x) \land Φ (y) \\ = {(1 + \exp (\frac{π (- x)}{\sqrt{3}}))}^{- 1} \\ \land {(1 + \exp (\frac{π (- y)}{2 \sqrt{3}}))}^{- 1} . \end{matrix} \end{matrix}$

Similarly, we have $\begin{matrix} \begin{matrix} Φ_{n} (c) & = ℳ {ξ_{n} \leq x} \land ℳ {η_{n} \leq y} \\ = {(1 + \exp (\frac{π (- x)}{\sqrt{3} (1 + \frac{1}{n})}))}^{- 1} \\ \land {(1 + \exp (\frac{π (- y)}{\sqrt{3} (2 + \frac{1}{n})}))}^{- 1} \end{matrix} \end{matrix}$ for n = 1, 2, ⋯ . On n→ ∞, we have Φ_n (c) → Φ (c) which implies {ζ_n} converges in distributionto ζ.

Example 5.4. Convergence in distribution does not imply convergence in measure. Take an uncertainty space (Γ, ℒ, ℳ) to be {γ₁, γ₂} with $ℳ {γ_{1}} = ℳ {γ_{2}} = \frac{1}{2}$ . We define a complex uncertain variable as $ζ (γ) = {\begin{matrix} i, & if γ = γ_{1} \\ - i, & if γ = γ_{2} . \end{matrix}$

We also define ζ_n = - ζ for n = 1, 2, ⋯. Then ζ_n and ζ have the same distribution $Φ_{n} (c) = Φ_{n} (a + ib) = {\begin{matrix} 0, & if a < 0, b < + \infty \\ 0, & if a \geq 0, b < - 1 \\ 1 / 2, & if a \geq 0, - 1 \leq b < 1 \\ 1, & if a \geq 0, b \geq 1 . \end{matrix}$

Then {ζ_n} converges in distribution to ζ. However, for some small number ɛ > 0, we have $ℳ {∥ ζ_{n} - ζ ∥ \geq ɛ} = ℳ {γ | ∥ ζ_{n} (γ) - ζ (γ) ∥ \geq ɛ} = 1 .$

That is, the sequence {ζ_n} does not converge in measure to ζ. By Theorem 5.2, the real part and imaginary part of {ζ_n} also do not converge in measure.

In addition, since ζ_n = - ζ for n = 1, 2, ⋯, the sequence {ζ_n} does not converge a.s. to ζ.

5.3 Convergence almost surely and convergence in measure

Example 5.5. Convergence a.s. does not imply convergence in measure. Take an uncertainty space (Γ, ℒ, ℳ) to be {γ₁, γ₂, ⋯} with $ℳ {Λ} = {\begin{matrix} sup_{γ_{n} \in Λ} \frac{n}{2 n + 1}, & if sup_{γ_{n} \in Λ} \frac{n}{2 n + 1} < \frac{1}{2} \\ 1 - sup_{γ_{n} \in Λ^{c}} \frac{n}{2 n + 1}, & if sup_{γ_{n} \in Λ^{c}} \frac{n}{2 n + 1} < \frac{1}{2} \\ \frac{1}{2}, & otherwise . \end{matrix}$

Then we define complex uncertain variables by $ζ_{n} (γ) = {\begin{matrix} in, & if γ = γ_{n} \\ 0, & otherwise \end{matrix}$ for n = 1, 2, ⋯ and ζ ≡ 0. Then the sequence {ζ_n} converges a.s. to ζ. However, for some small number ɛ > 0, we have $\begin{matrix} \begin{matrix} ℳ {∥ ζ_{n} - ζ ∥ \geq ɛ} & = ℳ {γ | ∥ ζ_{n} (γ) - ζ (γ) ∥ \geq ɛ} \\ = ℳ {γ_{n}} \\ = \frac{n}{2 n + 1} \to \frac{1}{2} \end{matrix} \end{matrix}$ as n→ ∞. That is, the sequence {ζ_n} does not converge in measure to ζ.

In addition, the complex uncertainty distributions of ζ_n are $\begin{matrix} Φ_{n} (c) = Φ_{n} (a + ib) \\ = {\begin{matrix} 0, & if a < 0, b < + \infty \\ 0, & if a \geq 0, b < 0 \\ 1 - \frac{n}{2 n + 1}, & if a \geq 0, 0 \leq b < n \\ 1, & if a \geq 0, b \geq n \end{matrix} \end{matrix}$ for n = 1, 2, ⋯, respectively. And the complex uncertainty distribution of ζ is $Φ (c) = {\begin{matrix} 0, & if a < 0, b < + \infty \\ 0, & if a \geq 0, b < 0 \\ 1, & if a \geq 0, b \geq 0 . \end{matrix}$

Clearly Φ_n (c) does not converge to Φ (c) at a ≥ 0, b ≥ 0. That is, the sequence {ζ_n} does not converge to ζ in distribution.

Example 5.6. Convergence in measure does not imply convergence a.s. Take an uncertainty space (Γ, ℒ, ℳ) to be [0,1] with Borel algebra and Lebesgue measure. For any positive integer n, there is an integer m such that n = 2^m + k where k is an integer between 0 and 2^m - 1. Then we define a complex uncertain variable by $ζ_{n} (γ) = {\begin{matrix} i, & if \frac{k}{2^{m}} \leq γ \leq \frac{(k + 1)}{2^{m}} \\ 0, & otherwise \end{matrix}$ for n = 1, 2, ⋯ and ζ ≡ 0. For some small number ɛ > 0, we obtain $\begin{matrix} \begin{matrix} ℳ {∥ ζ_{n} - ζ ∥ \geq ɛ} & = ℳ {γ | ∥ ζ_{n} (γ) - ζ (γ) ∥ \geq ɛ} \\ = \frac{1}{2^{m}} \to 0 \end{matrix} \end{matrix}$ as n→ ∞. So the sequence {ζ_n} converges in measure to ζ. In addition, we have $E [∥ ζ_{n} - ζ ∥] = \frac{1}{2^{m}} \to 0$ as n→ ∞. Thus the sequence {ζ_n} also converges in mean to ζ.

However, for any γ ∈ [0, 1], there is an infinite number of intervals of the form $[\frac{k}{2^{m}}, \frac{k + 1}{2^{m}}]$ containing γ. Thus ζ_n (γ) does not converge to 0. In other words, the sequence {ζ_n} does not converge a.s. to ζ.

5.4 Convergence almost surely and convergence in mean

Example 5.7. Convergence a.s. does not imply convergence in mean. Take an uncertainty space (Γ, ℒ, ℳ) to be {γ₁, γ₂ ⋯} with $ℳ {Λ} = \sum_{γ_{n} \in Λ} 1 / 2^{n} .$

The complex uncertain variables are defined by $ζ_{n} (γ) = {\begin{matrix} i 2^{n}, & if γ = γ_{n} \\ 0, & otherwise \end{matrix}$ for n = 1, 2, ⋯ and ζ ≡ 0. Then {ζ_n} converges a.s. to ζ. However, the uncertainty distributions of ∥ζ_n∥ are $Φ_{n} (x) = {\begin{matrix} 0, & if x < 0 \\ 1 - \frac{1}{2^{n}}, & if 0 \leq x < 2^{n} \\ 1, & if x \geq 2^{n} \end{matrix}$ for n = 1, 2, ⋯, respectively. Then we have $E [∥ ζ_{n} - ζ ∥] = \int_{0}^{2^{n}} 1 - (1 - \frac{1}{2^{n}}) dx = 1 .$

So the sequence {ζ_n} does not converge in meanto ζ.

By Example 5.6, we can obtain that convergence in mean does not imply convergence a.s.

5.5 Convergence almost surely and convergence uniformly almost surely

Proposition 5.1.Let ζ, ζ₁, ζ₂, ⋯ be complex uncertain variables. Then {ζ_n} converges a.s. to ζ if and only if for any ɛ > 0, we have $ℳ (⋂_{k = 1}^{\infty} ⋃_{n = k}^{\infty} {∥ ζ_{n} - ζ ∥ \geq ɛ}) = 0 .$

Proof: By the definition of convergence a.s., we have that there exists an event Λ with ℳ {Λ} =1 such that $lim_{n \to \infty} ∥ ζ_{n} - ζ ∥ = 0$ . Then for any ɛ > 0, there exists k such that ∥ζ_n - ζ ∥ < ɛ where n > k and for any γ ∈ Λ, that is equivalent to $ℳ (⋃_{k = 1}^{\infty} ⋂_{n = k}^{\infty} {∥ ζ_{n} - ζ ∥ < ɛ}) = 1 .$

It follows from the duality axiom of uncertain measure that $ℳ (⋂_{k = 1}^{\infty} ⋃_{n = k}^{\infty} {∥ ζ_{n} - ζ ∥ \geq ɛ}) = 0 .$

The proposition is thus proved. □

Proposition 5.2.Let ζ, ζ₁, ζ₂, ⋯ be complex uncertain variables. Then {ζ_n} converges uniformly almost surely to ζ if and only if $lim_{n \to \infty} ℳ (⋃_{n = k}^{\infty} {∥ ζ_{n} - ζ ∥ \geq ɛ}) = 0 .$

Proof: If {ζ_n} converges uniformly almost surely to ζ, then for any δ > 0, there exists B such that ℳ {B} < δ and {ζ_n} uniformly converges to ζ on Γ - B. Thus, for any ɛ > 0, there exists k > 0 such that ∥ζ_n - ζ ∥ < ɛ where n ≥ k and γ ∈ Γ - B. That is $⋃_{n = k}^{\infty} {∥ ζ_{n} - ζ ∥ \geq ɛ} \subset B .$

It follows from the subadditivity axiom of uncertain measure that $ℳ (⋃_{n = k}^{\infty} {∥ ζ_{n} - ζ ∥ \geq ɛ}) \leq ℳ {B} < δ .$

Then $lim_{n \to \infty} ℳ (⋃_{n = k}^{\infty} {∥ ζ_{n} - ζ ∥ \geq ɛ}) = 0 .$

On the contrary, if $lim_{n \to \infty} ℳ (⋃_{n = k}^{\infty} {∥ ζ_{n} - ζ ∥ \geq ɛ}) = 0$ for any ɛ > 0, then for any given δ > 0 and m ≥ 1, there exists m_k such that $ℳ (⋃_{n = m_{k}}^{\infty} {∥ ζ_{n} - ζ ∥ \geq \frac{1}{m}}) < \frac{δ}{2^{m}} .$

Let $B = ⋃_{m = 1}^{\infty} ⋃_{n = m_{k}}^{\infty} {∥ ζ_{n} - ζ ∥ \geq \frac{1}{m}}$ . Then $\begin{matrix} \begin{matrix} ℳ {B} & \leq \sum_{m = 1}^{\infty} ℳ (⋃_{n = m_{k}}^{\infty} {∥ ζ_{n} - ζ ∥ \geq \frac{1}{m}}) \\ \leq \sum_{m = 1}^{\infty} \frac{δ}{2^{m}} = δ . \end{matrix} \end{matrix}$

Furthermore, we have $sup_{γ \in Γ - B} ∥ ζ_{n} - ζ ∥ < \frac{1}{m}$ for any m = 1, 2, ⋯ and n > m_k. The proposition is proved. □

Theorem 5.4. If the complex uncertain sequence {ζ_n} converges uniformly almost surely to ζ, then {ζ_n} converges almost surely to ζ.

Proof: It follows from Proposition 5.2 that if {ζ_n} converges uniformly almost surely to ζ, then $lim_{n \to \infty} ℳ (⋃_{n = k}^{\infty} {∥ ζ_{n} - ζ ∥ \geq ɛ}) = 0 .$

Since $\begin{matrix} ℳ (⋂_{k = 1}^{\infty} ⋃_{n = k}^{\infty} {∥ ζ_{n} - ζ ∥ \geq ɛ}) \\ \leq ℳ (⋃_{n = k}^{\infty} {∥ ζ_{n} - ζ ∥ \geq ɛ}), \end{matrix}$ taking the limit as n→ ∞ on both side of above inequality, we obtain $ℳ (⋂_{k = 1}^{\infty} ⋃_{n = k}^{\infty} {∥ ζ_{n} - ζ ∥ \geq ɛ}) = 0 .$

By Proposition 5.1, {ζ_n} converges a.s. to ζ. The theorem is verified. □

As is seen from Example 5.1, since $lim_{n \to \infty} ℳ (⋃_{n = k}^{\infty} {∥ ζ_{n} - ζ ∥ \geq ɛ}) = 0,$ the sequence {ζ_n} converges uniformly almost surely to ζ, and thus {ζ_n} converges a.s. to ζ by Theorem 5.4. □

5.6 Convergence uniformly almost surely and convergence in measure

Theorem 5.5. If the complex uncertain sequence {ζ_n} converges uniformly almost surely to ζ, then {ζ_n} converges in measure to ζ.

Proof: If {ζ_n} converges uniformly almost surely to ζ, then from Proposition 5.2 we have $lim_{n \to \infty} ℳ (⋃_{n = k}^{\infty} {∥ ζ_{n} - ζ ∥ \geq ɛ}) = 0 .$

And $ℳ {∥ ζ_{n} - ζ ∥ \geq ɛ} \leq ℳ {⋃_{n = k}^{\infty} ∥ ζ_{n} - ζ ∥ \geq ɛ} .$

Letting n→ ∞, we can obtain {ζ_n} converges in measure to ζ. □

As is seen from Example 5.6, {ζ_n} converges in measure to ζ. However, it does not converges a.s. to ζ. It follows from Theorem 5.5 that {ζ_n} does not converges uniformly almost surely to ζ.

In conclusion, the relationships among the five convergence concepts are displayed in Table 1.

6 Conclusion

In this paper, some convergence concepts of complex uncertain sequences have been introduced. The relationships among convergence in measure, convergence in mean, convergence almost surely, convergence in distribution and convergence uniformly almost surely are studied.

Footnotes

Acknowledgments

This work is supported by Natural Science Foundation of Shandong Province (ZR2014GL002).

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