# Convergence Properties for Uncertain Sequence

Authors: Cuilian You; Lijuan Yan
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

Uncertainty theory is a branch of axiomatic mathematics for modeling belief degrees. Since convergence for uncertain sequences plays a very important role in uncertainty theory and applications, we investigated some properties of convergence in measure in this paper. In addition, some theorems and properties of convergence inp-distance were studied.


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## Body

## 1. Introduction

Uncertainty theory which is founded by Liu [1] has been used to model uncertain phenomena for a long time. During the past years, many researchers contributed a lot to this area. For instance, Liu [2] introduced a classical application of uncertainty theory. In order to indicate the belief degree that an event happens, Liu [1] defined a definition of uncertain measure as a set function satisfying normality, duality, subadditivity, and product axiom. Liu [1] put forward the concept of uncertain variable. Then Liu and Ha [3] presented a formula to calculate the expected value of a function of uncertain variables. After that, Liu [4] introduced uncertain programming, and Li and Liu [5] studied uncertain logic. Liu [6] founded uncertain process which is essentially a sequence of uncertain variables indexed by time, and then the uncertain distribution and independence of uncertain process were studied by Liu [7] in 2014. Furthermore, uncertain differential equation was also given by Liu [6] and further researched by Chen and Liu [8] whose main content is existence and uniqueness theorem for uncertain differential equation. To date, uncertainty theory has been already applied to uncertain risk measure (Liu [2]), uncertain set (Liu [9]), uncertain inference (Gao et al. [10]), and so on.Since sequence convergence is a useful tool in the fundamental theory of mathematics, some definitions of sequences convergence of uncertain sequences were proposed by Liu [1] and You [11]. The Cauchy convergence of uncertain sequences and the sufficient conditions of convergence almost surely were presented by Xia [12]. H. Wang and Z. Wang [13] studied the dual limits of uncertain variables sequence in 2013. Then, Gao [14] discussed the convergence theorems for expected value of uncertain variables. A necessary and sufficient condition of convergence in measure for uncertain sequence was presented by Guo et al. [15]. Therefore,r-order convergence and dual convergence of uncertain variable sequences were researched by Yuan [16] and Yuan et al. [17]. Chen et al. [18] introduced the convergence concepts of complex uncertain sequence. In addition, You and Yan [19] gave a new concept of sequence convergence for uncertain variables that is convergence inp-distance; then the relationships among concepts of convergence for uncertain variables sequences were studied by You and Yan [20] in 2016.The main objective of this paper is to research the properties of convergence for uncertain sequence. The structure of the paper is as follows: we shall briefly introduce uncertain space and some basic contents and theorems of uncertainty theory in Section2. Then some new properties of convergence in measure and convergence inp-distance will be investigated in Section 3. Finally, a brief summary will be given.


## 2. Preliminary

Some definitions and theorems of uncertainty theory are presented in this section.LetΓ be a nonempty set, and let L be a σ-algebra over Γ. Each element Λ in L is called an event. To describe the belief degree that each event will happen, Liu [1] proposed the set function M, which is called uncertain measures if it satisfies the following three axioms.Axiom 1 (normality).
M
Γ
=
1
.Axiom 2 (self-duality).
M
Λ
+
M
Λ
c
=
1, for any event Λ.Axiom 3 (subadditivity).
For every countable sequence of eventsΛi, we have (1)M⋃i=1∞Λi≤∑i=1∞MΛi.Next, we will introduce the definition of uncertainty space.Definition 1 (see Liu [1]).
LetΓ be a nonempty set, let L be a σ-algebra over Γ, and let M be an uncertain measure. Then the triplet Γ,L,M is called an uncertainty space.For uncertain measure, we have the following theorem.Theorem 2 (monotonicity, Liu [1]).
Uncertain measureM is a monotone increasing set function. That is, for any events Λ1⊂Λ2, we have M{Λ1}≤M{Λ2}.Definition 3 (see Liu [1]).
An uncertain variable is a functionξ from an uncertainty space (Γ,L,M) to the set of real numbers such that {ξ∈B} is an event for any set B of real numbers.Then some convergence concepts and theorems of uncertain sequence were given.Definition 4 (see Liu [1]).
The uncertain sequenceξi is said to be convergent in measure to ξ if (2)limi→∞⁡Mξi-ξ≥ε=0,for every ε>0.Remark 5.
According to Definition  2.3 and Corollary  3.2 in You [11], we know there exists subsequence ξik such that limk→∞⁡Mξik-ξ<ε=1. That is to say, if uncertain sequence ξi converges in measure to ξ, we can get limk→∞⁡ξik-ξ=0, a.s.Definition 6 (see You and Yan [19]).
The uncertain sequence{ξi} is said to be convergent inp-distance to ξ if (3)limi,j→∞⁡dpξi,ξj=limi,j→∞⁡Eξi-ξjp1/p+1=0.Theorem 7 (see You and Yan [19]).
Letξ, η, and τ be uncertain variables, and let dp(·,·) be thep-distance. Then we have the following:(a)
(Nonnegativity)dp(ξ,η)≥0.(b)
(Identification)dp(ξ,η)=0 if and only if ξ=η.(c)
(Symmetry)dp(ξ,η)=dp(η,ξ).(d)
(Triangle Inequality)dp(ξ,η)≤dp(ξ,τ)+dp(τ,η).Theorem 8 (see You and Yan [20]).
Letξ, η be uncertain variables defined on metric space F,dp, and let ξi, ηi be uncertain variables sequences defined on metric space F,dp. Thus we obtain the following conclusions:(a)
Iflimi→∞⁡dp(ξi,ξ)=0, then limi→∞⁡dp(|ξi|,|ξ|)=0.(b)
Iflimi→∞⁡dp(ξi,ξ)=0 and limi→∞⁡dp(ξi,η)=0, then ξ=η.(c)
Iflimi→∞⁡dp(ξi,ξ)=0 and limi→∞⁡dp(ηi,η)=0, then limi→∞⁡dpaξi+bηi,aξ+bη=0, for any real numbers a,b.(d)
Iflimi→∞⁡dp(ξi,ξ)=0, limi→∞⁡dp(ηi,η)=0, and there exists a real number M>0 such that |ξ|≤M and |η|≤M, then limi→∞⁡dp(ξiηi,ξη)=0.


## 3. Some Properties of Convergence

This section will describe some properties of convergence in measure and convergence inp-distance.Theorem 9.
Suppose that uncertain variables sequences{ξn}, {ηn} converge in measure to ξ, η, respectively. If there exist positive numbers M1, M, N1, and N such that M1≤|ξn|≤M and N1≤|ηn|≤N, for any n, then we have the following:(1)
ξn+ηn converges in measure to ξ+η.(2)
ξn-ηn converges in measure to ξ-η.(3)
ξnηn converges in measure to {ξη}.(4)
ξn/ηn converges in measure to ξ/η.Proof.
Since uncertain sequenceξi converges in measure to ξ, it follows from Remark 5 that there exists subsequence ξik such that limk→∞⁡ξik-ξ=0, a.s. Since M1≤ξn≤M, we have M1≤ξ≤M. In a similar way, if N1≤ηn≤N, then N1≤|η|≤N.(1)
It follows from Axiom3 that(4)Mξn+ηn-ξ+η≥ε=Mξn-ξ+ηn-η≥ε≤Mξn-ξ≥ε2+Mηn-η≥ε2⟶0,as n→0. Thus uncertain variables sequence ξn+ηn converges in measure to ξ+η.(2)
According to Axiom3, we obtain(5)Mξn-ηn-ξ-η≥ε=Mξn-ξ-ηn-η≥ε≤Mξn-ξ≥ε2+Mηn-η≥ε2⟶0,as n→0. Therefore, uncertain variables sequence {ξn-ηn} converges in measure to {ξ-η}.(3)
By Axiom3, we have(6)Mξnηn-ξη≥ε=Mξnηn-ξnη+ξnη-ξη≥ε≤Mξnηn-ξnη≥ε2+Mξnη-ξη≥ε2≤MMηn-η≥ε2+MNξn-ξ≥ε2=Mηn-η≥ε2M+Mξn-ξ≥ε2N⟶0,as n→0. Thus uncertain variables sequence {ξnηn} converges in measure to {ξη}.(4)
It follows from Axiom3 that(7)Mξnηn-ξη≥ε=Mξnηn-ξnη+ξnη-ξη≥ε=Mξnη-ηnηnη+ξn-ξη≥ε≤Mξnηn-ηηnη≥ε2+Mξn-ξη≥ε2≤MMN12ηn-η≥ε2+M1N1ξn-ξ≥ε2=Mηn-η≥N1N1ε2M+Mξn-ξ≥N1ε2⟶0,as n→0. Then we get that uncertain variables sequence ξn/ηn converges in measure to ξ/η.The following theorem is a generalization of Theorem9.Theorem 10.
Suppose that uncertain variables sequences{ξ1n},{ξ2n},…,{ξkn} converge in measure to ξ1,ξ2,…,ξk, respectively. Assume Q(x1,x2,…,xk) is rational function. If there exist positive numbers Mi′, Mi such that Mi′≤|ξi|≤Mi(i=1,2,…,k) and Q(ξ1,ξ2,…,ξk)≠±∞, then Q(ξ1n,ξ2n,…,ξkn) converges in measure to Q(ξ1,ξ2,…,ξk).Proof.
According to Theorem9, we have (8)MQξ1n,ξ2n,…,ξkn-Qξ1,ξ2,…,ξk≥ε≤Mξ1n-ξ1≥εk+Mξ2n-ξ2≥εk+⋯+Mξkn-ξk≥εk⟶0,as n→∞. The proof is completed.Theorem 11.
Assume uncertain variables sequence{ξn} converges in measure to ξ. If f(x) is a continuous function, then f(ξn) converges in measure to f(ξ).Proof.
(1) Iffx=∑i=1kaixi is k-order polynomial function, by Theorem 10, we can get that f(ξn) converges in measure to fξ.
(2) Since{ξn} converges in measure to ξ, there exists a positive number M0 such that M{|ξ|>M0}<δ, for any 0<δ<1. Since {ξn} converges in measure to ξ, there exists positive number N1 such that M{|ξ-ξn|>1}<δ, for any n≥N1. Thus(9)Mξn≥M0+1≤Mξ>M0+Mξ-ξn>1<2δ.
Sincef(x) is a continuous function, there exists ak-order polynomial function gkx such that (10)fx-gkx<ε3.
It follows from (1) thatgkξn converges in measure to gkξ. That is to say, there exists a positive integer N2 such that(11)Mgkξn-gkξ≥ε3<δ,for any n≥N2.
Then(12)Mfξ-fξn≥ε≤Mfξ-fξn≥ε∩A∪B+Mfξ-fξn≥ε∩A∪B¯=I1+I2,where (13)A∪B=ξ>M0∪ξn>M0+1,A∪B¯=A¯∩B¯=ξ≤M0∩ξn≤M0+1,I1=Mfξ-fξn≥ε∩A∪B,I2=Mfξ-fξn≥ε∩A∪B¯.
Furthermore, we have(14)I1=Mfξ-fξn≥ε∩ξ>M0∪ξn>M0+1≤Mξ>M0∪ξn>M0+1≤Mξ>M0+Mξn>M0+1<3δ.
Since(15)fξ-fξn≥ε=fξ-gkξ+gkξ-gkξn+gkξn-fξn≥ε,we obtain fξ-gkξ≥ε/3 or gkξ-gkξn≥ε/3 or gkξn-fξn≥ε/3. That is,(16)fξ-fξn≥ε⊂fξ-gkξ≥ε3∪gkξ-gkξn≥ε3∪gkξn-fξn≥ε3,
By the reason that(17)Mfξ-gkξ≥ε3∩A∪B¯=Mfξ-gkξ≥ε3∩ξ≤M0∩ξn≤M0+1=0,Mgkξn-fξn≥ε3∩A∪B¯=Mgkξn-fξn≥ε3∩ξ≤M0∩ξn≤M0+1=0.
we have(18)I2≤Mgkξ-gkξn≥ε3∩ξ≤M0∩ξn≤M0+1≤Mgkξ-gkξn≥ε3<δ.
Therefore,(19)Mfξ-gkξ≥ε≤I1+I2<4δ.
By the arbitrariness ofε, δ, we can get that f(ξn) converges in measure to f(ξ).Based on Theorems10 and 11, the following theorem was acquired.Theorem 12.
If uncertain variables sequences{ξin},i=1,2,…,k, converge in measure to ξi, respectively, and Qg1ξ1,g2ξ2,…,gkξk≠±∞, for any rational function Q(x1,x2,…,xk) and continuous function gi(x),i=1,2,…,k, then Qg1ξ1n,g2ξ2n,…,gkξkn converges in measure to Qg1ξ1,g2ξ2,…,gkξk.Proof.
The theorem can be proved easily by using Theorems10 and 11.Theorem 13.
If uncertain variables sequence{ξn} converges in measure to ξ, η, respectively, then M{ξ=η}=1.Proof.
According to Axiom3, we obtain (20)0≤Mξ-η≥ε=Mξ-ξn+ξn-η≥ε≤Mξ-ξn≥ε2+Mη-ξn≥ε2⟶0,as n→∞. Thus the theorem is proved.Next, the results of convergence in measure will be extended to the case of convergence inp-distance.Theorem 14.
Suppose that uncertain variables sequences{ξn}, {ηn} converge inp-distance to ξ, η, respectively. If there exist positive numbers M1, M, N1, and N such that M1≤|ξn|≤M and N1≤ηn≤N, for any n, then(1)
limi→∞⁡dpξi-ηi,ξ-η=0,(2)
limi→∞⁡dpξi/ηi,ξ/η=0.Proof.
(1) It follows from Triangle Inequality that(21)0≤limi→∞⁡dpξi-ηi,ξ-η≤limi→∞⁡dpξi-ηi,ξi-η+limi→∞⁡dpξi-η,ξ-η=limi→∞Eξi-ηi-ξi-ηp1/p+1+limi→∞⁡Eξi-η-ξ-ηp1/p+1=limi→∞Eηi-ηp1/p+1+limi→∞⁡Eξi-ξp1/p+1=limi→∞⁡dpηi,η+limi→∞⁡dpξi,ξ=0.
Thus,limi→∞⁡dp(ξi-ηi,ξ-η)=0.
(2) Suppose thatΓ+ is the set of points which have positive uncertain measure. For any γ∈Γ+,(22)limi→∞⁡ξiγ-ξγ≤limi→∞⁡Eξi-ξpMγ=limi→∞⁡dpp+1ξi,ξMγ=0.
Since0<M1≤ξn≤M, we have M1≤ξ≤M. In a similar way, if N1≤ηn≤N, then N1≤η≤N.
By Triangle Inequality, we can get(23)0≤limi→∞⁡dpξiηi,ξη≤limi→∞⁡dpξiηi,ξiη+limi→∞⁡dpξiη,ξη=limi→∞⁡Eξiηi-ξiηp1/p+1+limi→∞⁡Eξiη-ξηp1/p+1=limi→∞⁡Eξiη-ηiηiηp1/p+1+limi→∞⁡Eξi-ξηp1/p+1≤MN12p/p+1limi→∞⁡dpηi,η+1N1p/p+1limi→∞⁡dpξi,ξ=0.
Thuslimi→∞⁡dpξi/ηi,ξ/η=0 is verified.Remark 15.
According to Theorems8 and 14, we know the arithmetic operation of uncertain sequences, which are convergent in p-distance, are still convergent in p-distance.Theorem 16.
Suppose that uncertain variables sequencesξ1n,ξ2n,…,ξkn converge inp-distance to ξ1,ξ2,…,ξk, respectively. Assume Qx1,x2,…,xk is rational function. If there exist positive numbers Mi′, Mi such that Mi′≤ξi≤Mii=1,2,…,k and Qξ1,ξ2,…,ξk≠±∞, then Qξ1n,ξ2n,…,ξkn converges inp-distance to Qξ1,ξ2,…,ξk.Proof.
It follows from Theorems14 and 8 that (24)limi→∞⁡dpQξ1n,ξ2n,…,ξkn,Qξ1,ξ2,…,ξk≤limi→∞⁡dpξ1n,ξ1+limi→∞⁡dpξ2n,ξ2+⋯+limi→∞⁡dpξkn,ξk=0.
Therefore,Q(ξ1n,ξ2n,…,ξkn) converges inp-distance to Q(ξ1,ξ2,…,ξk).Theorem 17.
If uncertain variables sequence{ξn} converges inp-distance to uncertain variable ξ, then f(ξn) converges in p-distance to f(ξ), for any continuous function f(x).Proof.
(1) Iff(x)=∑i=1kaixi isk-order polynomial function, according to Theorem 10, we know that f(ξn) converges inp-distance to f(ξ).
(2) Iff(x) is arbitrary continuous function, there exists ak-order polynomial function gk(x) such that(25)fx-gkx<ε3p+1/p,for any ε>0. That is to say,(26)dpfx,gkx=Efx-gkxp1/p+1<ε3.
From (1), we obtain thatgk(ξn) converges inp-distance to gk(ξ), for any positive number k. That is to say, there exists positive integer N, such that(27)dpgkξn,gkξ<ε3,for any n>N.
Thus, we get(28)dpfξn,fξ≤dpfξn,gnξn+dpgnξn,fξ≤dpfξn,gnξn+dpgnξn,gnξ+dpgnξ,fξ<ε3+ε3+ε3=ε,for any n>N.
Therefore,fξn converges inp-distance to ξ.


## 4. Conclusions

The main point of this paper is to discuss properties of convergence for uncertain sequence. To begin with, some properties of convergence in measure were investigated in this paper; for example, the arithmetic operation of uncertain sequence, which are convergent in measure, are still convergent in measure. If an uncertain sequence is convergent in measure, then the sequence formed by the continuous function of the uncertain sequence is convergent in measure too and so forth. Then the results of convergence in measure were extended to the case of convergence inp-distance.Next, we will investigate the corresponding convergence properties of uncertain sequence in the sense of convergence almost surely and convergence in distribution and other convergence concepts. Furthermore, we will apply these convergence properties into the real problems in engineering and mathematical finance.



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