On the Convergence of Numerical Series of Uncertain Variables

Shan Lu,Lifen Jia

doi:10.1109/access.2019.2949118

Abstract

In this paper, within the framework of uncertainty theory, two kinds of concepts about uncertain numerical series and uncertain positive numerical series are introduced. Besides, several convergence theorems in terms of above two concepts are presented respectively, and some corresponding examples are also given.

Highlights

It is well know that, for modelling indeterminacy, there are two most common approaches, one is probability theory and the other is fuzzy set theory
An uncertain numerical series is defined as the form ξ1 + ξ2 + . . . + ξi + . . . , for shorthand, we use the symbol ξi to denote the series and called the uncertain numerical series
Proof: (i) If the uncertain numerical series ξi is convergent in measure, according to Definition 2, we can obtain that for any ε > 0, there exists a positive integer N, when m > N, we have

Summary

INTRODUCTION

It is well know that, for modelling indeterminacy, there are two most common approaches, one is probability theory and the other is fuzzy set theory. Sometimes we may face the situations that no or small samples can be gotten, due to technical difficulties and the occurrence of small probability events Therefor, in these cases, probability theory may not suit for modelling indeterminacy. In order to model indeterminacy phenomena, especially experts’ subjective estimation, Liu [2] founded uncertainty theory in 2007; it is based on four axioms, namely normality, duality, subadditivity and product axioms. In this theory, uncertain variable represent quantities with uncertainty, and belief degree is considered as its uncertainty distribution.

CONVERGENCE OF UNCERTAIN NUMERICAL SERIES

CONCLUSION