# Limit Theorems and Applications of Set-Valued and Fuzzy Set-Valued Random Variables

• Shoumei Li
• Yukio Ogura
Book

Part of the Theory and Decision Library book series (TDLB, volume 43)

1. Front Matter
Pages i-xii
2. ### Limit Theorems of Set-Valued and Fuzzy Set-Valued Random Variables

1. Front Matter
Pages xiii-xiii
2. Shoumei Li, Yukio Ogura, Vladik Kreinovich
Pages 1-39
3. Shoumei Li, Yukio Ogura, Vladik Kreinovich
Pages 41-85
4. Shoumei Li, Yukio Ogura, Vladik Kreinovich
Pages 87-115
5. Shoumei Li, Yukio Ogura, Vladik Kreinovich
Pages 117-160
6. Shoumei Li, Yukio Ogura, Vladik Kreinovich
Pages 161-190
7. Shoumei Li, Yukio Ogura, Vladik Kreinovich
Pages 191-219
8. Shoumei Li, Yukio Ogura, Vladik Kreinovich
Pages 221-234
3. ### Practical Applications of Set-Valued Random Variables

1. Front Matter
Pages 251-251
2. Shoumei Li, Yukio Ogura, Vladik Kreinovich
Pages 253-293
3. Shoumei Li, Yukio Ogura, Vladik Kreinovich
Pages 295-354
4. Shoumei Li, Yukio Ogura, Vladik Kreinovich
Pages 355-372
4. Back Matter
Pages 387-394

### Introduction

After the pioneering works by Robbins {1944, 1945) and Choquet (1955), the notation of a set-valued random variable (called a random closed set in literatures) was systematically introduced by Kendall {1974) and Matheron {1975). It is well known that the theory of set-valued random variables is a natural extension of that of general real-valued random variables or random vectors. However, owing to the topological structure of the space of closed sets and special features of set-theoretic operations ( cf. Beer [27]), set-valued random variables have many special properties. This gives new meanings for the classical probability theory. As a result of the development in this area in the past more than 30 years, the theory of set-valued random variables with many applications has become one of new and active branches in probability theory. In practice also, we are often faced with random experiments whose outcomes are not numbers but are expressed in inexact linguistic terms.

### Keywords

Martingal Martingale Random variable fuzzy optimization

#### Authors and affiliations

• Shoumei Li
• 1
• Yukio Ogura
• 2
• 3
1. 1.Beijing Polytechnic UniversityBeijingThe Peoples Republic of China
2. 2.Saga UniversitySagaJapan
3. 3.University of Texas El PasoEl PasoUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-94-015-9932-0