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

1. Front Matter

   Pages i-xii

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2. Limit Theorems of Set-Valued and Fuzzy Set-Valued Random Variables

   1. Front Matter

      Pages xiii-xiii

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   2. The Space of Set-Valued Random Variables

      • Shoumei Li, Yukio Ogura, Vladik Kreinovich

      Pages 1-39

   3. The Aumann Integral and the Conditional Expectation of a Set-Valued Random Variable

      • Shoumei Li, Yukio Ogura, Vladik Kreinovich

      Pages 41-85

   4. Strong Laws of Large Numbers and Central Limit Theorems for Set-Valued Random Variables

      • Shoumei Li, Yukio Ogura, Vladik Kreinovich

      Pages 87-115

   5. Convergence Theorems for Set-Valued Martingales

      • Shoumei Li, Yukio Ogura, Vladik Kreinovich

      Pages 117-160

   6. Fuzzy Set-Valued Random Variables

      • Shoumei Li, Yukio Ogura, Vladik Kreinovich

      Pages 161-190

   7. Convergence Theorems for Fuzzy Set-Valued Random Variables

      • Shoumei Li, Yukio Ogura, Vladik Kreinovich

      Pages 191-219

   8. Convergences in the Graphical Sense for Fuzzy Set-Valued Random Variables

      • Shoumei Li, Yukio Ogura, Vladik Kreinovich

      Pages 221-234

3. Practical Applications of Set-Valued Random Variables

   1. Front Matter

      Pages 251-251

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   2. Mathematical Foundations for the Applications of Set-Valued Random Variables

      • Shoumei Li, Yukio Ogura, Vladik Kreinovich

      Pages 253-293

   3. Applications to Imaging

      • Shoumei Li, Yukio Ogura, Vladik Kreinovich

      Pages 295-354

   4. Applications to Data Processing

      • Shoumei Li, Yukio Ogura, Vladik Kreinovich

      Pages 355-372

4. Back Matter

   Pages 387-394

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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.

• Martingal
• Martingale
• Random variable
• fuzzy
• optimization

From the reviews:

"The book under review is devoted to set-valued and fuzzy set-valued random variables which are generalizations of ordinary random variables … . the book is a useful reference for mathematicians who are working on set-valued or fuzzy set-valued random variables and related topics. Here one can find in one place results that are scattered throughout the literature. All the theorems are proven and the historical comments give the reader a wider perspective." (Osmo Kaleva, Mathematical Reviews, Issue 2005 b)

• Beijing Polytechnic University, Beijing, The Peoples Republic of China

  Shoumei Li

• Saga University, Saga, Japan

  Yukio Ogura

• University of Texas El Paso, El Paso, USA

  Vladik Kreinovich

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