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

ISBN-13: 9789048161393 / Angielski / Miękka / 2010 / 394 str.

Shoumei Li; Y. Ogura; V. Kreinovich

Opis

This book presents a clear, systematic treatment of convergence theorems of set-valued random variables (random sets) and fuzzy set-valued random variables (random fuzzy sets). Topics such as strong laws of large numbers and central limit theorems, including new results in connection with the theory of empirical processes are covered. The author's own recent developments on martingale convergence theorems and their applications to data processing are also included. The mathematical foundations along with a clear explanation such as Holmander's embedding theorem, notions of various convergence of sets and fuzzy sets, Aumann integrals, conditional expectations, selection theorems, measurability and integrability arguments for both set-valued and fuzzy set-valued random variables and newly obtained optimizations techniques based on invariant properties are also given."

Recenzje

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)

Spis treści

Preface. Part I: Limit Theorems of Set-Valued and Fuzzy Set-Valued Random Variables. 1. The Space of Set-Valued Random Variables. 2. The Aumann Integral and the Conditional Expectation of a Set-Valued Random Variable. 3. Strong Laws of Large Numbers and Central Limit Theorems for Set-Valued Random Variables. 4. Convergence Theorems for Set-Valued Martingales. 5. Fuzzy Set-Valued Random Variables. 6. Convergence Theorems for Fuzzy Set-Valued Random Variables. 7. Convergences in the Graphical Sense for Fuzzy Set-Valued Random Variables. References for Part I. Part II: Practical Applications of Set-Valued Random Variables. 8. Mathematical Foundations for the Applications of Set-Valued Random Variables. 9. Applications to Imaging. 10. Applications to Data Processing. References for Part II. Index.