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

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.

---

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 H?lmander'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.

---

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 H?lmander'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.
Content:
Front Matter....Pages i-xii
Front Matter....Pages xiii-xiii
The Space of Set-Valued Random Variables....Pages 1-39
The Aumann Integral and the Conditional Expectation of a Set-Valued Random Variable....Pages 41-85
Strong Laws of Large Numbers and Central Limit Theorems for Set-Valued Random Variables....Pages 87-115
Convergence Theorems for Set-Valued Martingales....Pages 117-160
Fuzzy Set-Valued Random Variables....Pages 161-190
Convergence Theorems for Fuzzy Set-Valued Random Variables....Pages 191-219
Convergences in the Graphical Sense for Fuzzy Set-Valued Random Variables....Pages 221-234
Front Matter....Pages 251-251
Mathematical Foundations for the Applications of Set-Valued Random Variables....Pages 253-293
Applications to Imaging....Pages 295-354
Applications to Data Processing....Pages 355-372
Back Matter....Pages 387-394

---

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 H?lmander'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.
Content:
Front Matter....Pages i-xii
Front Matter....Pages xiii-xiii
The Space of Set-Valued Random Variables....Pages 1-39
The Aumann Integral and the Conditional Expectation of a Set-Valued Random Variable....Pages 41-85
Strong Laws of Large Numbers and Central Limit Theorems for Set-Valued Random Variables....Pages 87-115
Convergence Theorems for Set-Valued Martingales....Pages 117-160
Fuzzy Set-Valued Random Variables....Pages 161-190
Convergence Theorems for Fuzzy Set-Valued Random Variables....Pages 191-219
Convergences in the Graphical Sense for Fuzzy Set-Valued Random Variables....Pages 221-234
Front Matter....Pages 251-251
Mathematical Foundations for the Applications of Set-Valued Random Variables....Pages 253-293
Applications to Imaging....Pages 295-354
Applications to Data Processing....Pages 355-372
Back Matter....Pages 387-394
....