Limit theorems and applications of set-valued and fuzzy set-valued random variables /

Limit theorems and applications of set-valued and fuzzy set-valued random variables /

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Bibliographic Details
Author / Creator:Li, Shoumei.
Imprint:Dordrecht ; Boston : Kluwer Academic Publishers, c2002.
Description:xii, 391 p. ; 25 cm.
Language:English
Series:Theory and decision library. Series B, Mathematical and statistical methods ; v. 43
Subject:
Limit theorems (Probability theory)
Random variables.
Limit theorems (Probability theory)
Random variables.
Format: Print Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/4829036
Hidden Bibliographic Details
Other authors / contributors:Ogura, Yukio.
Kreinovich, Vladik.
ISBN:1402009186 (acid-free paper)
Notes:Includes bibliographical references and index.
Table of Contents:
  • Preface
  • Part I. Limit Theorems of Set-Valued and Fuzzy Set-Valued Random Variables
  • 1.. The Space of Set-Valued Random Variables
  • 1. Hyperspaces of a Banach Space
  • 1.1. The Hausdorff Metric in Hyperspaces and An Embedding Theorem
  • 1.2. Convergences in Hyperspaces
  • 2. Set-Valued Random Variables
  • 3. The Set of Integrable Selections
  • 4. The Spaces of Integrably Bounded Set-Valued Random Variables
  • 2.. The Aumann Integral and the Conditional Expectation of a Set-Valued Random Variable
  • 1. The Aumann Integral and Its Properties
  • 2. Sufficient Conditions for the Aumann Integrals To Be Closed
  • 3. Conditional Expectation and Its Properties
  • 4. Fatou's Lemmas and Lebesgue's Dominated Convergence Theorems
  • 5. Radon-Nikodym Theorems for Set-Valued Measures
  • 5.1. Set-Valued Measures
  • 5.2. Radon-Nikodym Theorems for Set-Valued Measures
  • 3.. Strong Laws of Large Numbers and Central Limit Theorems for Set-Valued Random Variables
  • 1. Limit Theorems for Set-Valued Random Variables in the Hausdorff Metric
  • 1.1. Strong Laws of Large Numbers in the Hausdorff Metric
  • 1.2. Central Limit Theorems
  • 2. Strong Laws of Large Numbers for Set-Valued Random Variables in the Kuratowski-Mosco Sense
  • 3. Gaussian Set-Valued Random Variables
  • 4. Appendix A of Subsection 3.1.2
  • 4.. Convergence Theorems for Set-Valued Martingales
  • 1. Set-Valued Martingales
  • 2. Representation Theorems for Closed Convex Set-Valued Martingales
  • 3. Convergence of Closed Convex Set-Valued Martingales in the Kuratowski-Mosco Sense
  • 4. Convergence of Closed Convex Set-Valued Submartingales and Supermartingales in the Kuratowski-Mosco Sense
  • 4.1. Convergence of Closed Convex Set-Valued Submartingales in the Kuratowski-Mosco Sense
  • 4.2. Convergence of Closed Convex Set-Valued Supermartingales in the Kuratowski-Mosco Sense
  • 5. Convergence of Closed Convex Set-Valued Supermartingales (Martingales) Whose Values May Be Unbounded
  • 6. Optional Sampling Theorems for Closed Convex Set-Valued Martingales
  • 7. Doob Decomposition of Set-Valued Submartingales
  • 5.. Fuzzy Set-Valued Random Variables
  • 1. Fuzzy Sets
  • 2. The Space of Fuzzy Set-Valued Random Variables
  • 3. Expectations of Fuzzy Set-Valued Random Variables
  • 4. Conditional Expectations of Fuzzy Random Sets
  • 5. The Radon-Nikodym Theorem for Fuzzy Set-Valued Measures
  • 6.. Convergence Theorems for Fuzzy Set-Valued Random Variables
  • 1. Embedding Theorems and Gaussian Fuzzy Random Sets
  • 1.1. Embedding Theorems
  • 1.2. Gaussian Fuzzy Set-Valued Random Variables
  • 2. Strong Laws of Large Numbers for Fuzzy Set-Valued Random Variables
  • 3. Central Limit Theorems for Fuzzy Set-Valued Random Variables
  • 4. Fuzzy Set-Valued Martingales
  • 7.. Convergences in the Graphical Sense for Fuzzy Set-Valued Random Variables
  • 1. Convergences in the Graphical Sense for Fuzzy Sets
  • 2. Separability for the Graphical Convergences and Applications to Strong Laws of Large Numbers
  • 3. Convergence in the Graphical Sense for Fuzzy Set-Valued Martingales and Smartingales
  • References for Part I
  • Part II. Practical Applications of Set-Valued Random Variables
  • 8.. Mathematical Foundations for the Applications of Set-Valued Random Variables
  • 1. How Can Limit Theorems Be Applied?
  • 2. Relevant Optimization Techniques
  • 2.1. Introduction: Optimization of Set Functions Is a Practically Important but Difficult Problem
  • 2.2. The Existing Methods of Optimizing Set Functions: Their Successes (In Brief) and the Territorial Division Problem as a Challenge
  • 2.3. A Differential Formalism for Set Functions
  • 2.4. First Application of the New Formalism: Territorial Division Problem
  • 2.5. Second Application of the New Formalism: Statistical Example--Excess Mass Method
  • 2.6. Further Directions, Related Results, and Open Problems
  • 3. Optimization Under Uncertainty and Related Symmetry Techniques
  • 3.1. Case Study: Selecting Zones in a Plane
  • 3.2. General Case
  • 9.. Applications to Imaging
  • 1. Applications to Astronomy
  • 2. Applications to Agriculture
  • 2.1. Detecting Trash in Ginned Cotton
  • 2.2. Classification of Insects in the Cotton Field
  • 3. Applications to Medicine
  • 3.1. Towards Foundations for Traditional Oriental Medicine
  • 3.2. Towards Optimal Pain Relief: Acupuncture and Spinal Cord Stimulation
  • 4. Applications to Mechanical Fractures
  • 4.1. Fault Shapes
  • 4.2. Best Sensor Locations for Detecting Shapes
  • 5. What Segments are the Best in Representing Contours?
  • 6. Searching For a 'Typical' Image
  • 6.1. Average Set
  • 6.2. Average Shape
  • 10.. Applications to Data Processing
  • 1. 1-D Case: Why Intervals? A Simple Limit Theorem
  • 2. 2-D Case: Candidate Sets for Complex Interval Arithmetic
  • 3. Multi-D Case: Why Ellipsoids?
  • 4. Conclusions
  • References for Part II
  • Index
Purchased from a Book Fund Established by The Class of 1927 bookplate
The John Crerar Library bookplate

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