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

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THEORY AND DECISION LIBRARY General Editors: W. Leinfellner (Vienna) and G. Eberlein (Munich) Series A: Philosophy and Methodology of the Social Sciences Series B: Mathematical and Statistical Methods Series C: Game Theory, Mathematical Programming and Operations Research

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LIMIT THEOREMS AND APPLICATIONS OF SET-VALUED AND FUZZY SET-VALUED RANDOM VARIABLES by

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

YUKIOOGURA Saga University, Saga, Japan

and

VLADIK KREINOVICH University ofTexas El Paso, El Paso, U.S.A.

SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-94-015-9932-0 (eBook) ISBN 978-90-481-6139-3 DOl 10.1007/978-94-015-9932-0

Printed on acid-free paper

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Contents

Preface

IX

Part I Limit Theorems of Set-Valued and Fuzzy SetValued Random Variables 1. THE SPACE OF SET-VALUED RANDOM VARIABLES Hyperspaces of a Banach Space 1 The Hausdorff Metric in Hyperspaces and An Em1.1

bedding Theorem Convergences in Hyperspaces Set-Valued Random Variables The Set of Integrable Selections The Spaces of Integrably Bounded Set-Valued Random Variables

1.2

2 3 4