Project/Area Number  09440085 
Research Category 
GrantinAid for Scientific Research (B).

Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  Saga University 
Principal Investigator 
OGURA Yukio Saga University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (00037847)

CoInvestigator(Kenkyūbuntansha) 
半田 賢司 佐賀大学, 理工学部, 助教授 (10238214)
MITOMA Itaru Saga University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (40112289)
SUGITA Hiroshi Kyushu University, Graduate School of Mathematics, Associate Professor, 大学院・数理学研究科, 助教授 (50192125)
NAGAI Hideo Osaka University, Graduate School of Fundamental Engineering, Professor, 基礎工学部, 教授 (70110848)
KUSUOKA Shigeo University of Tokyo, Graduate School of Mathematical Science, Professor, 大学院・数理科学研究科, 教授 (00114463)
LI Shoumei Saga University, Faculty of Science and Engineering, Associate Professor, 理工学部, 助教授 (00304874)

Project Fiscal Year 
1997 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥8,300,000 (Direct Cost : ¥8,300,000)
Fiscal Year 1999 : ¥2,600,000 (Direct Cost : ¥2,600,000)
Fiscal Year 1998 : ¥2,700,000 (Direct Cost : ¥2,700,000)
Fiscal Year 1997 : ¥3,000,000 (Direct Cost : ¥3,000,000)

Keywords  fuzzy analysis / fuzzy martingale / random set / random fuzzy set / graph onvergence / separability for convergence / law of large numbers / optional stopping theorem / ファジィ解析 / ファジィマルチンゲール / ランダム集合 / ランダムファジィ集合 / グラフ収束 / 収束の可分性 / 大数の法則 / 任意抽出定理 / デファジィフィケイション / ファジィマルチシゲール / ファジィマルチンゲル / KudoAumann積分 / 優マルチンゲル / 劣マルチンゲル 
Research Abstract 
Former study on limit theorems for random fuzzy sets was mainly for the case of compact level sets. This is due to its method which is based on the embedding theorem of convex compact sets into the space of bounded continuous functions on the unit closed ball in the dual space. The object of this project is to obtain limit theorems for wider class of random fuzzy sets, extending the former study. There are two cases where the compactness condition fails. One is the case of bounded but non compact sets, which occurs in infinite dimension cases. The other is that of unbounded sets, which occurs even in finite dimensional cases. In the former case, the first problem one may encounter is the closedness of the Aumann integrals. This integral is fundamental in the study of random sets or random fuzzy sets, and although a counter example for the closeness was known, a good sufficient condition was not known. In this project, we obtained that the Aumann integral is closed if the underlying spac
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e is a reflexive Banach space. We next dealt with the concept of convergence, which is significant in both former and latter cases. In most of former works, the fuzzy sets were treated as simple collections of level sets and the topologies sometimes differs from intuition. In this project, we introduced a new concept called graph convergence which fits intuition more, and improved various convergence theorems. Finally, we also obtained separability theorem for graph convergence, which ensures that one have only to check the convergence of level sets for countably many levels for a proof of convergence of random fuzzy sets. Based on the former theoretical advance, we obtained or improved various limit theorems on random sets and/or random fuzzy sets, which covers law of large numbers, limit theorems for martingale and smartingales, regular representation theorem for martingales, optional sampling theorems, approximation of random fuzzy sets and representation theorem of Gaussian random fuzzy sets. Less
