Study on convergence theorems and modeling for fuzzy random sets
Project/Area Number  10640137 
Research Category 
GrantinAid for Scientific Research (C).

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

Research Institution  Science University of Tokyo 
Principal Investigator 
INOUE Hiroshi Science University of Tokyo, Management, Professor, 経営学部, 教授 (90096694)

CoInvestigator(Kenkyūbuntansha) 
SHI Jiunming Science University of Tokyo, Management, Instructor, 経営学部, 助手 (70287465)

Project Fiscal Year 
1998 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥2,000,000 (Direct Cost : ¥2,000,000)
Fiscal Year 1999 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 1998 : ¥1,300,000 (Direct Cost : ¥1,300,000)

Keywords  random set / fuzzy random set / convergence theorem / dependency / exchangeability / weight sums / average behavior / KuratowskiMosco convergence / ランダム集合 / ファジィランダム集合 / 収束定理 / 従属性 / 交換可能性 / 重み付き総和 / 平均挙動 / KuratowskiMosco収束 / 大域最適化問題 / 変換可能性 / 負の従属性 / 重み付け総和 
Research Abstract 
In many statistical analysis some kind of dependency of random sets or fuzzy random sets may be required, and exchangeability as an alternative to the random sample with independent, identically distributed random sets gives the study of convergence theorems. For proving convergence theorems most of the authors use reverse martingale techniques. In this study we focus on the behavior of weighted sums for fuzzy random sets, in particular, arrays of fuzzy random sets which is an extension of laws of large numbers, and some other conditions including the extended Hausdorff metric is required to prove the theorems. Further, KuratowskiMosco convergence which is weaker than Hausdorff sense is interesting and can be used with exchangeability in constructing convergence theorems. For modeling of process of behaviors of fuzzy random sets there are some difficult in using the property of exchangeability so that some small change of our initial plan for this study was needed. However, the properties of exchangeability are applicable to many situations such as allocation problems, optimization with fuzzy constraints etc. Thus in the future work it may be possible to construct convergence theorems for exchangeable fuzzy random sets under KuratowskiMosco sense of which unbounded case for random sets are most interesting and essential.

Report
(4results)
Research Output
(4results)