1999 Fiscal Year Final Research Report Summary
Study on convergence theorems and modeling for fuzzy random sets
| Project/Area Number |
10640137
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Science University of Tokyo |
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Principal Investigator |
INOUE Hiroshi Science University of Tokyo, Management, Professor, 経営学部, 教授 (90096694)
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| Co-Investigator(Kenkyū-buntansha) |
SHI Jiunming Science University of Tokyo, Management, Instructor, 経営学部, 助手 (70287465)
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| Project Period (FY) |
1998 – 1999
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| Keywords | random set / fuzzy random set / convergence theorem / dependency / exchangeability / weight sums / average behavior / Kuratowski-Mosco convergence |
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| 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, Kuratowski-Mosco 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 Kuratowski-Mosco sense of which unbounded case for random sets are most interesting and essential.
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