2006 Fiscal Year Final Research Report Summary
Study on limit theorems for fuzzy set-valued random variables
| Project/Area Number |
17540123
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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 | Saga University |
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Principal Investigator |
OGURA Yukio Saga University, Department of Mathematics, Professor, 理工学部, 教授 (00037847)
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| Co-Investigator(Kenkyū-buntansha) |
MATSUMOTO Hiroyuki Nagoya University, Graduate School of Information Science, Professor, 大学院・情報科学研究科, 教授 (00190538)
SHIOYA Takashi Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (90235507)
TOMISAKI Matsuyo Nara Women's University, Department of Mathematics, Professor, 理学部, 教授 (50093977)
MITOMA Itaru Saga University, Department of Mathematics, Professor, 理工学部, 教授 (40112289)
HANDA Kenji Saga University, Department of Mathematics, Associate Professor, 理工学部, 助教授 (10238214)
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| Project Period (FY) |
2005 – 2006
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| Keywords | Markov property / generalized diffusion processes / fuzzy set-valued random variables / large deviation principles / Fell-Matheron topology / Choquet theorem / Choquet capacity / law of large numbers |
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| Research Abstract |
1. We gave a necessary and sufficient condition for stochastic processes in a class of one-dimensional Markovian (not necessarily strong Markovian) processes with continuous paths to be bi-generalized diffusion process which were introduced by the head investigator. It is worth to note that the scale functions of the processes in our class are no more continuous in general and may miss the strong Markov property at the discontinuous points of scales.
2. We gave large deviation principles for sums of independent identically distributed fuzzy set-valued random variables with compact level sets. More precisely, we gave Cramer type large deviation principles for such sums with respect to the topologies induced by the mean convergence of order p, Levy's metric and graph convergence, and the relative topology of a countable direct product topology.
3. We extended a Choquet theorem on the construction of set-valued random variables to that on the construction of fuzzy set-valued random variables. That is, we showed existence of a fuzzy set-valued random variable associated with an alternating Choquet capacity of infinite order with respect to the graph topology which coincides the relative topology of product Fell-Matheron topology.
4. We gave a concrete metric compatible with the Fell-Matheron topology on the space of closed sets in a locally compact second countable Hausdorff space. We had to modify Hausdorff-Buseman which is given in the book "Random sets" by I. Molchanov introduced as a compatible metric.
5. We gave a strong law of large numbers for sums of independent fuzzy set-valued random variables, which are not necessarily indentically distributed.
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[Book] 確率論入門I2006
Author(s)
池田信行
Total Pages
323+x
Publisher
培風館
Description
「研究成果報告書概要(和文)」より
