2002 Fiscal Year Final Research Report Summary
determinisms of Set-Indexed Random Field and their Applications
| Project/Area Number |
13640145
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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 | Okayama University of Science |
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Principal Investigator |
TAKENAKA Shigeo Okayama University of Science, Dept. of Science, Professor, 理学部, 教授 (80022680)
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| Co-Investigator(Kenkyū-buntansha) |
KOJO Katsuya Niihama College of Technology, Mathematics and Science, Lecturer, 理数科, 講師 (10280471)
TAKASHIMA Keizo Okayama University of Science, Dept. of Science, Professor, 理学部, 教授 (00137184)
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| Project Period (FY) |
2001 – 2002
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| Keywords | stable random fields / random processes / determinsim / multi-dimensional time |
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| Research Abstract |
Random fields with parameter in subsets of a certain measure space are called Set-Indexed Random Fields and have a special property called determinism. A Gaussian system has 2-dimensional determinism, that is, the higher dimensional distributions of the system are completely determined by their own 1-dimensional and 2-dimensional (covariances) marginal distributions. On the contrary, there are some examples of non-Gaussian set-indexed random fields which are determined by their 3-dimensional marginals but not determined by 2-dimensional marginals.
Let fix a convex cone V in the n-dimensional Euclidean space. A curve L is called time-like if for any point x in L, the "future" of the line is included in V+x, and the "past" is included in -V+x. An n-parameter random field X is called multi-parameter additive process if the restriction X|L on any time-like curve L is an additive process, that is X|L has independent and uniform increments. In this research, we obtained a characterization of n-parameter additie processes as set-indexed processes.
Also, a characterization of 1-parameter processes valued in the space of time-like curves is obtained.
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