2000 Fiscal Year Final Research Report Summary
Study of fractional Brownian motion
| Project/Area Number |
10640107
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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 | Ochanomizu University |
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Principal Investigator |
KASAHARA Yuji Faculty of Science, Ochanomizu University, Prof., 理学部, 教授 (60108975)
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| Co-Investigator(Kenkyū-buntansha) |
KOSUGI Nobuko Faculty of Science, Ochanomizu University, Assistant, 理学部, 助手 (20302995)
MAEJIMA Makoto Faculty of Sci.Eng., Keio Univ., Prof., 理工学部, 教授 (90051846)
KANEKO Akira Faculty of Science, Ochanomizu University, Prof., 理学部, 教授 (30011654)
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| Project Period (FY) |
1998 – 2000
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| Keywords | fractional Brownian motion / Brownian motion / arc-sine law / diffusion process / self-similar process |
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| Research Abstract |
1. Fractional Brownian motions are Gaussian processes having self-similarity. We studied the relationship between the Hearst index and the asymptotic behavior of the tail probabilities of their local times. In relation to this problem we studied the order of infinitesimal of the determinant of the covariance matrix as the dimension goes to infinity. We proved that it decreases exponentially and we found the relation between the exponent and the Hearst index. We also generalized the above results for more general Gaussian processes.
2. When we studied the above problem we noticed that Tauberian theorems of exponential types are essential, and we obtained some useful theorems on this subject. As an application we studied the distribution function of the sums of independent random variables which are positive and identically distributed.
3. We studied on some properties of self-similar processes.
4. It is well known that the amount of time that a Brownian motion spends on the half line obeys the arc-sine law. We tried to find similar results for fractional Brownian motions but failed. Instead, however, we obtained an interesting result for linear diffusions : We found a relation between the so-called speed measure of the diffusion and the asymptotic behavior of the occupation time on the half line.
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