Project/Area Number  10640107 
Research Category 
GrantinAid for Scientific Research (C).

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

Research Institution  Ochanomizu University 
Principal Investigator 
KASAHARA Yuji Faculty of Science, Ochanomizu University, Prof., 理学部, 教授 (60108975)

CoInvestigator(Kenkyūbuntansha) 
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)

Project Fiscal Year 
1998 – 2000

Project Status 
Completed(Fiscal Year 2000)

Budget Amount *help 
¥2,900,000 (Direct Cost : ¥2,900,000)
Fiscal Year 2000 : ¥1,000,000 (Direct Cost : ¥1,000,000)
Fiscal Year 1999 : ¥1,100,000 (Direct Cost : ¥1,100,000)
Fiscal Year 1998 : ¥800,000 (Direct Cost : ¥800,000)

Keywords  fractional Brownian motion / Brownian motion / arcsine law / diffusion process / selfsimilar process / ブラウン運動 / 逆正弦法則 / 拡散過程 / 自己相似確率過程 / self similarity / local time / tail probability / Gaussian process / Tauberian theorem / occupation time 
Research Abstract 
1. Fractional Brownian motions are Gaussian processes having selfsimilarity. 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 selfsimilar processes. 4. It is well known that the amount of time that a Brownian motion spends on the half line obeys the arcsine 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 socalled speed measure of the diffusion and the asymptotic behavior of the occupation time on the half line.
