Study of fractional Brownian motion

1.分数布朗运动是具有自相似性的高斯过程。我们研究了赫斯特指数与其当地时间的尾概率的渐近行为之间的关系。针对这一问题，我们研究了协方差矩阵的行列式的无穷小阶。我们证明了它是指数递减的，并且找到了指数与赫斯特指数之间的关系。我们还将上述结果推广到更一般的高斯过程。当我们研究上述问题时，我们注意到指数型的Tauberian定理是必不可少的，并得到了一些关于这个问题的有用的定理。作为应用，研究了正同分布的独立随机变量和的分布函数。研究了自相似过程的一些性质。众所周知，布朗运动在半直线上所花费的时间服从反正弦定律。我们试图为分数布朗运动找到类似的结果，但失败了。然而，我们得到了一个关于线性扩散的有趣的结果：我们发现了所谓的扩散速度度量和占据时间在半直线上的渐近行为之间的关系。

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.