Limit theorems for U- and V-statistics for dependent random variables and their applications

应用Banach空间值I.I.D.的极限定理，用新的技巧研究了相依随机变量对称统计量(U-统计量和V-统计量)的极限定理.随机变量。通常众所周知的霍夫丁对对称学派的分解不能用于具有非退化核的对称统计量。此外，我们还尝试了一些对满足某些混合条件或强相依的相依随机变量的模拟。在一定的相依条件下，该模拟可以应用于数学金融领域。本文重点研究了伪随机数的分布，并讨论了当基础随机变量的分布不同于正态分布时，Euler-Maruyama近似的误差估计。在此基础上，得到了多维区域上具有边界条件的随机微分方程(所谓的Skorohod SDE)的一些结果。利用罚函数法定义了具有反射障碍的随机微分方程的近似解，并给出了近似解的误差估计。在这本笔记中，我们有两个目标。一种是既利用独立的正态分布的布朗运动的增量序列，又利用不服从正态分布的相依序列来定义近似解。另一种是，为了显示惩罚方法的优点，我们观察了具有软边界的布朗运动的样本路径，即布朗运动的任何路径不立即在边界上反射，而是根据离开边界的路径的强度被短时间吸收。

The author investigated limit theorems for symmetric statistics(U-statistics and V-statistics) for dependent random variables using new technique by applying limit theorems for Banach space valued i.i.d. random variables. Usually well known Hoeffding's decomposition for symmetric scholastics cannot be used for symmetric statistics with non-degenerate kernels. Furthermore we tried some simulations of such dependent random variables satisfying some mixing conditions or strongly dependence. The simulation can be applied to mathematical finance under some dependent conditions.The author focused on the distribution of pseudo-random numbers which are used for numerical application of such approximate solutions and consider the error estimation of the Euler-Maruyama approximation when the distribution of underlying random variables is different from the normal distribution. Furthermore some results for stochastic differential equations with boudary conditions on multi-dimensional domains(so-called Skorohod SDE) are obtained. We define an approximate solution of stochastic differential equation(SDE) with a reflecting barrier using the penalty method and estimate error of the approximate solution. In this note we have two aims. One is to define the approximate solution using not only a sequence of increments of Brownian motion which is independent and has normal distribution but also dependent sequence that does not obey normal distribution. Another one is, to show the advantage of the penalty method, we observe sample paths of Brownian motion with a soft boundary, i.e. any path of the Brownian motion does not reflect at the bundary immediately but is absorbed for a short period according to the strength of the path getting out of the boundary.