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

因随机变量的 U 和 V 统计量的极限定理及其应用

基本信息

批准号：
16540124
负责人：
KANAGAWA Shuya
金额：
$ 2.46万
依托单位：
Musashi Institute of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2007
项目状态：
已结题

项目摘要

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.

作者通过应用 Banach 空间独立同分布的极限定理，利用新技术研究了因随机变量的对称统计（U 统计和 V 统计）的极限定理。随机变量。通常，众所周知的对称经院学 Hoeffding 分解不能用于具有非简并核的对称统计。此外，我们尝试了对满足某些混合条件或强依赖性的此类相关随机变量进行一些模拟。该模拟在某些依赖条件下可以应用于数学金融。作者重点研究了用于此类近似解的数值应用的伪随机数的分布，并考虑了当底层分布时欧拉-丸山近似的误差估计随机变量与正态分布不同。此外，还得到了多维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。我们利用罚分法定义了带有反射障碍的随机微分方程（SDE）的近似解，并估计了近似解的误差。在这篇文章中，我们有两个目标。一是不仅使用独立且具有正态分布的布朗运动增量序列，而且使用不服从正态分布的相关序列来定义近似解。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都不会立即在边界处反射，而是根据强度在短时间内被吸收离开边界的路径。