Limit theorems for U-statistics with degenerate kernels and applications

利用Banach空间值i.i.d随机变量的极限定理，利用新技术研究了对称统计的极限定理。通常我们熟知的Hoeffding对称量的分解不能用于具有非简并核的对称统计。由于我们考虑u统计量的大偏差原理的一些应用，我们需要找到速率函数的具体值。然而，一般情况下很难得到它，因为速率函数包含概率测度的Radon-Nikodym导数。因此，我们研究了在Eudidean空间上定义的速率函数的另一种表示，不仅用于对称统计的数学分析，而且用于对称统计的数值分析。另一方面，对称统计量与伊藤随机微分方程（SDE）的近似解之间存在一定的关系。本文重点讨论了数值应用中的伪随机数的分布，进一步讨论了这类近似解，并考虑了当底层随机变量的分布不同于正态分布时，欧拉-丸山近似的误差估计。在此基础上，得到了多维域上具有边界条件的随机微分方程（Skorohod SDE）的一些结果。利用惩罚法和近似解的估计误差，定义了具有反射障的随机微分方程的近似解。在本文中，我们有两个目的。一种是用独立的、具有正态分布的布朗运动增量序列和不服从正态分布的依赖序列来定义近似解。另一种是，为了显示惩罚法的优点，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，而是根据出边界路径的强度在短时间内被吸收。少

The author investigated limit theorems for symmetric statistics 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. Since we consider some applications of large deviation principles for U-statistics we need to find the concrete value of the rate function. However in general it is difficult to obtain it because the rate function contains the Radon-Nikodym derivative of probability measures. Therefore we investigate another representation of the rate function defined on Eudidean space for not only mathematical but also numerical analysis of symmetric statistics.On the other hand there are some relations between symmetric statistics and approximate solutions of Ito's stochastic differential equation (SDE). The author focused on the distribution of pseudo-random numbers which are used for numerical applicati … More on 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 boundary conditions on mulii-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 boundary immediately but is absorbed for a short period according to the strength of the path getting out of the boundary. Less