Limit theorems for U-statistics with degenerate kernels and applications

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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