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

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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.