Mathematical Sciences: Extreme Values and Stochastic Models

提出的各种问题分为三类，极值理论、点过程和时间序列。极值序列或记录时间序列形成点过程，点过程理论为分析相关问题提供了有力的工具。首席调查员已在这一领域取得了一些结果，并将继续进一步调查。在证明滑动平均过程泛函收敛方面非常有用的点过程方法将推广到连续时间过程。由于在许多相关情况下产生的随机点具有非常复杂的依赖结构，因此需要开发新的点过程方法。一组点的凸壳的思想导致了定义多元极值的一种可能的方法。因此，多元有序统计量依赖于凸壳的分布理论。

The various problems proposed fall in three categories, extreme value theory, point processes and time series. Sequences of extreme values or record times form a point process and hence the theory of point processes offers a powerful tool to analyze the related problems. The principal investigator has a number of results in this area and will continue his investigation further. The point processes methods that proved to be so useful in showing the convergence of functionals of moving average processes will be extended to the continuous time processes. New point processes methods need to be developed since the random points generated in many relevant situations have a very complicated dependence structure. The ideas of convex hulls of a set of points lead to one possible way of defining multivariate extremes. Thus the multivariate order statistics depend on the distribution theory of the convex hulls.