Mathematical Sciences: Extreme Values and Stochastic Models

提出的各种问题分为三大类：极值理论、点过程和时间序列。极值序列或记录时间序列构成点过程，因此点过程理论为分析相关问题提供了有力的工具。首席研究员在这一领域取得了一些成果，并将继续他的进一步研究。在证明移动平均过程函数的收敛性方面非常有用的点过程方法将被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，需要开发新的点处理方法。一组点的凸包概念导致了一种定义多元极值的可能方法。因此多元序统计量依赖于凸包的分布理论。

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.