Mathematical Sciences: Partial Sums, Partial Maxima and Random Sets

本研究将学习独立随机变量的部分和与部分极大值理论、Banach空间中的概率论以及随机集理论。要考察的三个主要问题中，将近似独立随机变量和的分布。重点将是获得广泛适用的结果。对于取实值或向量值或无穷维Banach空间中的值的随机变量。第二个问题是研究部分极大值及其与部分和的关系。第三个问题将是研究独立或相依随机集的联合及其在模拟增长现象中的作用。这些模型将纳入一种手段，根据一个地区受到污染的次数来改变该地区的活动。为了补充理论工作，将在图形化计算机上进行模拟。

This research will study the theory of partial sums and partial maxima of independent random variables, probability in Banach spaces and the theory of random sets. The of the three major questions to be viewed will to approximate the distributions of sums of independent random variables. The emphasis will be to obtain results which are widely applicable eg. for random variables taking real or vector values or values in infinite dimensional Banach spaces. The second problem will be to study the partial maxima and their relationship to partial sums. The third problem will be to study unions of independent or dependent random sets and their role in modelling growth phenomena. These models will incorporate a means to alter the activity in a region based on the number of times that a region has been contaminated. Simulations on graphical computer will be done to complement the theoretical work.