Mathematical Sciences: Probability, Statistics and Functional Analysis

关于可微统计泛函和算子的研究将继续进行。设P是概率度量。设P(N)是与分布P无关的n个观测值的质量1/n之和的经验概率测度，T是定义在这种概率测度上的一个非线性算子。对于合适的T，人们寻找这样的可微性：TP(N)=TP+A P(N)-P+r(N)，其中A是线性算子，r(N)是在适当意义上比|P(N)-P|小数量级的余数。该项目将调查在其上定义测量的一维和多维空间。对于独立的、同分布的观测值的大数和，概率极限定理，如大数定律和中心极限定理，显示了渐近行为。在最近的工作中，这些事实正在被扩展到适当的非线性函数，而不是和。在本项目中，观测的样本空间正在从一维空间扩展到多维空间。

Research will continue on differentiable statistical functionals and operators. Let P be a probability measure. Let P(n) be the empirical probability measure which is the sum of masses 1/n at each of n observations independent with distribution P. Let T be a nonlinear operator defined on such probability measures. For suitable T, one is looking for differentiability properties such that T P(n) = T P + A P(n)-P + r(n) where A is a linear operator, and r(n) is a remainder of smaller order of magnitude than |P(n)-P| in a suitable sense. The project will investigate both one-dimensional and multidimensional spaces on which the measures are defined. For sums of large numbers of independent, identically distributed observations, probability limit theorems such as the law of large numbers and central limit theorems show the asymptotic behavior. In recent work, such facts are being extended to suitable non- linear functions in place of sums. In the current project, the sample spaces of observations are being extended from one- dimensional to multidimensional spaces.