Mathematical Sciences: Probability, Statistics and Functional Analysis

将继续研究可微统计泛函和算子。 P是一个概率测度。 设P（n）是经验概率测度，它是与分布P无关的n个观测值的质量之和1/n。设T是定义在这种概率测度上的非线性算子。 对于合适的T，人们正在寻找可微性，使得T P（n）= T P + A P（n）-P + r（n），其中A是线性算子，r（n）是比|P（n）-P|在适当的意义上。 该项目将调查一维和多维空间上的措施被定义。 对于大量独立同分布观测值的和，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，这样的事实被推广到合适的非线性函数，而不是求和. 在当前的项目中，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.