Mathematical Sciences: Probability, Statistics and Functional Analysis

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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.