The proposed research centers on stochastic inequalities, including correlation inequalities, monotonicity of certain expectations with respect to parameters of distributions, inequalities which quantify dependence structures, and related notions of multivariate total positivity. The emphasis is on multivariate distributions and processes. The proposal also refers to dependence concepts arising in sequential analysis and the study of optimal stopping value comparisons under various dependence structures, and prophet inequalities which provide upper bounds on optimal stopping values. Limit theorems for various types of dependence structures will be studied, with initial emphasis on approximation rates. The resulting improved convergence rates may be applicable to the study of asymptotics of multivariate statistics which can be expressed as sums of dependent random variables and various random counts arising in problems of combinatorial nature. The assumptions of statistical independence of sampled measurements is basic to most methods of data analysis, and is ubiquitous in problems of stochastic optimization. However, one encounters many situations in which this assumption is violated. The proposed research centers on the study of certain aspects of samples which consist of observations which are not independent.
