Asymptotic Results for Stochastic Processes via New Projective Methods

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An important area of research in probability theory, with rich applications in statistics, is the asymptotic theory of stochastic processes, also known as large sample theory. This is a framework to assess properties of estimators and statistical tests, necessary for a high level of confidence in predictions. The large sample theory, first developed for independent variables, is considerably more difficult when data is dependent, that is, when quantities have dependencies that relate their values. The questions become even more challenging when the random variations are changing with time. The goal of this research is to address these questions by developing new methods to analyze large samples selected from classes of dependent structures arising in many applied fields, such as data from economics or engineering. The aim of this project is to develop new techniques for studying sequences and fields of dependent random variables, which will lead to sharp inequalities and general limit theorems for both stationary and non-stationary additive functionals of Markov chains and other dependent structures. The PI plans to develop a new type of approximation with martingales for Markov chains and fields based on the fruitful idea of conditioning with respect to both past and future of the process. The advantage of this new, surprising method is that no restrictions on the rate of convergence to zero of the dependence coefficients is required for obtaining various limit theorems. The PI also aims to develop operator perturbation theory for non-stationary Markov chains and functions of independent random elements. This approach will lead to new, deep local limit theorems for Markov chains, random fields, and the left random walk on the group of invertible d-dimensional real matrices.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.