Problems at the Interface of Probability and Convex Geometry: Random Projections and Constrained Processes


Convex sets in high-dimensional linear spaces are classical objects of study that have long enjoyed rich connections with probability theory. Interest in these connections has been further fueled by modern applications in statistics and engineering. In this thesis, we study two problems at the interface of convex geometry and probability theory: first, we consider the large deviation behavior of random projections of high-dimensional probability measures, as a complement to the central limit theorem for log-concave probability measures; secondly, we define a novel class of stochastic processes that includes those constrained to lie in a convex domain, and expose the crucial role played by the geometry induced by an associated norm on Euclidean space. In the large deviation setting, we first establish a large deviation principle (LDP) for one-dimensional projections of $n$-dimensional product measures as $n$ goes to infinity, and demonstrate how, given this geometric perspective, the classical Cramer's theorem for sums of independent and identically distributed random variables is in a sense "atypical". We then go beyond product measures and establish an LDP for the sequence of one-dimensional projections of random vectors drawn uniformly from an $n$-dimensional $\ell^p$ ball, and observe stark changes in large deviation behavior as $p$ varies. We consider both "quenched" LDPs (where we fix a particular sequence of projection directions) and "annealed" LDPs (where we incorporate randomness of the projection directions as contributors to large deviations), and establish a variational principle that relates the associated rate functions. Along the way, as a result of independent interest, we strengthen an existing LDP for the empirical measure of coordinates drawn from an $\ell^p$ sphere, and establish a related conditional limit theorem. As a final contribution in the large deviation setting, we extend the aforementioned one-dimensional LDP to $k_n$-dimensional projections of random vectors, for $k_n > 1$ (including the case where the lower dimension $k_n$ grows with the ambient dimension $n$). Furthermore, we generalize beyond $\ell^p$ balls to a larger class of sequences of random vectors satisfying a particular norm condition. In the second part of this thesis, we show that stochastic processes with a wide range of apparent degeneracies (such as constraint within a domain, singular drift, or discontinuous dynamics) may fall within a common framework. The formulation of our common framework relies on so-called accretive operators, which are defined with respect to a normed space. One of our primary goals is to expose the crucial role played by the geometry of the associated normed space.
Thesis (Ph. D.)--Brown University, 2017

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Kim, Steven Soon, "Problems at the Interface of Probability and Convex Geometry: Random Projections and Constrained Processes" (2017). Applied Mathematics Theses and Dissertations. Brown Digital Repository. Brown University Library.