Sampling From Stratified Spaces

Abstract

This dissertation studies Central Limit Theorems (CLTs) of Frechet means on stratified spaces. The broad goal of this work is to answer the following question: What information one should expect to get by sampling from a stratified space? In particular, this work explores relationships between geometry and different forms of CLT, namely classic, smeary and sticky. The work starts with explicit forms of CLTs for spaces of constant sectional curvature. As a consequence, we explain the effect of sectional curvature on the behaviors of Frechet means. We then give a sufficient condition for a smeary CLT to occur on spheres. In the later part, we propose a general form of CLT for star shaped Riemannian stratified spaces. The general CLT we propose is universal in the sense that it contains all of the different forms of aforementioned CLTs. The proposed CLT is verified on manifolds and on any flat 2-dimensional spaces with an isolated singularity.