Tangential CLT on Stratified Manifolds and Nonstationary Gaussian Process Modeling

Abstract

This dissertation addresses three interconnected problems in the theory and simulation of Gaussian processes, with a particular emphasis on geometric and probabilistic methods.First, we establish a convergence rate for the Central Limit Theorem (CLT) in the setting of tangential random fields on smoothly stratified manifolds. In many statistical applications, data lie in non-planar spaces, making the usual Euclidean CLT inapplicable. To address this, we map the sample points to a conical tangent space. A qualitative version of the CLT for tangent random fields on conical tangent spaces was proved recently.We adapt Stein’s method to prove that a suitably scaled sum of these tangential random fields converges to a Gaussian random field at a rate (under some assumptions) on the order of O(n^{-1/22 + \epsilon} \sqrt{log n}). Our analysis uses Gaussian integration by parts and spectral decompositions of covariance operators to manage the infinite-dimensional nature of these tangential fields.Second, we study the concentration of local maxima in conditional, nonstationary Gaussian processes, a framework widely used in Gaussian Process Regression (GPR). While existing results focus on the stationary case, real-world data often show non-stationarity. We derive a concentration inequality that gives a practical criterion for determining how far GPR predictions remain reliable, effectively identifying a threshold distance beyond which predictive performance declines.Third, we propose a new algorithm for approximating the excursion probabilities of Gaussian processes, extending to the conditional case. By combining Gaussian Process Regression with a local application of the Expected Euler Characteristic (EEC) method, we obtain computationally stable estimates of high-level exceedances. This local approximation bypasses numerical instabilities that can arise when directly applying classical excursion probability formulas to non-stationary processes.