Convergence Rates for Realizations of Gaussian Random Variables

TL;DR

This paper provides non-asymptotic convergence rates for approximating Gaussian processes in Banach spaces using finitely many linear observations, with implications for statistics and machine learning.

Contribution

It derives new finite-sample error bounds for Gaussian process approximation in Banach spaces without spectral methods.

Findings

Error bounds relate covariance approximation to process convergence

Approach uses finite linear observations instead of spectral methods

Results applicable to nonparametric statistics and Bayesian inference

Abstract

This paper investigates the approximation of Gaussian random variables in Banach spaces, focusing on the high-probability bounds for the approximation of Gaussian random variables using finitely many observations. We derive non-asymptotic error bounds for the approximation of a Gaussian process $ X $ by its conditional expectation, given finitely many linear functionals. Specifically, we quantify the difference between the covariance of $ X $ and its finite-dimensional approximation, establishing a direct relationship between the quality of the covariance approximation and the convergence of the process in the Banach space norm. Our approach avoids the reliance on spectral methods or eigenfunction expansions commonly used in Hilbert space settings, and instead uses finite, linear observations. This makes our result particularly suitable for practical applications in nonparametric statistics, machine learning, and Bayesian inference.