Almost Sure Uniform Convergence of a Random Gaussian Field Conditioned on a Large Linear Form to a Non Random Profile

TL;DR

This paper proves that a Gaussian field conditioned on a large linear functional converges almost surely and uniformly to a deterministic profile, extending previous results from weaker convergence modes.

Contribution

It establishes almost sure uniform convergence of Gaussian fields conditioned on large linear functionals, improving upon earlier $L^2$-convergence results.

Findings

Almost sure uniform convergence to a deterministic profile

Convergence depends only on covariance and linear functional

Improves previous $L^2$-convergence results

Abstract

We investigate the realizations of a random Gaussian field on a finite domain of ${\mathbb R}^d$ in the limit where a given linear functional of the field is large. We prove that if its variance is bounded, the field converges uniformly and almost surely to a non random profile depending only on the covariance and the considered linear functional of the field. This is a significant improvement of the weaker $L^2$-convergence in probability previously obtained in the case of conditioning on a large quadratic functional.