Extremes of Gaussian non-stationary processes and maximal deviation of projection density estimates

TL;DR

This paper derives new asymptotic results for the maximum of non-stationary Gaussian processes and applies them to construct confidence bands for density estimates with polynomial accuracy.

Contribution

It provides a novel asymptotic representation for the distribution of the supremum of non-stationary Gaussian processes, improving understanding of maximal deviations in density estimation.

Findings

Asymptotic distribution with exponentially decaying remainder for Gaussian process supremum

Construction of polynomial-rate approximating laws for maximal deviations

Development of honest confidence bands for density estimates

Abstract

In this paper, we consider the distribution of the supremum of non-stationary Gaussian processes, and present a new theoretical result on the asymptotic behaviour of this distribution. Unlike previously known facts in this field, our main theorem yields the asymptotic representation of the corresponding distribution function with exponentially decaying remainder term. This result can be efficiently used for studying the projection density estimates, based, for instance, on Legendre polynomials. More precisely, we construct the sequence of accompanying laws, which approximates the distribution of maximal deviation of the considered estimates with polynomial rate. Moreover, we construct the confidence bands for densities, which are honest at polynomial rate to a broad class of densities.