High excursion probabilities for Gaussian fields on smooth manifolds

TL;DR

This paper derives precise asymptotic formulas for the probability that Gaussian fields on smooth manifolds exceed high thresholds, with applications to various stochastic processes.

Contribution

It provides the first exact asymptotic analysis of large excursion probabilities for Gaussian fields on smooth manifolds, extending previous results to more general settings.

Findings

Exact asymptotic behaviors of large excursion probabilities are established.

Applications to vector Gaussian processes, chi-square processes, and Bessel-related processes demonstrate the results.

Theoretical framework advances understanding of Gaussian fields on manifolds.

Abstract

Gaussian random fields on finite dimensional smooth manifolds whose variances reach their maximum value at smooth submanifolds are considered. Exact asymptotic behaviors of large excursion probabilities have been evaluated. Vector Gaussian processes, chi-square processes, Bessel process, fractional Bessel process, Bessel bridge are examples of application of this result.