Tail approximations of integrals of Gaussian random fields

TL;DR

This paper derives asymptotic approximations for the tail probability of the integral of an exponential Gaussian random field over a domain, which is relevant for spatial and financial models, as the threshold tends to infinity.

Contribution

It introduces a two-step method to approximate tail probabilities of integrals of Gaussian fields, focusing on small domains and their relation to the entire set.

Findings

Provides asymptotic formulas for tail probabilities as threshold increases.

Shows the integral's tail probability can be approximated by a scaled version over a small domain.

Applicable to models in spatial processes, finance, and risk analysis.

Abstract

This paper develops asymptotic approximations of $P(\int_Te^{f(t)}\,dt>b)$ as $b\rightarrow\infty$ for a homogeneous smooth Gaussian random field, $f$, living on a compact $d$-dimensional Jordan measurable set $T$. The integral of an exponent of a Gaussian random field is an important random variable for many generic models in spatial point processes, portfolio risk analysis, asset pricing and so forth. The analysis technique consists of two steps: 1. evaluate the tail probability $P(\int_{\Xi}e^{f(t)}\,dt>b)$ over a small domain $\Xi$ depending on $b$, where $\operatorname {mes}(\Xi)\rightarrow0$ as $b\rightarrow \infty$ and $\operatorname {mes}(\cdot)$ is the Lebesgue measure; 2. with $\Xi$ appropriately chosen, we show that $P(\int_Te^{f(t)}\,dt>b)=(1+o(1))\operatorname{mes}(T)\times \operatorname{mes}^{-1}(\Xi)P(\int_{\Xi}e^{f(t)}\,dt>b)$.