Tail approximations of integrals of Gaussian random fields

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 $P(\int_Te^{f(t)}\,dt>b)=(1+o(1))\operatorname{mes}(T)\times \operatorname{mes}^{-1}(\Xi)P(\int_{\Xi}e^{f(t)}\,dt>b)$.