Tail approximations of integrals of Gaussian random fields (Q428142)

Let \(f=\{f(t), t\in T\}\) be a homogeneous centered Gaussian random field such that \(\operatorname{E}f^2(t)=1\) for \(t\in T\), where \(T\) is a \(d\)-dimensional Jordan-measurable compact subset of \(\mathbb{R}^d\). Assume that \(f\) is almost surely three times continuously differentiable. Moreover, let the Hessian matrix of the field covariance function at the origin be \(-I\), where \(I\) is the \(d\times d\) identity matrix. Consider NEWLINE\[NEWLINEI_{\sigma}(A):=\int_A \exp(\sigma f(t))\, dt,NEWLINE\]NEWLINE where \(\sigma >0\) and \(A\) is a Jordan-measurable subset of \(T\). For a given \(\sigma >0\) and \(b\) large enough, introduce \(u\) as the unique solution of the equation NEWLINE\[NEWLINE(2\pi/\sigma)^{d/2}u^{-d/2}e^{\sigma u}=b.NEWLINE\]NEWLINE Then NEWLINE\[NEWLINE \operatorname{P}(I_{\sigma}(T) >b) = (1+o(1))H\operatorname{mes}(T) u^{d-1}\exp(-u^2/2),\quad b\to \infty. NEWLINE\]NEWLINE Here, \(\operatorname{mes}\) is the Lebesgue measure and the explicit formula for \(H\) (depending on \(\sigma\) and other parameters) is provided. The study of the behavior of such integrals is important, e.g., for models involving spatial point processes and for portfolio risk analysis.NEWLINENEWLINEAt first, the author evaluates \(\operatorname{P}(I_{\sigma}(\Xi) >b)\) for a small domain \(\Xi\) depending on \(b\) where \(\operatorname{mes}(\Xi)\to 0\) as \(b\to \infty\). After that, it is demonstrated that for an appropriate choice of \(\Xi\) the following relation holds NEWLINE\[NEWLINE\operatorname{P}(I_{\sigma}(T) >b) =(1+o(1))\frac{\operatorname{mes}(T)}{\operatorname{mes}(\Xi)} \operatorname{P}( I_{\sigma}(\Xi) >b),\quad b\to \infty. NEWLINE\]