On the density functions of integrals of Gaussian random fields (Q2837753)

Let \(f:=\{f(t):t\in T\}\) be a Gaussian random field defined on a compact set \(T\subset \mathbb{R}^d\). For a finite measure \(\nu\) on \(T\) and a positive function \(\sigma(t)\), \(t\in T\), such that \(\sigma_T:=\sup_{t\in T}\sigma(t)<\infty\), set NEWLINE\[NEWLINE H_f:= \log\left(\int_Te^{\sigma(t)f(t)}d\nu(t)\right),\;\;F(a):=\mathrm{P}(H_f\leq a), NEWLINE\]NEWLINE where \(a\in\mathbb{R}\). The authors prove that if the process \(f\) is a.s. continuous with zero mean and unit variance, the boundary of \(T\) is a piecewise smooth manifold, the measure \(\nu\) is positive and \(\nu(T)=1\), then \(F'(a)\) exists almost everywhere and \( \limsup_{a\to\infty} \sigma_T^2a^{-1}e^{a^2/2\sigma^2_T}F'(a) \leq 1. \) Under additional conditions imposed on the covariance function of the process \(f\), the result formulated above is improved. An exact approximation of \(F'(a)\) as \(a\to \infty\) is given when \(f(t)\) is three times continuously differentiable. The basic technique is to use the Karhunen-Loève expansion \(f(t)=\sum_{i=1}^{\infty}x_i\phi_i(t)\) to establish the corresponding bounds for \(f_N(t)=\sum_{i=1}^{N}x_i\phi_i(t)\) and to let \(N\to \infty\). The Tsirel'son bound [\textit{V. S. Cirel'son}, Theory Probab. Appl. 20, 847--856 (1975); translation from Teor. Veroyatn. Primen. 20, 865--873 (1975; Zbl 0348.60050)] of the density of \(\sup_{t\in T}f(t)\) is employed as well. However, the analysis is more complicated since \(H_f\) is not a sublinear function of \(f\) which is a crucial condition in the mentioned paper.