Small deviations for Matérn random fields and processes in the Hilbert norm
The Matérn process \(X^{(\nu )}(t)\) (\(\nu>{1\over 2}\), \(t\in[0,1]\)) is defined as the Gaussian process with mean zero and covariance function \[ G_{X^{(\nu)}}(s, t)={2^{3/2-\nu}\over \Gamma(\nu-1/2)}|s-t|^{\nu-1/2}K_{\nu-1/2}(|s-t|), \quad s, t\in [0,1], \] where \(K_\alpha\) is the modified Bessel function with index \(\alpha\), and the Matérn fields are tensor products of Matérn processes. In this paper, the author investigates the asymptotics as \(\varepsilon\to 0\) of the probability \(P\{\|X^{(\nu)}\|_\rho\leq \varepsilon\}\), where \[ \|X\|_\rho=\left(\int_\Omega X^2(x)\rho(x)dx\right)^{1/2} \] for some random function \(X(x)\) and a nonnegative integrable function \(\rho(x)\), \(x\in\Omega\) (\(\Omega\) is a bounded domain in \(\mathbb R^d\)). More precisely, first, he gives the logarithmic asymptotics for the probabilities of small deviations for Matérn processes and fields. For example, if \(\rho\) is a nonnegative integrable function on \([0, 1]\), then \[ \begin{multlined} \lim_{\varepsilon\to 0}\varepsilon^{2/(2\nu-1)}\cdot \ln P\{\|X^{(\nu)}\|_\rho\leq \varepsilon\}\\ =-\left({2\Gamma(\nu)\over \pi^{2\nu-1/2}\Gamma(\nu-1/2)}\right)^{1/(2\nu-1)} {2\nu-1\over 2} \left({\pi J_{2\nu}\over 2\nu\sin{\pi\over 2\nu}}\right)^{2\nu/(2\nu-1)}, \end{multlined} \] where \(J_h=\int_{\Omega}\rho(x)^{1/h}dx\). Moreover, he obtains an exact asymptotics with explicitly given constants for small deviation probability of the Matérn process \(X^{(2)}\).
scientific article; zbMATH DE number 5656101