Small deviations for Matérn random fields and processes in the Hilbert norm
type: Item
instance of: scholarly article
title: Small deviations for Matérn random fields and processes in the Hilbert norm
zbMATH Open document ID: 1193.60037
DOI: 10.1134/S1064562408050372
author: R. S. Pusev
published in: Doklady Mathematics
publication date: 2010-01-12T00:00:00Z
review text: 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 \(X^{(2)}\).
Mathematics Subject Classification ID: 60F10
zbMATH DE Number: 5656101
zbMATH Keywords: Matérn process
zbMATH Keywords: Matérn fields
zbMATH Keywords: Hilbert norm
reviewed by: Renata Macaitienė
MaRDI profile type: Publication
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full work available at URL: s1064562408050372
OpenAlex ID: W1969286992
description: scientific article; zbMATH DE number 5656101