Modeling Gaussian isotropic random fields on a sphere (Q2739950)

This paper deals with the problem of simulation of Gaussian isotropic random fields NEWLINE\[NEWLINE\xi(x)=\sum_{m=0}^{\infty}\sum_{l=1}^{h(m,n)} \xi_{m}^{l}s_{m}^{l}(x)NEWLINE\]NEWLINE on the sphere \(S_{n}\), where \(\xi_{m}^{l}\) is a sequence of independent Gaussian random variables, \(E\xi_{m}^{l}=0, E\xi_{m}^{l} \xi_{r}^{s}=b_{m}\delta_{m}^{r}\delta_{l}^{s}\) \((m=0,1,2,\ldots; l=1,\ldots,h(m,n))\), \(\delta_{m}^{r}\) is the Kronecker symbol, \(b_{m}>0\); \(s_{m}^{l}(x)=s_{m}^{l}(\theta_1,\ldots,\theta_{n-2},\varphi)\) is the orthonormal spherical harmonics of order \(m\); \(h(m,n)=(2m+n-2)(m+n-3)!/ ((n-2)!m!)\) is a quantity of such harmonics. The authors construct a random field \(\widehat \xi\) such that for given \(\varepsilon\) and \(\delta\) the inequality NEWLINE\[NEWLINEP\left\{\left(\int_{S_{n}}|\widehat\xi(x)- \xi(x)|^{p} dx\right)^{1/p}>\varepsilon\right\}<\deltaNEWLINE\]NEWLINE holds true.