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A uniform metric estimate of the accuracy of modelling of Gaussian random fields on a sphere. (English. Ukrainian original) Zbl 0955.60056
Theory Probab. Math. Stat. 60, 165-174 (2000); translation from Teor. Jmovirn. Mat. Stat. 60, 149-157 (1999).
The author considers an isotropic Gaussian random field on a sphere \(S\) in \(R^n\) with the spectral representation \(\zeta(x)=\sum_{m=0}^\infty\sum_{l=1}^{h(m,n)}\sigma_m\xi_m^l S_m^l(x),\) where \(S_m^l(x)\) are spherical harmonics, \(\xi_m^l\) are i.i.d. standard Gaussian random variables. This field is approximated by the field \(\zeta_N(x)=\sum_{m=0}^N\sum_{l=1}^{h(m,n)}\sigma_m\xi_m^l S_m^l(x).\) Estimates of the approximation accuracy of the form \[ P\left\{\sup_{x\in S}\mid \xi(x)-\xi_N(x)|\geq\varepsilon\right\}\leq\delta \] are obtained.
60G60 Random fields
60H30 Applications of stochastic analysis (to PDEs, etc.)