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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.
##### MSC:
 60G60 Random fields 60H30 Applications of stochastic analysis (to PDEs, etc.)