## The problem of the interpolation of a random field that is homogeneous and isotropic in space and stationary in time from observations on an infinite cylindrical surface. I.(English. Ukrainian original)Zbl 1232.60040

Theory Probab. Math. Stat. 82, 139-148 (2011); translation from Teor. Jmovirn. Mat. Stat. No. 82, 137-145.
Summary: We solve the problem of interpolation of a homogeneous, space-isotropic, and time-stationary random field in the case of a finite sample observed on an infinite cylindrical surface. An explicit formula for the corresponding mean square error of interpolation is obtained. The asymptotic behavior of the error is studied as the number of observations is increasing. Conditions for the error-free approximation are given. For the problem of the error-free approximation, we find an optimal distribution of the weight coefficients in the interpolation formula.

### MSC:

 60G60 Random fields 62M20 Inference from stochastic processes and prediction
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### References:

 [1] Спектрал$$^{\приме}$$ная теория случайных полей, ”Вища Школа”, Киев, 1980 (Руссиан). [2] Yu. D. Popov, Some problems of linear extrapolation for homogeneous, space isotropic, and time stationary random fields, Dopovidi Akad. Nauk Ukrain. RSR Ser. A (1968), no. 12, 166-177. (Russian) [3] M. V. Kartashov, Finite-dimensional interpolation of a random field on the plane, Teor. Ĭmovīr. Mat. Stat. 51 (1994), 53 – 61 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 51 (1995), 53 – 61 (1996). · Zbl 0939.60041 [4] N. Semenovs$$^{\prime}$$ka, A problem of the interpolation of a homogeneous and isotropic random field, Teor. Ĭmovīr. Mat. Stat. 74 (2006), 150 – 158 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 74 (2007), 171 – 179.
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