Asymptotic properties of correlation bounds in functional spaces. I.

A bound is constructed for the correlation function of a uniform Gaussian random field in the scheme of series with respect to the many samples. Exact properties are established for the bound. It is proved that it is strongly consistent and asymptotically normal in the Hilbert space of functions which are square integrable on R^m with same weight function.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 2, pp. 179–187, February, 1991.