Ukrainian MathematicalJournal, Vot. 51, No. 1, 1999 LOCAL PROPERTIES OF GAUSSIAN RANDOM FIELDS ON COMPACT SYMMETRIC SPACES AND THEOREMS OF THE JACKSON-BERNSTEIN TYPE A. A. Malyarenko We consider local properties of sample functions of Gaussian isotropic random fields on compact Riemannian symmetric spaces M of rank 1. We give conditions under which the sample functions of a field almost surely possess logarithmic and power modulus of continuity. As a corollary, we prove a theorem of the Bemstein type for optimal approximations of functions of this sort by harmonic poly- nomials in the metric of the space L2(M). We use theorems of the Jackson-Bernstein-type to obtain sufficient conditions for the sample functions of a field to almost surely belong to the classes of func- tions associated with the Riesz and Ces~tro means. UDC 517.5 1. Introduction Let M = G / K be an N-dimensional compact symmetric space of rank one with group of motions G and stationary subgroup K [1]. Let ~(x) be a Gaussian field, continuous in mean square, on M with mean value zero and the correlation function B(x, y) = E~(x)~(y). The field ~(x) is called isotropic in a wide sense [2] if B(gx, gy) =B(x,y), g~ G, x, yeM. In Theorem 1, we give sufficient conditions for the continuity of the random field ~(x), which generalize the results of [3], where the corresponding assertions were proved for the case of an N-dimensional sphere, M = S N. Using the notions of a model of a random function and the kernel of a Gaussian measure in the space C (M) of continuous functions on M [4], we prove a theorem of the Bemstein type (Theorem 2): If the mean-square error of the optimal approximation of a function by harmonic polynomials whose degree does not exceed n is majorized by a numerical sequence a n that satisfies one of the conditions (9a)-(9c), then the function satisfies the correspond- ing condition from (1 la)-(1 lc). In Sec. 3, we use theorems of the Jackson-Bernstein type from [5] to obtain sufficient conditions for the sample functions of a random field ~ (x) to almost surely belong to the classes of continuous functions associated with the Riesz and Ces~tro means (Theorem 3). 2. Conditions of Continuity of a Random Field on a Compact Symmetric Space and a Bernstein-Type Theorem Let p (x, y) be the length of the geodesic that connects the points x, y ~ M and let L = SUPx,y ~M p(X, y) be the diameter of the space M. To simplify subsequent calculations, we fix the distance unit in the metric p so that L=7~. International Mathematical Center, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 1, pp. 60--68, January, 1999. Original article submitted September 17, 1996. 66 0041-5995/99/5101-0066 $22.00 9 1999 Kiuwer Academic/Plenum Publishers