ON EXPANSION OF ESTIMATES FOR MEAN VALUE FUNCTIONS OF HOMOGENEOUS RANDOM FIELDS ON COMPACT HOMOGENEOUS SPACES

Abstract

§ 2. Expansions of estimates for mean value functions. Let T= {t} be an arbitrary compact homogeneous space, that is, a compact space which admits a transitive transformation group G = {g}. We denote by K= {k} a stationary subgroup of G, that is, a subgroup which leaves invariant a point to e T. Let {X(t), t ET} be a real-valued homogeneous random field on T having the mean value function m(t)= E{X(t)} , t ET(1) and satisfying the conditions : (C.1)El X(01 2} < cx) , for all t E T. (C.2) The covariance functions R(t, s) = E{(X(t)—m(t))(X(s)—m(s))} is a continuous positive definite function on T x T. (C.3) For all g E G, R(t, s) = R(gt, gs), t, s ET . We denote by L2(X) a Hilbert space consisting of all random variables which may be represented either as a finite linear combinations U= E ciX(t;), for some integer n, points t1, t2, ••• , t7, in T and scalars c1, C2, ••• or as a limit in quadratic mean of such finite linear combinations under the scalar product defined by (U, W) = E{U • W}. We denote by H(R) a reproducing kernel Hilbert space generated by the kernel