OPTIMAL DESIGNS FOR APPROXIMATING A STOCHASTIC-PROCESS WITH RESPECT TO A MINIMAX CRITERION

We study the problem of approximating a stochastic process Y = {Y(t):t is an element of T} with known and continuous covariance function R o n the basis of finitely many observations Y(t(1)),..., Y(t(n)). Depend ent on the knowledge about the mean function, we use different approxi mations (Y) over cap and measure their performance by the correspondin g maximum mean squared error sup(t is an element of T) E(Y(t) - (Y) ov er cap(t))(2). For a compact T subset of R(P), we prove sufficient con ditions for the existence of optimal designs. For the class of covaria nce functions on T-2 = [0,1](2) which satisfy generalized Sacks/Ylvisa ker regularity conditions of order zero or are of product type, we con struct sequences of designs for which the proposed approximations perf orm asymptotically optimal.