One-step prediction for P_n-weakly stationary processes

The one-step prediction problem is studied in the context of P_n-weakly stationary stochastic processes {Mathematical expression}, where {Mathematical expression} is an orthogonal polynomial sequence defining a polynomial hypergroup on {Mathematical expression}. This kind of stochastic processes appears when estimating the mean of classical weakly stationary processes. In particular, it is investigated whether these processes are asymptotic P_n-deterministic, i.e. the prediction mean-squared error tends to zero. Sufficient conditions on the covariance function or the spectral measure are given for {Mathematical expression} being asymptotic P_n-deterministic. For Jacobi polynomials P_n(x) the problem of {Mathematical expression} being asymptotic P_n-deterministic is completely solved.