One-step prediction forP n-weakly stationary processes

The one-step prediction problem is studied in the context ofP_n-weakly stationary stochastic processes, where is an orthogonal polynomial sequence defining a polynomial hypergroup on. This kind of stochastic processes appears when estimating the mean of classical weakly stationary processes. In particular, it is investigated whether these processes are asymptoticP_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 being asymptoticP_n-deterministic. For Jacobi polynomialsP_n(x) the problem of being asymptoticP_n-deterministic is completely solved.