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One-step prediction for \(P_ n\)-weakly stationary processes. (English) Zbl 0764.60070

Summary: The one-step prediction problem is studied in the context of \(P_ n\)- weakly stationary stochastic processes \((X_ n)_{n\in\mathbb{N}_ 0}\), where \((P_ n(x))_{n\in\mathbb{N}_ 0}\) is an orthogonal polynomial sequence defining a polynomial hypergroup on \(\mathbb{N}_ 0\). 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 \((X_ n)_{n\in\mathbb{N}_ 0}\) being asymptotic \(P_ n\)-deterministic. For Jacobi polynomials \(P_ n(x)\) the problem of \((X_ n)_{n\in\mathbb{N}_ 0}\) being asymptotic \(P_ n\)-deterministic is completely solved.

MSC:

60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60G25 Prediction theory (aspects of stochastic processes)
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
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