Solution of the extrapolation problem for a class of stochastic processes (Q1118253)

Let Q(\(\lambda)\), \(Q_ 1(\lambda)\) and \(Q_ 2(\lambda)\) be polynomials of degree n, \(n_ 1\), and \(n_ 2\), respectively. Consider a stationary process x(t) with vanishing mean and with the spectral density \[ f(\lambda)=| Q_ 1(\lambda)+Q_ 2(\lambda)e^{i\lambda g}|^ 2| Q(\lambda)|^{-2} \] where \(g>0\). It is assumed that \(Q_ 1(\lambda)\) and \(Q_ 2(\lambda)\) as well as \(\bar Q_ 1(\lambda)\) and \(\bar Q_ 2(\lambda)\) have no common roots and that \(n_ 2\geq n_ 1\). The author derives a formula for extrapolation of the random variable \(x(t+\tau)\) when a realization of the process on an interval [t-T,t] is given.