Gaussian noise: prediction based on its value and N derivatives

An explicit form for the Slepian (1963) model of Gaussian noise X(t+τ) conditioned on the value X(t ) and any desired number N of its derivatives X\´( t), X"(t), . . ., X^(N)( t) at a given time t is obtained by using determinants for the Gram-Schmidt orthogonalisation of linear combinations of random variables, and then applying them to the least-mean-squared-error estimation of any zero-mean stationary random process. In this way a number of widely useful results are unified, clarified, simplified and extended. Finally, the application to a random process with a spectral density of Gaussian shape is studied