Some Results on Vandermonde Matrices with an Application to Time Series Analysis

In this paper we study Stein equations where the coefficient matrices are in companion form. Solutions to such equations are relatively easy to compute as soon as one knows how to invert a Vandermonde matrix (in the generic case where all eigenvalues have multiplicity one) or a confluent Vandermonde matrix (in the general case). As an application we present a way to compute the Fisher information matrix of an ARMA process. The computation is based on the fact that this matrix can be decomposed into blocks where each block satisfies a certain Stein equation.