Matrix representations of spectral coefficients of randomly sampled ARMA models

The paper is devoted to the spectrum of univariate randomly sampled autoregressive moving-average (ARMA) models. We determine precisely matrix representations for the spectrum numerator coefficients of the randomly sampled ARMA models. We give results when the poles of the initial ARMA model are simple and when they are multiple. We first prove the results when the probability generating function of the random sampling law is injective, then we precise the results when it is not injective.