Existence of a Periodic and Seasonal INAR Process

A spectral criterion involving the model parameters is given for the existence and uniqueness of a periodically correlated and seasonal non-negative integer-valued autoregressive process. The structure of the mean and covariance functions of the periodically stationary distribution of the model is derived using its implicit state-space representation. Two infinite series representations for the process, the moving average, and the immigrant generation, are established. Based on the latter representation, a novel and parallelizable simulation method is proposed to generate the process.