A UNIFIED APPROACH TO THE STUDY OF SUMS, PRODUCTS, TIME-AGGREGATION AND OTHER FUNCTIONS OF ARMA PROCESSES

Abstract. Conditions under which sums, products and time-aggregation of ARMA processes follow ARMA models are derived from a single theorem. This characterizes these processes in terms of difference equations satisfied by their autocovariance function. From this we obtain necessary and sufficient conditions for a function of a Gaussian ARMA process and the product of two possibly dependent Gaussian ARMA processes to be ARMA. We show that the sum and product of two ARMA processes related by a Box and Jenkins transfer function model belong to the ARMA family.