On the probabilistic structure of power threshold generalized arch stochastic processes

The aim of this paper is to develop a probabilistic study on a large and general class of conditionally heteroscedastic models, namely the δδ-TGARCH processes. For this class of processes we establish necessary and sufficient conditions of strict stationarity, ergodicity and existence of moments. A discussion on the weak stationarity of an associated vectorial process, moments and weak stationarity up to the order δδ of those processes is also presented. Finally, the minimal representation of a δδ-TGARCH process is obtained developing, in a unique way, the corresponding conditional moment of order δδ in terms of present and past observations.