Stationarity of a family of GARCH processes

P>A natural generalization of the first-order GARCH processes family introduced in 1999 to allow for higher-order past errors and conditional variances on the current conditional variance equation is proposed. This new family of GARCH processes includes many well-known GARCH processes. A sufficient and necessary condition for the existence of stationary solution of the family of GARCH processes is given. In particular, we proved the stationarity of the so-called family of IGARCH processes.