Asymptotic stationarity of Markov-switching time-frequency GARCH processes

Conditions for asymptotic wide-sense stationarity of generalized autoregressive conditional heteroscedasticity (GARCH) processes with regime-switching are necessary for ensuring finite second moments. In this paper, we introduce a stationarity analysis for the Markov-switching time-frequency GARCH (MSTF-GARCH) model which has been recently introduced for modeling nonstationary signals in the time-frequency domain. We obtain a recursive vector form for the unconditional variance by using a representative matrix which is constructed from both the GARCH parameters of each regime, and the regimes' transition probabilities. We show that constraining the spectral radius of that matrix to be less than one is both necessary and sufficient for asymptotic wide-sense stationarity. The generated matrix is also shown to be useful for deriving the asymptotic covariance matrix of the process.