A note on the stationarity of a threshold first-order bilinear process (Q1305226)

Let \(\{e(t)\}\) be a sequence of i.i.d. absolutely continuous random variables. Consider the threshold first-order bilinear model \[ X(t)=aX(t-1)+(b_1 {\mathbf 1}_{\{X(t-1)<c\}} +b_2 {\mathbf 1}_{\{X(t-1)\geq c\}}) X(t-1)e(t-1)+e(t). \] The authors give sufficient conditions for the existence of a stationary process \(\{X(t)\}\) satisfying the model and of its finite moments of order \(p\). The proofs are based on some properties of the Markov chain \[ Z(t)=[a+(b_1 {\mathbf 1}_{\{X(t)<c\}} +b_2 {\mathbf 1}_{\{X(t)\geq c\}})e(t)]X(t). \]