On probabilistic properties of nonlinear \(\text{ARMA}(p,q)\) models
The nonlinear ARMA model \(X_{n+1}=h(e_{n-q+1},\dots,e_n, X_{n-p+1},\dots,X_n)+e_{n+1}\) is considered where \(h\) is a measurable function and \(\{e_t\}\) is a sequence of i.i.d. random variables with \(E|e_n|<\infty\) such that \(e_n\) is independent of \(X_{n-1},\dots,X_0\). Sufficient conditions for stationarity and geometric ergodicity of \(\{X_n\}\) are derived using properties of the associated Markov chain. Technically, the method is based on an approximation of \(h({\mathbf x})\) by \(\sum_{i=1}^q \psi_i({\mathbf x})z_i+\sum_{i=1}^p \varphi_i({\mathbf x})x_i\) where \({\mathbf x}=(z_1,\dots,z_q,x_1,\dots, x_p)\).
scientific article; zbMATH DE number 1422608
scientific article