Asymptotic properties of nonlinear autoregressive Markov processes with state-dependent switching
The paper studies asymptotic properties of nonlinear autoregressive (AR) processes with state-dependent switching \(({X_n},{Z_n})\), defined by \({X_n} = {f_{{Z_n}}}({X_{n - 1}}) + {\varepsilon _n}\), \({X_n} \in {\mathbb{R}^d}\) and \(P[{Z_n} = l\left| {{Z_{n - 1}} = k,{X_{n - 1}} = x] = {p_{kl}}(x)} \right.\), \(k,l \in \{ 1,2,\dots,{n_0}\} ,x \in {\mathbb{R}^d}\). The error process \(\{ {\varepsilon _n}\} \)is assumed to be a sequence of independent and identically distributed random vectors. Section 2 proves Feller and strong Feller continuity for AR processes. Section 3 proves its aperiodicity and investigates geometric ergodicity. A number of illustrative examples are provided in Sections 3-5.
scientific article; zbMATH DE number 5704004