Verifying irreducibility and continuity of a nonlinear time series (Q1305269)

Let \(\{e_t\}\) be i.i.d. random vectors. Define an \(m\)-dimensional nonlinear autoregressive time series \(\xi_t = \zeta(e_t;\xi_{t-1},\dots,\xi_{t-p})\), \(t\geq 1\), where \(\zeta\) is a Borel measurable mapping. Then \(\{\xi_t\}\) is embedded in a Markov chain \(\{X_t\}\) with \(X_t=(\xi_t,\dots,\xi_{t-p+1})\). The authors present conditions under which the chain is irreducible, or aperiodic, or has smooth transitions. These properties can be used for establishing stability of time series. It is shown that conditionally heteroscedastic (ARCH) processes, bilinear processes, SETAR processes, and mixture transition distribution processes are special cases of the general model investigated in this paper.