In this paper we discuss how the notion of subgeometric ergodicity in Markov chain theory can be exploited to study the stability of nonlinear time series models. Subgeometric ergodicity means that the transition probability measures converge to the stationary measure at a rate slower than geometric. Specifically, we consider  higher-order nonlinear autoregressions that may exhibit rather arbitrary behavior for moderate values of the observed series and that behave in a near unit root manner for large values of the observed series. Generalizing existing first-order results, we show that these autoregressions are, under appropriate conditions, subgeometrically ergodic. As useful implications we also obtain stationarity and \beta-mixing with subgeometrically decaying mixing coefficients.
In this paper we discuss how the notion of subgeometric ergodicity in Markov chain theory can be exploited to study stationarity and ergodicity of nonlinear time series models. Subgeometric ergodicity means that the transition probability measures converge to the stationary measure at a rate slower than geometric. Specifically, we consider suitably defined higher-order nonlinear autoregressions that behave similarly to a unit root process for large values of the observed series but we place almost no restrictions on their dynamics for moderate values of the observed series. Results on the subgeometric ergodicity of nonlinear autoregressions have previously appeared only in the first-order case. We provide an extension to the higher-order case and show that the autoregressions we consider are, under appropriate conditions, subgeometrically ergodic. As useful implications we also obtain stationarity and \beta-mixing with subgeometrically decaying mixing coefficients.
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Results on the subgeometric ergodicity of nonlinear autoregressions have previously appeared only in the
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