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Published online by Cambridge University Press:  09 November 2020

Mika Meitz
University of Helsinki
Pentti Saikkonen*
University of Helsinki
Address correspondence to Pentti Saikkonen, Department of Mathematics and Statistics, University of Helsinki, P. O. Box 68, FI-00014 Helsinki, Finland; e-mail:


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.

© The Author(s), 2020. Published by Cambridge University Press

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The authors thank the Academy of Finland for financial support, and Peter Phillips, Donald Andrews, and two anonymous referees for useful comments and suggestions.



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