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.

在本文中，我们讨论了如何利用马尔可夫链理论中的亚几何遍历性概念来研究非线性时间序列模型的平稳性和遍历性。亚几何遍历性意味着转移概率测量以比几何慢的速率收敛到固定测量。具体来说，我们考虑适当定义的高阶非线性自回归，其行为类似于观察序列的大值的单位根过程，但我们几乎不限制观察序列的中等值的动态。非线性自回归的次几何遍历性的结果以前只出现在一阶情况下。我们提供了对高阶情况的扩展，并表明我们考虑的自回归在适当的条件下是亚几何遍历的。 $\beta $ -mixing with subgeometrically decaying mixing coefficients.