We study ergodic properties of some Markov chains models in random environments when the random Markov kernels that define the dynamic satisfy some usual drift and small set conditions but with random coefficients. In particular, we adapt a standard coupling scheme used for getting geometric ergodic properties for homogeneous Markov chains to the random environment case and we prove the existence of a process of randomly invariant probability measures for such chains, in the spirit of the approach of Kifer for chains satisfying some Doeblin type conditions. We then deduce ergodic properties of such chains when the environment is itself ergodic. Our results complement and sharpen existing ones by providing quite weak and easily checkable assumptions on the random Markov kernels. As a by-product, we obtain a framework for studying some time series models with strictly exogenous covariates. We illustrate our results with autoregressive time series with functional coefficients and some threshold autoregressive processes.
翻译:当随机的马可夫内核决定了动态中某些通常的漂移和小设定条件,但使用随机系数时,我们随机地研究某些马可夫链模型的异性特性。特别是,我们调整了一种标准混合办法,将同质马可夫链的几何异性特性用于随机环境案例,并证明存在一种随机的随机异性概率测量过程,这是Kifer对符合多布林型某些条件的链的随机方法的精神。当环境本身是异性时,我们再推导出这些链的异性特性。我们的结果补充和强化了现有的这些结果,在随机马可夫内核上提供了非常弱和容易核对的假设。我们作为副产品,我们获得了一个框架,用严格的外生共变性来研究一些时间序列模型。我们用功能系数和一些临界自转递递递减过程来说明我们的结果。