We discuss the existence and uniqueness of stationary and ergodic nonlinear autoregressive processes when exogenous regressors are incorporated into the dynamic. To this end, we consider the convergence of the backward iterations of dependent random maps. In particular, we give a new result when the classical condition of contraction on average is replaced with a contraction in conditional expectation. Under some conditions, we also discuss the dependence properties of these processes using the functional dependence measure of Wu (2005, Proceedings of the National Academy of Sciences 102, 14150–14154) that delivers a central limit theorem giving a wide range of applications. Our results are illustrated with conditional heteroscedastic autoregressive nonlinear models, Generalized AutoRegressive Conditional Heteroskedasticity (GARCH) processes, count time series, binary choice models, and categorical time series for which we provide many extensions of existing results.

中文翻译：

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非线性动力学中依赖随机映射的迭代和外生性

我们讨论了当外生回归量被纳入动态时，平稳和遍历非线性自回归过程的存在和唯一性。为此，我们考虑依赖随机映射的反向迭代的收敛性。特别是，当经典的平均收缩条件被条件期望的收缩代替时，我们给出了一个新的结果。在某些情况下，我们还使用 Wu (2005,美国国家科学院院刊102, 14150–14154），它提供了一个中心极限定理，提供了广泛的应用。我们的结果用条件异方差自回归非线性模型、广义自回归条件异方差 (GARCH) 过程、计数时间序列、二元选择模型和分类时间序列来说明，我们为此提供了许多现有结果的扩展。