  We discuss existence and uniqueness of stationary and ergodic nonlinear
autoregressive processes when exogenous regressors are incorporated in 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 derive an explicit
control of the functional dependence of Wu (2005) which guarantees a wide range
of statistical applications. Our results are illustrated with CHARN models,
GARCH processes, count time series, binary choice models and categorical time
series for which we provide many extensions of existing results.
