  We derive strong mixing conditions for many existing discrete-valued time
series models that include exogenous covariates in the dynamic. Our main
contribution is to study how a mixing condition on the covariate process
transfers to a mixing condition for the response. Using a coupling method, we
first derive mixing conditions for some Markov chains in random environments,
which gives a first result for some autoregressive categorical processes with
strictly exogenous regressors. Our result is then extended to some infinite
memory categorical processes. In the second part of the paper, we study
autoregressive models for which the covariates are sequentially exogenous.
Using a general random mapping approach on finite sets, we get explicit mixing
conditions that can be checked for many categorical time series found in the
literature, including multinomial autoregressive processes, ordinal time series
and dynamic multiple choice models. We also study some autoregressive count
time series using a somewhat different contraction argument. Our contribution
fill an important gap for such models, presented here under a more general
form, since such a strong mixing condition is often assumed in some recent
works but no general approach is available to check it.
