  We study the problem of stationarity and ergodicity for autoregressive
multinomial logistic time series models which possibly include a latent process
and are defined by a GARCH-type recursive equation. We improve considerably
upon the existing results related to stationarity and ergodicity conditions of
such models. Proofs are based on theory developed for chains with complete
connections. This approach is based on a useful coupling technique which is
utilized for studying ergodicity of more general finite-state stochastic
processes. Such processes generalize finite-state Markov chains by assuming
infinite order models of past values. For finite order Markov chains, we also
discuss ergodicity properties when some strongly exogenous covariates are
considered in the dynamics of the process.
