  This paper examines the impact of discrete marginal distributions on
copula-based Markov chains. We present results on mixing and parameter
estimation for a copula-based Markov chain model with Bernoulli($p$) marginal
distribution and highlight the differences between continuous and discrete
state-space Markov chains. We derive estimators for model parameters using the
maximum likelihood approach and discuss other estimators of $p$ that are
asymptotically equivalent to its maximum likelihood estimator. The asymptotic
distributions of the parameter estimators are provided. A simulation study
showcases the performance of the different estimators of $p$. Additionally,
statistical tests for model parameters are included.
