  In this paper, we study a notion of local stationarity for discrete time
Markov chains which is useful for applications in statistics. In the spirit of
some locally stationary processes introduced in the literature, we consider
triangular arrays of time-inhomogeneous Markov chains, defined by some families
of contracting Markov kernels. Using the Dobrushin's contraction coefficients
for various metrics, we show that the distribution of such Markov chains can be
approximated locally with the distribution of ergodic Markov chains and we also
study some mixing properties. From our approximation results in Wasserstein
metrics, we recover several properties obtained for autoregressive processes.
Moreover, using the total variation distance or more generally some distances
induced by a drift function, we consider new models, such as finite state space
Markov chains with time-varying transition matrices or some time-varying
versions of integer-valued autoregressive processes. For these two examples,
nonparametric kernel estimation of the transition matrix is discussed.
\end{abstract}
