Local stationarity and time-inhomogeneous Markov chains

TL;DR

This paper introduces a notion of local stationarity for discrete time Markov chains, enabling approximation by ergodic chains and analyzing mixing properties, with applications to time-varying models and nonparametric estimation.

Contribution

It develops a framework for local stationarity in Markov chains using contraction coefficients, extending to models with time-varying transitions and estimation methods.

Findings

Distribution of Markov chains can be approximated locally by ergodic chains.

Established mixing properties for locally stationary Markov chains.

Applied results to models with time-varying transition matrices and autoregressive processes.

Abstract

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}