Open Access
August 2019 Local stationarity and time-inhomogeneous Markov chains
Lionel Truquet
Ann. Statist. 47(4): 2023-2050 (August 2019). DOI: 10.1214/18-AOS1739


A primary motivation of this contribution is to define new locally stationary Markov models for categorical or integer-valued data. For this initial purpose, we propose a new general approach for dealing with time-inhomogeneity that extends the local stationarity notion developed in the time series literature. We also introduce a probabilistic framework which is very flexible and allows us to consider a much larger class of Markov chain models on arbitrary state spaces, including most of the locally stationary autoregressive processes studied in the literature. We consider triangular arrays of time-inhomogeneous Markov chains, defined by some families of contracting and slowly-varying Markov kernels. The finite-dimensional distribution of such Markov chains can be approximated locally with the distribution of ergodic Markov chains and some mixing properties are also available for these triangular arrays. As a consequence of our results, some classical geometrically ergodic homogeneous Markov chain models have a locally stationary version, which lays the theoretical foundations for new statistical modeling. Statistical inference of finite-state Markov chains can be based on kernel smoothing and we provide a complete and fast implementation of such models, directly usable by the practitioners. We also illustrate the theory on a real data set. A central limit theorem for Markov chains on more general state spaces is also provided and illustrated with the statistical inference in INAR models, Poisson ARCH models and binary time series models. Additional examples such as locally stationary regime-switching or SETAR models are also discussed.


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Lionel Truquet. "Local stationarity and time-inhomogeneous Markov chains." Ann. Statist. 47 (4) 2023 - 2050, August 2019.


Received: 1 October 2017; Revised: 1 June 2018; Published: August 2019
First available in Project Euclid: 21 May 2019

zbMATH: 07082278
MathSciNet: MR3953443
Digital Object Identifier: 10.1214/18-AOS1739

Primary: 62G08 , 62M05
Secondary: 62M10

Keywords: local stationarity , Markov chains

Rights: Copyright © 2019 Institute of Mathematical Statistics

Vol.47 • No. 4 • August 2019
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