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Local stationarity and time-inhomogeneous Markov chains. (English) Zbl 1429.62372

In the paper, a probabilistic framework is introduced which allows us to consider a large class of Markov chain models on arbitrary state spaces, including most of the locally stationary autoregressive processes.
Let \((E,d)\) be a metric space with its corresponding Borel \(\sigma\)-field \(\mathcal{B}(E)\), and let \(\{Q_u, u\in[0,1]\}\) be a family of Markov kernels on \((E,\mathcal{B}(E))\). The author considers triangular arrays \(\{X_{n,j},n\in\mathbb{N}, 1\leqslant j\leqslant n\}\) such that for all \(n\), the sequence \(\{X_{n,1}, X_{n,2},\dots, X_{n,n}\}\) is a nonhomogeneous Markov chain under condition \[ \mathbb{P}(X_{n,k}\in A\,|\,X_{n,k-1}=x)=Q_{k/n}(x,A),\ k\in\{ 1,2,\dots, n\}. \] In particular, he derives an array of interesting properties related with the local stationarity of the described nonhomogeneous Markov chain. In order to satisfy various properties the generating Markov kernels family should satisfy some regularity conditions and some contraction assumptions.

MSC:

62M05 Markov processes: estimation; hidden Markov models
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
62G08 Nonparametric regression and quantile regression
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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References:

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