Some dichotomy results for functionals of Harris recurrent Markov chains
type: Item
instance of: scholarly article
title: Some dichotomy results for functionals of Harris recurrent Markov chains
zbMATH Open document ID: 0998.60009
DOI: 10.1016/S0304-4149(99)00038-1
author: Xia Chen
published in: Stochastic Processes and their Applications
publication date: 2002-08-29T00:00:00Z
review text: Let \((E,\mathfrak E)\) be a measurable space with a countably generated \(\sigma \)-algebra \(\mathfrak E\), \(\{X_{n}\}\) a Harris recurrent Markov chain in \((E,\mathfrak E)\) with a transition probability \(P\) and an invariant measure \(\pi \). Let \(B\) be a separable Banach space, \(\xi : E\rightarrow B\) a measurable map, set \(S_k = \sum ^{k}_{j=1} \xi (X_{j})\), \(k\geq 1\). In the study of limit behaviour of the sequence \(\{S_{k}\}\), various integrability properties and tail estimates appear, involving a set \(A\) from a suitable class (small sets, \(D\)-sets), cf. the author's book ``Limit theorems for functionals of ergodic Markov chains with general state space'' (1999; Zbl 0952.60014). In the paper, it is shown that several conditions of this type are satisfied either for all sets in the considered class, or for none. To be more explicit, let us quote one of the obtained results. Set \(\mathfrak E^{+} = \{A\in \mathfrak E; \pi (A)>0\}\), \(\tau _{A} = \inf \{n\geq 1; X_{n}\in A\}\) for any \(A\in \mathfrak E^{+}\), denote by \(\mathfrak S\) the class of all finite unions of small sets (\(S\in \mathfrak E^{+}\) is called a small set, if \(P^{m}\geq \beta \mathbf 1_{S}\otimes \nu \) for some \(m\geq 1\), \(\beta >0\) and a probability measure \(\nu \)), and by \(\widetilde {\mathfrak S}\) the subset of all \(A\in \mathfrak S\) such that \(\int _{A} E_{x}\varphi ( |S_{\tau _{A}} |) \text d\pi (x) < \infty \), where \(\varphi : [0,\infty [\to [0,\infty [\) is a nondecreasing function, \(\varphi (cs) \leq c^\lambda \varphi (s)\) for some \(\lambda >0\) and all \(s\geq 0\), \(c\geq 2\). If \(\int _{E} \varphi ( |\xi (x) |) \text d\pi (x) <\infty \), then either \(\widetilde {\mathfrak S} =\emptyset \), or \(\widetilde {\mathfrak S} = \mathfrak S\). Moreover, a sufficient condition (uniform ergodicity) for the latter case to occur is given.
reviewed by: Jan Seidler
Mathematics Subject Classification ID: 60B12
Mathematics Subject Classification ID: 60J05
zbMATH DE Number: 1792538
zbMATH Keywords: Harris recurrence
zbMATH Keywords: small sets
zbMATH Keywords: \(D\)-sets
zbMATH Keywords: Markov chains
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1792538