Some dichotomy results for functionals of Harris recurrent Markov chains
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.
scientific article; zbMATH DE number 1792538