Limiting distributions of functionals of Markov chains (Q1056975)

Let (\(\Omega\),\({\mathcal F},P)\) be a probability space, (E,\({\mathcal E})\) a measurable space, \(\{X_ n: n\geq 0\}\) be a sequence of (E,\({\mathcal E})\)- valued random variables, S be a complete separable metric space equipped with its Borel field \({\mathcal B}\), and let \({\mathcal M}(S)\) be the space of probability measures on (S,\({\mathcal B})\). Let \(\{Y_ n: n\geq 0\}\) be a sequence of (S,\({\mathcal B})\)-valued random variables such that \(p(Y_ n\in A| \sigma (X_ n))=p_{X_ n}(A)\) where \(p_ x(\cdot)\in {\mathcal M}(S)\) for each \(x\in E\) and \(p_.(A)\) is \({\mathcal E}\)-measurable for each \(A\in {\mathcal B}\). Suppose \(S^*={\mathcal M}(S)\) has the topology of weak convergence and let \({\mathcal B}^*\) and \({\mathcal M}^*\) denote the Borel field of \(S^*\) and the set of probability measures on \((S^*,{\mathcal B}^*)\), respectively. Finally let \(\Gamma_ n\) be the distribution of \(p_{X_ n}(\cdot)\), that is, for \(B\in {\mathcal B}^*\), \(\Gamma_ n(B)=P(p_{X_ n}(\cdot)\in B).\) The authors prove that if \(\Gamma_ n\to \Gamma\) weakly in \({\mathcal M}^*\) then \(\{Y_ n\}\) converges in distribution to the law \(\nu \in S^*\), where \(\nu (A)=\int_{S^*}\mu (A)\Gamma (d\mu)\). Several examples are given. Suppose E is a metric space and \({\mathcal E}\) is its Borel field and that the map \(x\to p_ x\in C(E,S^*)\). If \(X_ n\) converges in distribution to a measure \(\pi\) then \(Y_ n\) converges in distribution to \(\nu\) and \(\nu (A)=\int_{E}p_ x(A)\pi (d\nu)\). This is used to prove the convergence in distribution of the inter-event times of a Poisson process with periodic rate function. A second particular case concerns the case where E is countable and \(P(X_ n\in B)\to 0\) (n\(\to \infty)\) for each finite subset \(B\subset E\).