Tightness and continuity of a family of invariant measures for Markov chains depending on a parameter (Q1358018)

A family of Markov chains with transition functions \((P^{(\delta)}(x,B))_{\delta\in R}\) (\(x\in S\): metric space, \(B\in{\mathcal B}(S)\)) is considered. Every kernel \(p^{(\delta)}\) has at least one invariant distribution \(\pi^{(\delta)}\): \(\pi^{(\delta)}(B)= \int_S P^{(\delta)}(x,B)\pi^{(\delta)}(dx)\). Two conditions are proved to be sufficient for \(\pi^{(\delta)}\) to converge weakly to \(\pi^{(0)}\) \((\delta\to 0)\). They are the following: 1) for all \(x\in S\) under \(y\to x\), \(P^{(\delta)}(y,\cdot)\) tends weakly to \(P^{(0)}(x,\cdot)\), 2) the family \((\pi^{(\delta)})\) is tight in \(S\). For investigation of tightness of the family a method of Lyapunov test functions is used. These are some real functions with a special property of their increments on the chain. In case of \(S=R_+\) the tightness is connected with distributions of increments of the chain itself.