Tightness for Non-irreducible Markov Chains

In this paper we develop Foster-type criteria guaranteeing tightness for Markov chains which are not necessarily irreducible. The results include criteria for both tightness of the marginal distributions and tightness of the Cesaroaveraged transition probabilities. In addition, we obtain results guaranteeing boundedness in expectation for real-valued functionals of the chain. Keywords: Markov chains, stochastic stability, tightness, Lyapunov functions. 1991 Mathematics Subject Classification: 60J05, 93E15. 1 Introduction Let \Phi = f\Phi 0 ; \Phi 1 ; : : :g be a time homogeneous Markov chain, and let P be its associated transition kernel, so that P (x; B) := Pf\Phi n+1 2 B j \Phi n = xg; x 2 X; B 2 B(X): We assume throughout that the state space X is a complete, separable metric space, with Borel oe-field B(X). In the stochastic systems literature, stability of a Markov chain is frequently equated with tightness of the underlying distributions, for each initial condition. This giv..