• DOI:10.2307/1427479
• Corpus ID: 18700628

Stability of Markovian processes I: criteria for discrete-time Chains

@article{Meyn1992StabilityOM,
  title={Stability of Markovian processes I: criteria for discrete-time Chains},
  author={Sean P. Meyn and Richard L. Tweedie},
  journal={Advances in Applied Probability},
  year={1992},
  volume={24},
  pages={542 - 574}
}

• S. Meyn, R. Tweedie
• Published 1 September 1992
• Mathematics
• Advances in Applied Probability

In this paper we connect various topological and probabilistic forms of stability for discrete-time Markov chains. These include tightness on the one hand and Harris recurrence and ergodicity on the other. We show that these concepts of stability are largely equivalent for a major class of chains (chains with continuous components), or if the state space has a sufficiently rich class of appropriate sets (‘petite sets'). We use a discrete formulation of Dynkin's formula to establish unified criteria for these stability concepts, through bounding of moments of first entrance times to petite sets. This gives a generalization of Lyapunov–Foster criteria for the various stability conditions to hold. Under these criteria, ergodic theorems are shown to be valid even in the non-irreducible case. These results allow a more general test function approach for determining rates of convergence of the underlying distributions of a Markov chain, and provide strong mixing results and new versions of the central limit theorem and the law of the iterated logarithm.