{
  "id": "1807.03568",
  "version": "v1",
  "published": "2018-07-10T11:07:07.000Z",
  "updated": "2018-07-10T11:07:07.000Z",
  "title": "On the ergodicity of certain Markov chains in random environments",
  "authors": [
    "Balazs Gerencser",
    "Miklos Rasonyi"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the ergodic behaviour of a discrete-time process $X$ which is a Markov chain in a stationary random environment. The laws of $X_t$ are shown to converge to a limiting law in (weighted) total variation distance as $t\\to\\infty$. Convergence speed is estimated and an ergodic theorem is established for functionals of $X$. Our hypotheses on $X$ combine the standard \"small set\" and \"drift\" conditions for geometrically ergodic Markov chains with conditions on the growth rate of a certain \"maximal process\" of the random environment. We are able to cover a wide range of models that have heretofore been untractable. In particular, our results are pertinent to difference equations modulated by a stationary Gaussian process. Such equations arise in applications, for example, in discretized stochastic volatility models of mathematical finance.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-10T11:07:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ergodicity",
      "stationary random environment",
      "discretized stochastic volatility models",
      "total variation distance",
      "stationary gaussian process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}