{
  "id": "0809.0177",
  "version": "v5",
  "published": "2008-09-01T06:32:07.000Z",
  "updated": "2009-12-15T19:14:32.000Z",
  "title": "Limit theorems for additive functionals of a Markov chain",
  "authors": [
    "Milton Jara",
    "Tomasz Komorowski",
    "Stefano Olla"
  ],
  "comment": "Accepted for the publication in Annals of Applied Probability",
  "journal": "The Annals of Applied Probability 19, 6 (2009) 2270-2300",
  "doi": "10.1214/09-AAP610",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Consider a Markov chain $\\{X_n\\}_{n\\ge 0}$ with an ergodic probability measure $\\pi$. Let $\\Psi$ a function on the state space of the chain, with $\\alpha$-tails with respect to $\\pi$, $\\alpha\\in (0,2)$. We find sufficient conditions on the probability transition to prove convergence in law of $N^{1/\\alpha}\\sum_n^N \\Psi(X_n)$ to a $\\alpha$-stable law. ``Martingale approximation'' approach and ``coupling'' approach give two different sets of conditions. We extend these results to continuous time Markov jump processes $X_t$, whose skeleton chain satisfies our assumptions. If waiting time between jumps has finite expectation, we prove convergence of $N^{-1/\\alpha}\\int_0^{Nt} V(X_s) ds$ to a stable process. In the case of waiting times with infinite average, we prove convergence to a Mittag-Leffler process.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2009-12-15T19:14:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "60F17",
      "76P05"
    ],
    "keywords": [
      "markov chain",
      "limit theorems",
      "additive functionals",
      "continuous time markov jump processes",
      "skeleton chain satisfies"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0809.0177J"
    }
  }
}