{
  "id": "0906.5322",
  "version": "v1",
  "published": "2009-06-29T17:23:15.000Z",
  "updated": "2009-06-29T17:23:15.000Z",
  "title": "Geometric Ergodicity and the Spectral Gap of Non-Reversible Markov Chains",
  "authors": [
    "Ioannis Kontoyiannis",
    "Sean P. Meyn"
  ],
  "categories": [
    "math.PR",
    "math.SP"
  ],
  "abstract": "We argue that the spectral theory of non-reversible Markov chains may often be more effectively cast within the framework of the naturally associated weighted-$L_\\infty$ space $L_\\infty^V$, instead of the usual Hilbert space $L_2=L_2(\\pi)$, where $\\pi$ is the invariant measure of the chain. This observation is, in part, based on the following results. A discrete-time Markov chain with values in a general state space is geometrically ergodic if and only if its transition kernel admits a spectral gap in $L_\\infty^V$. If the chain is reversible, the same equivalence holds with $L_2$ in place of $L_\\infty^V$, but in the absence of reversibility it fails: There are (necessarily non-reversible, geometrically ergodic) chains that admit a spectral gap in $L_\\infty^V$ but not in $L_2$. Moreover, if a chain admits a spectral gap in $L_2$, then for any $h\\in L_2$ there exists a Lyapunov function $V_h\\in L_1$ such that $V_h$ dominates $h$ and the chain admits a spectral gap in $L_\\infty^{V_h}$. The relationship between the size of the spectral gap in $L_\\infty^V$ or $L_2$, and the rate at which the chain converges to equilibrium is also briefly discussed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-06-29T17:23:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J05",
      "60J10",
      "37A30",
      "37A25"
    ],
    "keywords": [
      "spectral gap",
      "non-reversible markov chains",
      "geometric ergodicity",
      "chain admits",
      "general state space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0906.5322K"
    }
  }
}