{
  "id": "1406.3768",
  "version": "v1",
  "published": "2014-06-14T20:27:33.000Z",
  "updated": "2014-06-14T20:27:33.000Z",
  "title": "Some limit results for Markov chains indexed by trees",
  "authors": [
    "Peter Czuppon",
    "Peter Pfaffelhuber"
  ],
  "comment": "12 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider a sequence of Markov chains $(\\mathcal X^n)_{n=1,2,...}$ with $\\mathcal X^n = (X^n_\\sigma)_{\\sigma\\in\\mathcal T}$, indexed by the full binary tree $\\mathcal T = \\mathcal T_0 \\cup \\mathcal T_1 \\cup ...$, where $\\mathcal T_k$ is the $k$th generation of $\\mathcal T$. In addition, let $(\\Sigma_k)_{k=0,1,2,...}$ be a random walk on $\\mathcal T$ with $\\Sigma_k \\in \\mathcal T_k$ and $\\widetilde{\\mathcal R}^n = (\\widetilde R_t^n)_{t\\geq 0}$ with $\\widetilde R_t^n := X_{\\Sigma_{[tn]}}$, arising by observing the Markov chain $\\mathcal X^n$ along the random walk. We present a law of large numbers concerning the empirical measure process $\\widetilde{\\mathcal Z}^n = (\\widetilde Z_t^n)_{t\\geq 0}$ where $\\widetilde{Z}_t^n = \\sum_{\\sigma\\in\\mathcal T_{[tn]}} \\delta_{X_\\sigma^n}$ as $n\\to\\infty$. Precisely, we show that if $\\widetilde{\\mathcal R}^n \\to \\mathcal R$ for some Feller process $\\mathcal R = (R_t)_{t\\geq 0}$ with deterministic initial condition, then $\\widetilde{\\mathcal Z}^n \\to \\mathcal Z$ with $Z_t = \\delta_{\\mathcal L(R_t)}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-14T20:27:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F15",
      "60F05"
    ],
    "keywords": [
      "markov chains",
      "limit results",
      "random walk",
      "full binary tree",
      "deterministic initial condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.3768C"
    }
  }
}