{
  "id": "1707.06129",
  "version": "v1",
  "published": "2017-07-19T14:48:42.000Z",
  "updated": "2017-07-19T14:48:42.000Z",
  "title": "Conditioned local limit theorems for random walks defined on finite Markov chains",
  "authors": [
    "Ion Grama",
    "Ronan Lauvergnat",
    "Emile Le Page"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $(X_n)_{n\\geq 0}$ be a Markov chain with values in a finite state space $\\mathbb X$ starting at $X_0=x \\in \\mathbb X$ and let $f$ be a real function defined on $\\mathbb X$. Set $S_n=\\sum_{k=1}^{n} f(X_k)$, $n\\geqslant 1$. For any $y \\in \\mathbb R$ denote by $\\tau_y$ the first time when $y+S_n$ becomes non-positive. We study the asymptotic behaviour of the probability $\\mathbb P_x \\left( y+S_{n} \\in [z,z+a] \\,,\\, \\tau_y > n \\right)$ as $n\\to+\\infty.$ We first establish for this probability a conditional version of the local limit theorem of Stone. Then we find for it an asymptotic equivalent of order $n^{3/2}$ and give a generalization which is useful in applications. We also describe the asymptotic behaviour of the probability $\\mathbb P_x \\left( \\tau_y = n \\right)$ as $n\\to+\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-19T14:48:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J10",
      "60F05"
    ],
    "keywords": [
      "conditioned local limit theorems",
      "finite markov chains",
      "random walks",
      "asymptotic behaviour",
      "probability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}