{
  "id": "1104.1554",
  "version": "v2",
  "published": "2011-04-08T11:52:04.000Z",
  "updated": "2013-04-12T03:08:17.000Z",
  "title": "A Local Limit Theorem for the Minimum of a Random Walk with Markovian Increasements",
  "authors": [
    "Yinna Ye"
  ],
  "comment": "40 pages, 3 figures; updated author's present address, corrected typos, unified notations",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $(\\Omega,\\mathcal{F}, \\mathbb{P})$ be a probability space and $E$ be a finite set. Assume that $X=(X_n)$ is an irreducible and aperiodic Markov chain, defined on $(\\Omega,\\mathcal{F}, \\mathbb{P})$, with values in $E$ and with transition probability $P=\\Big(p_{i,j}\\Big)_{i,j}$. Let $(F(i,j,\\d x))_{i,j\\in E}$ be a family of probability measures on $\\mathbb{R}$. Consider a semi-markovian chain $(Y_n,X_n)$ on $\\mathbb{R}\\times E$ with transition probability $\\widetilde{P}$, defined by $\\widetilde{P}\\Big((u,i),A\\times\\{j\\}\\Big)=\\mathbb{P}(Y_{n+1}\\in A,X_{n+1}=j|Y_n= u,X_n=i)=p_{i,j}F(i,j,A)$, for any $(u,i)\\in\\mathbb{R}\\times E$, any Borel set $A\\subset\\mathbb{R}$ and any $j\\in E$. We study the asymptotic behavior of the sequence of Laplace transforms of $(X_n,m_n)$, where $m_n=\\min(S_0,S_1,...,S_n)$ and $S_n=Y_0+...+Y_{n-1}$. Under quite general assumptions on $F(i,j,dx)$, we prove that for all $(i,j)\\in E\\times E$, $\\sqrt{n}\\E_i[\\exp(\\lambda m_n), X_n=j]$ converges to a positive function $H_{i,j}(\\lambda)$ and we obtain further informations on this limit function as $\\lambda\\to 0^+$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-04-12T03:08:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "local limit theorem",
      "random walk",
      "markovian increasements",
      "transition probability",
      "aperiodic markov chain"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1104.1554Y"
    }
  }
}