{
  "id": "1206.6953",
  "version": "v1",
  "published": "2012-06-29T06:43:38.000Z",
  "updated": "2012-06-29T06:43:38.000Z",
  "title": "Return Probabilities for the Reflected Random Walk on $\\mathbb N_0$",
  "authors": [
    "Rim Essifi",
    "Marc Peigné"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $(Y_n)$ be a sequence of i.i.d. $\\mathbb Z$-valued random variables with law $\\mu$. The reflected random walk $(X_n)$ is defined recursively by $X_0=x \\in \\mathbb N_0, X_{n+1}=|X_n+Y_{n+1}|$. Under mild hypotheses on the law $\\mu$, it is proved that, for any $ y \\in \\mathbb N_0$, as $n \\to +\\infty$, one gets $\\mathbb P_x[X_n=y]\\sim C_{x, y} R^{-n} n^{-3/2}$ when $\\sum_{k\\in \\mathbb Z} k\\mu(k) >0$ and $\\mathbb P_x[X_n=y]\\sim C_{y} n^{-1/2}$ when $\\sum_{k\\in \\mathbb Z} k\\mu(k) =0$, for some constants $R, C_{x, y}$ and $C_y >0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-06-29T06:43:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "reflected random walk",
      "return probabilities",
      "valued random variables",
      "mild hypotheses"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1206.6953E"
    }
  }
}