{
  "id": "1505.07622",
  "version": "v1",
  "published": "2015-05-28T10:05:06.000Z",
  "updated": "2015-05-28T10:05:06.000Z",
  "title": "Strong renewal theorems with infinite mean beyond local large deviations",
  "authors": [
    "Zhiyi Chi"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/14-AAP1029 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Applied Probability 2015, Vol. 25, No. 3, 1513-1539",
  "doi": "10.1214/14-AAP1029",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $F$ be a distribution function on the line in the domain of attraction of a stable law with exponent $\\alpha\\in(0,1/2]$. We establish the strong renewal theorem for a random walk $S_1,S_2,\\ldots$ with step distribution $F$, by extending the large deviations approach in Doney [Probab. Theory Related Fileds 107 (1997) 451-465]. This is done by introducing conditions on $F$ that in general rule out local large deviations bounds of the type $\\mathbb{P}\\{S_n\\in(x,x+h]\\}=O(n)\\overline{F}(x)/x$, hence are significantly weaker than the boundedness condition in Doney (1997). We also give applications of the results on ladder height processes and infinitely divisible distributions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-05-28T10:05:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "strong renewal theorem",
      "infinite mean",
      "local large deviations bounds",
      "ladder height processes",
      "large deviations approach"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150507622C"
    }
  }
}