{
  "id": "0904.1167",
  "version": "v3",
  "published": "2009-04-07T15:21:43.000Z",
  "updated": "2010-09-29T15:33:52.000Z",
  "title": "Martingales and Rates of Presence in Homogeneous Fragmentations",
  "authors": [
    "Nathalie Krell",
    "Alain Rouault"
  ],
  "comment": "In this version, we correct a misprint in Assumption A from the previous one",
  "categories": [
    "math.PR"
  ],
  "abstract": "The main focus of this work is the asymptotic behavior of mass-conservative homogeneous fragmentations. Considering the logarithm of masses makes the situation reminiscent of branching random walks. The standard approach is to study {\\bf asymptotical} exponential rates. For fixed $v > 0$, either the number of fragments whose sizes at time $t$ are of order $\\e^{-vt}$ is exponentially growing with rate $C(v) > 0$, i.e. the rate is effective, or the probability of presence of such fragments is exponentially decreasing with rate $C(v) < 0$, for some concave function $C$. In a recent paper, N. Krell considered fragments whose sizes decrease at {\\bf exact} exponential rates, i.e. whose sizes are confined to be of order $\\e^{-vs}$ for every $s \\leq t$. In that setting, she characterized the effective rates. In the present paper we continue this analysis and focus on probabilities of presence, using the spine method and a suitable martingale.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2010-09-29T15:33:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "homogeneous fragmentations",
      "martingale",
      "exponential rates",
      "standard approach",
      "main focus"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0904.1167K"
    }
  }
}