{
  "id": "2410.20799",
  "version": "v1",
  "published": "2024-10-28T07:38:12.000Z",
  "updated": "2024-10-28T07:38:12.000Z",
  "title": "Sample-Path Large Deviations for Lévy Processes and Random Walks with Lognormal Increments",
  "authors": [
    "Zhe Su",
    "Chang-Han Rhee"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "The large deviations theory for heavy-tailed processes has seen significant advances in the recent past. In particular, Rhee et al. (2019) and Bazhba et al. (2020) established large deviation asymptotics at the sample-path level for L\\'evy processes and random walks with regularly varying and (heavy-tailed) Weibull-type increments. This leaves the lognormal case -- one of the three most prominent classes of heavy-tailed distributions, alongside regular variation and Weibull -- open. This article establishes the \\emph{extended large deviation principle} (extended LDP) at the sample-path level for one-dimensional L\\'evy processes and random walks with lognormal-type increments. Building on these results, we also establish the extended LDPs for multi-dimensional processes with independent coordinates. We demonstrate the sharpness of these results by constructing counterexamples, thereby proving that our results cannot be strengthened to a standard LDP under $J_1$ topology and $M_1'$ topology.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-28T07:38:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F10"
    ],
    "keywords": [
      "random walks",
      "sample-path large deviations",
      "lévy processes",
      "lognormal increments",
      "sample-path level"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}