{
  "id": "2102.01459",
  "version": "v1",
  "published": "2021-02-02T12:24:50.000Z",
  "updated": "2021-02-02T12:24:50.000Z",
  "title": "The Method of Cumulants for the Normal Approximation",
  "authors": [
    "Hanna Döring",
    "Sabine Jansen",
    "Kristina Schubert"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "The survey is dedicated to a celebrated series of quantitave results, developed by the Lithuanian school of probability, on the normal approximation for a real-valued random variable. The key ingredient is a bound on cumulants of the type $|\\kappa_j(X)| \\leq j!^{1+\\gamma} /\\Delta^{j-2}$, which is weaker than Cram\\'er's condition of finite exponential moments. We give a self-contained proof of some of the \"main lemmas\" in a book by Saulis and Statulevi\\v{c}ius (1989), and an accessible introduction to the Cram\\'er-Petrov series. In addition, we explain relations with heavy-tailed Weibull variables, moderate deviations, and mod-phi convergence. We discuss some methods for bounding cumulants such as summability of mixed cumulants and dependency graphs, and briefly review a few recent applications of the method of cumulants for the normal approximation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-02T12:24:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "60F10",
      "60G70"
    ],
    "keywords": [
      "normal approximation",
      "finite exponential moments",
      "main lemmas",
      "lithuanian school",
      "cramers condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}