{
  "id": "2004.10152",
  "version": "v1",
  "published": "2020-04-21T16:59:31.000Z",
  "updated": "2020-04-21T16:59:31.000Z",
  "title": "Cumulant-cumulant relations in free probability theory from Magnus' expansion",
  "authors": [
    "A. Celestino",
    "K. Ebrahimi-Fard",
    "F. Patras",
    "D. Perales Anaya"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Relations between moments and cumulants play a central role in both classical and non-commutative probability theory. The latter allows for several distinct families of cumulants corresponding to different types of independences: free, Boolean and monotone. Relations among those cumulants have been studied recently. In this work we focus on the problem of expressing with a closed formula multivariate monotone cumulants in terms of free and Boolean cumulants. In the process we introduce various constructions and statistics on non-crossing partitions. Our approach is based on a pre-Lie algebra structure on cumulant functionals. Relations among cumulants are described in terms of the pre-Lie Magnus expansion combined with results on the continuous Baker-Campbell-Hausdorff formula due to A. Murua.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-21T16:59:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16T30",
      "17A30",
      "46L53",
      "46L54"
    ],
    "keywords": [
      "free probability theory",
      "cumulant-cumulant relations",
      "closed formula multivariate monotone cumulants",
      "pre-lie algebra structure",
      "pre-lie magnus expansion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}