{
  "id": "2303.06489",
  "version": "v2",
  "published": "2023-03-11T19:58:35.000Z",
  "updated": "2023-05-03T18:02:08.000Z",
  "title": "Weighted Sums and Berry-Esseen type estimates in Free Probability Theory",
  "authors": [
    "Leonie Neufeld"
  ],
  "comment": "55 pages; in v2: Theorem 1.8 added",
  "categories": [
    "math.PR",
    "math.OA"
  ],
  "abstract": "We study weighted sums of free identically distributed random variables with weights chosen randomly from the unit sphere and show that the Kolmogorov distance between the distribution of such a weighted sum and Wigner's semicircle law is of order $(\\log n)^{\\frac{1}{2}}n^{-\\frac{1}{2}}$ with high probability. In the special case of bounded random variables the rate can be improved to $n^{-\\frac{1}{2}}$. Replacing the Kolmogorov distance by a weaker pseudometric, we obtain a rate of convergence of order $(\\log n)n^{-1}$. Our results can be seen as a free analogue of the Klartag-Sodin result in classical probability theory. Moreover, we show that our ideas generalise to the setting of sums of free non-identically distributed bounded random variables providing a new rate of convergence in the free central limit theorem.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2023-05-03T18:02:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46L54",
      "60E05"
    ],
    "keywords": [
      "berry-esseen type estimates",
      "free probability theory",
      "weighted sum",
      "identically distributed random variables",
      "distributed bounded random variables"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 55,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}