{
  "id": "2001.02793",
  "version": "v1",
  "published": "2020-01-09T00:42:05.000Z",
  "updated": "2020-01-09T00:42:05.000Z",
  "title": "Central Limit Theorems on Compact Metric Spaces",
  "authors": [
    "Steven Rosenberg",
    "Jie Xu"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We produce a series of Central Limit Theorems (CLTs) associated to compact metric measure spaces $(K,d,\\eta)$, with $\\eta$ a reasonable probability measure. For the first CLT, we can ignore $\\eta$ by isometrically embedding $K$ into $\\mathcal{C}(K)$, the space of continuous functions on $K$ with the sup norm, and then applying known CLTs for sample means on Banach space. However, the sample mean makes no sense back on $K$, so using $\\eta$ we develop a CLT for the sample Fr\\'echet mean. To work in the easier Hilbert space setting of $L^2(K,\\eta)$, we have to modify the metric $d$ to a related metric $d_\\eta$. We then obtain an $L^2$-CLT for both the sample mean and the sample Fr\\'echet mean. Since the $L^2$ and $L^\\infty$ norms play important roles, in the last section we develop a metric-measure criterion relating $d$ and $\\eta$ under which all $L^p$ norms are equivalent.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-09T00:42:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B12"
    ],
    "keywords": [
      "central limit theorems",
      "compact metric spaces",
      "sample frechet mean",
      "sample mean",
      "norms play important roles"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}