{
  "id": "2403.16920",
  "version": "v2",
  "published": "2024-03-25T16:38:34.000Z",
  "updated": "2024-09-19T15:15:04.000Z",
  "title": "Optimal convergence rates of MCMC integration for functions with unbounded second moment",
  "authors": [
    "Julian Hofstadler"
  ],
  "comment": "The title has changed, the proofs have been simplified, and some typos were corrected",
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "We study the Markov chain Monte Carlo (MCMC) estimator for numerical integration for functions that do not need to be square integrable w.r.t. the invariant distribution. For chains with a spectral gap we show that the absolute mean error for $L^p$ functions, with $p \\in (1,2)$, decreases like $n^{1/p -1}$, which is known to be the optimal rate. This improves currently known results where an additional parameter $\\delta>0$ appears and the convergence is of order $n^{(1+\\delta)/p-1}$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-09-19T15:15:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65C05",
      "60J22",
      "65C20"
    ],
    "keywords": [
      "optimal convergence rates",
      "unbounded second moment",
      "mcmc integration",
      "markov chain monte carlo",
      "absolute mean error"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}