{
  "id": "1212.2015",
  "version": "v4",
  "published": "2012-12-10T10:04:11.000Z",
  "updated": "2015-01-11T06:32:04.000Z",
  "title": "Concentration inequalities for Markov chains by Marton couplings and spectral methods",
  "authors": [
    "Daniel Paulin"
  ],
  "comment": "38 pages. Minor changes in this version",
  "categories": [
    "math.PR"
  ],
  "abstract": "We prove a version of McDiarmid's bounded differences inequality for Markov chains, with constants proportional to the mixing time of the chain. We also show variance bounds and Bernstein-type inequalities for empirical averages of Markov chains. In the case of non-reversible chains, we introduce a new quantity called the \"pseudo spectral gap\", and show that it plays a similar role for non-reversible chains as the spectral gap plays for reversible chains. Our techniques for proving these results are based on a coupling construction of Katalin Marton, and on spectral techniques due to Pascal Lezaud. The pseudo spectral gap generalises the multiplicative reversiblication approach of Jim Fill.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-04-10T12:34:24.000Z",
      "comment": "46 pages. Major revision",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2015-01-11T06:32:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E15",
      "60J05",
      "60J10",
      "28A35",
      "05C81",
      "68Q87"
    ],
    "keywords": [
      "markov chains",
      "spectral methods",
      "marton couplings",
      "concentration inequalities",
      "pseudo spectral gap generalises"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1212.2015P"
    }
  }
}