{
  "id": "1808.04964",
  "version": "v1",
  "published": "2018-08-15T04:19:38.000Z",
  "updated": "2018-08-15T04:19:38.000Z",
  "title": "A Probabilistic Proof of the Perron-Frobenius Theorem",
  "authors": [
    "Peter W. Glynn",
    "Paritosh Y. Desai"
  ],
  "categories": [
    "math.PR",
    "stat.OT"
  ],
  "abstract": "The Perron-Frobenius theorem plays an important role in many areas of management science and operations research. This paper provides a probabilistic perspective on the theorem, by discussing a proof that exploits a probabilistic representation of the Perron-Frobenius eigenvalue and eigenvectors in terms of the dynamics of a Markov chain. The proof provides conditions in both the finite-dimensional and infinite-dimensional settings under which the Perron-Frobenius eigenvalue and eigenvectors exist. Furthermore, the probabilistic representations that arise can be used to produce a Monte Carlo algorithm for computing the Perron-Frobenius eigenvalue and eigenvectors that will be explored elsewhere.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-15T04:19:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "probabilistic proof",
      "perron-frobenius eigenvalue",
      "probabilistic representation",
      "eigenvectors",
      "perron-frobenius theorem plays"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}