{
  "id": "1408.2278",
  "version": "v1",
  "published": "2014-08-10T22:43:44.000Z",
  "updated": "2014-08-10T22:43:44.000Z",
  "title": "Kolmogorov complexity and strong approximation of Brownian motion",
  "authors": [
    "Bjørn Kjos-Hanssen",
    "Tamás Szabados"
  ],
  "journal": "Proceedings of the American Mathematical Society 139 (2011) no. 9, 3307--3316",
  "doi": "10.1090/S0002-9939-2011-10741-X",
  "categories": [
    "math.LO"
  ],
  "abstract": "Brownian motion and scaled and interpolated simple random walk can be jointly embedded in a probability space in such a way that almost surely the $n$-step walk is within a uniform distance $O(n^{-1/2}\\log n)$ of the Brownian path for all but finitely many positive integers $n$. Almost surely this $n$-step walk will be incompressible in the sense of Kolmogorov complexity, and all {Martin-L\\\"of random} paths of Brownian motion have such an incompressible close approximant. This strengthens a result of Asarin, who obtained the bound $O(n^{-1/6} \\log n)$. The result cannot be improved to $o(n^{-1/2}{\\sqrt{\\log n}})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-10T22:43:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "brownian motion",
      "kolmogorov complexity",
      "strong approximation",
      "step walk",
      "interpolated simple random walk"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AMS",
      "journal": "Proc. Amer. Math. Soc."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.2278K"
    }
  }
}