{
  "id": "1408.4444",
  "version": "v1",
  "published": "2014-08-19T19:55:59.000Z",
  "updated": "2014-08-19T19:55:59.000Z",
  "title": "Rate of convergence of the mean for sub-additive ergodic sequences",
  "authors": [
    "Antonio Auffinger",
    "Michael Damron",
    "Jack Hanson"
  ],
  "comment": "38 pages, 2 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "For sub-additive ergodic processes $\\{X_{m,n}\\}$ with weak dependence, we analyze the rate of convergence of $\\mathbb{E}X_{0,n}/n$ to its limit $g$. We define an exponent $\\gamma$ given roughly by $\\mathbb{E}X_{0,n} \\sim ng + n^\\gamma$, and, assuming existence of a fluctuation exponent $\\chi$ that gives $ X_{0,n} \\sim n^{2\\chi}$, we provide a lower bound for $\\gamma$ of the form $\\gamma \\geq \\chi$. The main requirement is that $\\chi \\neq 1/2$. In the case $\\chi=1/2$ and under the assumption $\\mathop{\\mathrm{Var}} X_{0,n} = O(n/(\\log n)^\\beta)$ for some $\\beta>0$, we prove $\\gamma \\geq \\chi - c(\\beta)$ for a $\\beta$-dependent constant $c(\\beta)$. These results show in particular that non-diffusive fluctuations are associated to non-trivial $\\gamma$. Various models, including first-passage percolation, directed polymers, the minimum of a branching random walk and bin packing, fall into our general framework, and the results apply assuming $\\chi$ exists.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-19T19:55:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sub-additive ergodic sequences",
      "convergence",
      "weak dependence",
      "general framework",
      "fluctuation exponent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.4444A"
    }
  }
}