{
  "id": "1001.2152",
  "version": "v1",
  "published": "2010-01-13T13:14:33.000Z",
  "updated": "2010-01-13T13:14:33.000Z",
  "title": "Rate of convergence of predictive distributions for dependent data",
  "authors": [
    "Patrizia Berti",
    "Irene Crimaldi",
    "Luca Pratelli",
    "Pietro Rigo"
  ],
  "comment": "Published in at http://dx.doi.org/10.3150/09-BEJ191 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)",
  "journal": "Bernoulli 2009, Vol. 15, No. 4, 1351-1367",
  "doi": "10.3150/09-BEJ191",
  "categories": [
    "math.ST",
    "stat.TH"
  ],
  "abstract": "This paper deals with empirical processes of the type \\[C_n(B)=\\sqrt{n}\\{\\mu_n(B)-P(X_{n+1}\\in B\\mid X_1,...,X_n)\\},\\] where $(X_n)$ is a sequence of random variables and $\\mu_n=(1/n)\\sum_{i=1}^n\\delta_{X_i}$ the empirical measure. Conditions for $\\sup_B|C_n(B)|$ to converge stably (in particular, in distribution) are given, where $B$ ranges over a suitable class of measurable sets. These conditions apply when $(X_n)$ is exchangeable or, more generally, conditionally identically distributed (in the sense of Berti et al. [Ann. Probab. 32 (2004) 2029--2052]). By such conditions, in some relevant situations, one obtains that $\\sup_B|C_n(B)|\\stackrel{P}{\\to}0$ or even that $\\sqrt{n}\\sup_B|C_n(B)|$ converges a.s. Results of this type are useful in Bayesian statistics.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-01-13T13:14:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dependent data",
      "predictive distributions",
      "convergence",
      "conditions",
      "paper deals"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}