{
  "id": "math/0410105",
  "version": "v1",
  "published": "2004-10-05T17:08:37.000Z",
  "updated": "2004-10-05T17:08:37.000Z",
  "title": "Limit theorems for a class of identically distributed random variables",
  "authors": [
    "Patrizia Berti",
    "Luca Pratelli",
    "Pietro Rigo"
  ],
  "comment": "Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/009117904000000676",
  "journal": "Annals of Probability 2004, Vol. 32, No. 3A, 2029-2052",
  "doi": "10.1214/009117904000000676",
  "categories": [
    "math.PR"
  ],
  "abstract": "A new type of stochastic dependence for a sequence of random variables is introduced and studied. Precisely, (X_n)_{n\\geq 1} is said to be conditionally identically distributed (c.i.d.), with respect to a filtration (G_n)_{n\\geq 0}, if it is adapted to (G_n)_{n\\geq 0} and, for each n\\geq 0, (X_k)_{k>n} is identically distributed given the past G_n. In case G_0={\\varnothing,\\Omega} and G_n=\\sigma(X_1,...,X_n), a result of Kallenberg implies that (X_n)_{n\\geq 1} is exchangeable if and only if it is stationary and c.i.d. After giving some natural examples of nonexchangeable c.i.d. sequences, it is shown that (X_n)_{n\\geq 1} is exchangeable if and only if (X_{\\tau(n)})_{n\\geq 1} is c.i.d. for any finite permutation \\tau of {1,2,...}, and that the distribution of a c.i.d. sequence agrees with an exchangeable law on a certain sub-\\sigma-field. Moreover, (1/n)\\sum_{k=1}^nX_k converges a.s. and in L^1 whenever (X_n)_{n\\geq 1} is (real-valued) c.i.d. and E[| X_1| ]<\\infty. As to the CLT, three types of random centering are considered. One such centering, significant in Bayesian prediction and discrete time filtering, is E[X_{n+1}| G_n]. For each centering, convergence in distribution of the corresponding empirical process is analyzed under uniform distance.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-10-05T17:08:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B10",
      "60G09",
      "60F05",
      "60F15"
    ],
    "keywords": [
      "identically distributed random variables",
      "limit theorems",
      "finite permutation",
      "kallenberg implies",
      "natural examples"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.....10105B"
    }
  }
}