{
  "id": "1105.3862",
  "version": "v2",
  "published": "2011-05-19T13:13:04.000Z",
  "updated": "2011-07-23T10:36:31.000Z",
  "title": "A BK inequality for randomly drawn subsets of fixed size",
  "authors": [
    "J. van den Berg",
    "J. Jonasson"
  ],
  "comment": "Revised version for PTRF. Equation (13) corrected. Several, mainly stylistic, changes; more compact",
  "categories": [
    "math.PR"
  ],
  "abstract": "The BK inequality (\\cite{BK85}) says that,for product measures on $\\{0,1\\}^n$, the probability that two increasing events $A$ and $B$ `occur disjointly' is at most the product of the two individual probabilities. The conjecture in \\cite{BK85} that this holds for {\\em all} events was proved by Reimer (cite{R00}). Several other problems in this area remained open. For instance, although it is easy to see that non-product measures cannot satisfy the above inequality for {\\em all} events,there are several such measures which, intuitively, should satisfy the inequality for all{\\em increasing} events. One of the most natural candidates is the measure assigning equal probabilities to all configurations with exactly $k$ 1's (and probability 0 to all other configurations). The main contribution of this paper is a proof for these measures. We also point out how our result extends to weighted versions of these measures, and to products of such measures.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-07-23T10:36:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60C05",
      "60K35"
    ],
    "keywords": [
      "randomly drawn subsets",
      "bk inequality",
      "fixed size",
      "probability",
      "measure assigning equal probabilities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1105.3862V"
    }
  }
}