{
  "id": "1511.00224",
  "version": "v1",
  "published": "2015-11-01T10:13:55.000Z",
  "updated": "2015-11-01T10:13:55.000Z",
  "title": "Linearity of regression for weak records, revisited",
  "authors": [
    "Rafał Karczewski",
    "Jacek Wesołowski"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Since many years characterization of distribution by linearity of regression of non-adjacent weak records E(W_{i+s}|W_i) = \\beta_1 W_i+\\beta_0 for discrete observations has been known to be a difficult question. Lopez- Blazquez (2004) proposed an interesting idea of reducing it to the adjacent case and claimed to have the characterization problem completely solved. We will explain that, unfortunately, there is a major aw in the proof given in that paper. This aw is related to fact that in some situations the operator responsible for reduction of the non-adjacent case to the adjacent one is not injective. The operator is trivially injective when 0<\\beta_1<1. We show that when \\beta_1>=1 the operator is injective when s = 2, 3, 4. Therefore in these cases the method proposed by Lopez-Blazquez is valid. We also show that the operator is not injective when \\beta_1 >=1 and s >= 5. Consequently, in this case the reduction methodology does not work and thus the characterization problem remains open.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-01T10:13:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "62E10",
      "62G30"
    ],
    "keywords": [
      "regression",
      "characterization problem remains open",
      "non-adjacent weak records",
      "discrete observations",
      "major aw"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}