{
  "id": "1806.07347",
  "version": "v1",
  "published": "2018-06-19T16:54:46.000Z",
  "updated": "2018-06-19T16:54:46.000Z",
  "title": "Couplings for determinantal point processes and their reduced Palm distributions with a view to quantifying repulsiveness",
  "authors": [
    "Jesper Møller",
    "Eliza O'Reilly"
  ],
  "comment": "30 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "For a determinantal point process $X$ with a kernel $K$ whose spectrum is strictly less than one, Andr\\'e Goldman has established a coupling to its reduced Palm process $X^u$ at a point $u$ with $K(u,u)>0$ so that in distribution $X^u$ is obtained by removing a finite number of points from $X$. The intensity function of the difference $X\\setminus X^u$ is known, but apart from special cases the distribution of $X\\setminus X^u$ is unknown. Considering the restriction $X_S$ of $X$ to any compact set $S$, we establish a coupling of $X_S$ and its reduced Palm process $X^u_S$ so that the difference is at most one point. Specifically, we assume $K$ restricted to $S\\times S$ is either (i) a projection or (ii) has spectrum strictly less than one. In case of (i), we have in distribution that $X^u_S$ is obtained by removing one point from $X_S$, and we can specify the distribution of this point. In case of (ii), in distribution we obtain $X^u_S$ either by moving one point in $X$ or by removing one point from $X_S$, and to a certain extent we can describe the distribution of these points. We discuss how Goldman's and our results can be used for quantifying repulsiveness in $X$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-19T16:54:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60D05",
      "60G55",
      "60K35"
    ],
    "keywords": [
      "determinantal point process",
      "reduced palm distributions",
      "quantifying repulsiveness",
      "reduced palm process",
      "finite number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}