{
  "id": "math/0503463",
  "version": "v1",
  "published": "2005-03-22T15:49:18.000Z",
  "updated": "2005-03-22T15:49:18.000Z",
  "title": "Large deviations for template matching between point processes",
  "authors": [
    "Zhiyi Chi"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/105051604000000576 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Applied Probability 2005, Vol. 15, No. 1A, 153-174",
  "doi": "10.1214/105051604000000576",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the asymptotics related to the following matching criteria for two independent realizations of point processes X\\sim X and Y\\sim Y. Given l>0, X\\cap [0,l) serves as a template. For each t>0, the matching score between the template and Y\\cap [t,t+l) is a weighted sum of the Euclidean distances from y-t to the template over all y\\in Y\\cap [t,t+l). The template matching criteria are used in neuroscience to detect neural activity with certain patterns. We first consider W_l(\\theta), the waiting time until the matching score is above a given threshold \\theta. We show that whether the score is scalar- or vector-valued, (1/l)\\log W_l(\\theta) converges almost surely to a constant whose explicit form is available, when X is a stationary ergodic process and Y is a homogeneous Poisson point process. Second, as l\\to\\infty, a strong approximation for -\\log [\\Pr{W_l(\\theta)=0}] by its rate function is established, and in the case where X is sufficiently mixing, the rates, after being centered and normalized by \\sqrtl, satisfy a central limit theorem and almost sure invariance principle. The explicit form of the variance of the normal distribution is given for the case where X is a homogeneous Poisson process as well.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-03-22T15:49:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F10",
      "60G55"
    ],
    "keywords": [
      "point processes",
      "template matching",
      "large deviations",
      "explicit form",
      "sure invariance principle"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......3463C"
    }
  }
}