{
  "id": "1708.02521",
  "version": "v1",
  "published": "2017-08-08T15:28:21.000Z",
  "updated": "2017-08-08T15:28:21.000Z",
  "title": "Stein's method for multivariate Brownian approximations of sums under dependence",
  "authors": [
    "Mikołaj Kasprzak"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We use Stein's method to obtain a bound on the distance between scaled $p$-dimensional random walks and a $p$-dimensional (correlated) Brownian Motion. We consider dependence schemes including those in which the summands in scaled sums are weakly dependent and their $p$ components are strongly correlated. We also find a bound on the rate of convergence of scaled U-statistics to Brownian Motion, representing an example of a sum of strongly dependent terms. Furthermore, we consider an $r$-regular dependency graph with independent compensated Poisson processes assigned to its edges. We calculate the distance between the $p$-dimensional vector of sums of those processes attached to $p$ vertices and a $p$-dimensional standard Brownian Motion.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-08T15:28:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B10",
      "60F17",
      "60B12",
      "60J65",
      "60E05",
      "60E15"
    ],
    "keywords": [
      "multivariate brownian approximations",
      "steins method",
      "dependence",
      "dimensional standard brownian motion",
      "independent compensated poisson processes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}