{
  "id": "2310.13983",
  "version": "v1",
  "published": "2023-10-21T12:11:42.000Z",
  "updated": "2023-10-21T12:11:42.000Z",
  "title": "Iterates of multidimensional Bernstein-type operators and diffusion processes in population genetics",
  "authors": [
    "Takatoshi Hirano",
    "Ryuya Namba"
  ],
  "comment": "41 pages",
  "categories": [
    "math.PR",
    "math.FA"
  ],
  "abstract": "The Bernstein operator is known as a typical example of positive linear operators which uniformly approximates continuous functions on $[0, 1]$. In the present paper, we introduce a multidimensional extension of the Bernstein operator which is associated with a transition probability of a certain discrete Markov chain. In particular, we show that the iterate of the multidimensional Bernstein-type operator uniformly converges to the Feller semigroup corresponding to the multidimensional Wright-Fisher diffusion process with mutation arising in the study of population genetics, together with its rate of convergence. The convergence of process-level is obtained as well. Moreover, by taking the limit as both the number of iterate and the dimension of the Bernstein-type operator tend to infinity simultaneously, we prove that the iterate of the multidimensional Bernstein-type operator uniformly converges to the Feller semigroup corresponding to a probability measure-valued Fleming-Viot process with mutation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-21T12:11:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J60",
      "41A36",
      "60J70",
      "60G53",
      "60F05"
    ],
    "keywords": [
      "population genetics",
      "multidimensional bernstein-type operator uniformly converges",
      "multidimensional wright-fisher diffusion process",
      "feller semigroup corresponding",
      "bernstein operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}