{
  "id": "1506.04925",
  "version": "v1",
  "published": "2015-06-16T11:25:53.000Z",
  "updated": "2015-06-16T11:25:53.000Z",
  "title": "Dispersion and limit theorems for random walks associated with hypergeometric functions of type BC",
  "authors": [
    "Michael Voit"
  ],
  "comment": "Extended version of arXiv:1205.4866",
  "categories": [
    "math.CA",
    "math-ph",
    "math.MP",
    "math.PR",
    "math.RT"
  ],
  "abstract": "The spherical functions of the noncompact Grassmann manifolds $G_{p,q}(\\mathbb F)=G/K$ over the (skew-)fields $\\mathbb F=\\mathbb R, \\mathbb C, \\mathbb H$ with rank $q\\ge1$ and dimension parameter $p>q$ can be described as Heckman-Opdam hypergeometric functions of type BC, where the double coset space $G//K$ is identified with the Weyl chamber $ C_q^B\\subset \\mathbb R^q$ of type B. The corresponding product formulas and Harish-Chandra integral representations were recently written down by M. R\\\"osler and the author in an explicit way such that both formulas can be extended analytically to all real parameters $p\\in[2q-1,\\infty[$, and that associated commutative convolution structures $*_p$ on $C_q^B$ exist. In this paper we introduce moment functions and the dispersion of probability measures on $C_q^B$ depending on $*_p$ and study these functions with the aid of this generalized integral representation. Moreover, we derive strong laws of large numbers and central limit theorems for associated time-homogeneous random walks on $(C_q^B, *_p)$ where the moment functions and the dispersion appear in order to determine drift vectors and covariance matrices of these limit laws explicitely. For integers $p$, all results have interpretations for $G$-invariant random walks on the Grassmannians $G/K$. Besides the BC-cases we also study the spaces $GL(q,\\mathbb F)/U(q,\\mathbb F)$, which are related to Weyl chambers of type A, and for which corresponding results hold. For the rank-one-case $q=1$, the results of this paper are well-known in the context of Jacobi-type hypergroups on $[0,\\infty[$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-16T11:25:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C67",
      "43A90",
      "43A62",
      "60B15",
      "33C80",
      "60F05",
      "60F15"
    ],
    "keywords": [
      "type bc",
      "dispersion",
      "weyl chamber",
      "moment functions",
      "invariant random walks"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150604925V"
    }
  }
}