Dispersion and Limit Theorems for Random Walks Associated with Hypergeometric Functions of Type BC

Michael Voit ()
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Michael Voit: Fakultät Mathematik, Technische Universität Dortmund

Journal of Theoretical Probability, 2017, vol. 30, issue 3, 1130-1169

Abstract: Abstract The spherical functions of the non-compact Grassmann manifolds $$G_{p,q}({\mathbb {F}})=G/K$$ G p , q ( F ) = G / K over the (skew-)fields $${\mathbb {F}}={\mathbb {R}}, {\mathbb {C}}, {\mathbb {H}}$$ F = R , C , H with rank $$q\ge 1$$ q ≥ 1 and dimension parameter $$p>q$$ 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 $$ C_q^B\subset {\mathbb {R}}^q$$ C q B ⊂ R q of type B. The corresponding product formulas and Harish-Chandra integral representations were recently written down by M. Rösler and the author in an explicit way such that both formulas can be extended analytically to all real parameters $$p\in [2q-1,\infty [$$ p ∈ [ 2 q - 1 , ∞ [ , and that associated commutative convolution structures $$*_p$$ ∗ p on $$C_q^B$$ C q B exist. In this paper, we study the associated moment functions and the dispersion of probability measures on $$C_q^B$$ C q B with the aid of this generalized integral representation. This leads to strong laws of large numbers and central limit theorems for associated time-homogeneous random walks on $$(C_q^B, *_p)$$ ( 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 explicitly. 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 $$GL(q,{\mathbb {F}})/U(q,{\mathbb {F}})$$ G L ( q , F ) / U ( q , F ) , which are related to Weyl chambers of type A, and for which corresponding results hold. For the rank-one-case $$q=1$$ q = 1 , the results of this paper are well known in the context of Jacobi-type hypergroups on $$[0,\infty [$$ [ 0 , ∞ [ .

Keywords: Hypergeometric functions associated with root systems; Heckman–Opdam theory; Non-compact Grassmann manifolds; Spherical functions; Random walks on symmetric spaces; Random walks on hypergroups; Dispersion; Moment functions; Central limit theorems; Strong laws of large numbers; 33C67; 43A90; 43A62; 60B15; 33C80; 60F05; 60F15 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s10959-016-0669-5

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