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

Dispersion and Limit Theorems for Random Walks Associated with Hypergeometric Functions of Type BC 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 , ∞ [ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Theoretical Probability Springer Journals

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

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Publisher
Springer US
Copyright
Copyright © 2016 by Springer Science+Business Media New York
Subject
Mathematics; Probability Theory and Stochastic Processes; Statistics, general
ISSN
0894-9840
eISSN
1572-9230
D.O.I.
10.1007/s10959-016-0669-5
Publisher site
See Article on Publisher Site

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 , ∞ [ .

Journal

Journal of Theoretical ProbabilitySpringer Journals

Published: Feb 4, 2016

References

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