Some central limit theorems for random walks associated with hypergeometric functions of type BC
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Date
2018-02
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Abstract
The spherical functions of the noncompact Grassmann manifolds $G_{p,q}(\mathbb F)=G/K$ over $\mathbb F=\mathbb R, \mathbb C, \mathbb H$ with rank $q\ge1$ and
dimension parameter $p>q$ are Heckman-Opdam hypergeometric functions of type BC, when the double coset spaces $G//K$ are identified with the Weyl chamber $C_q^B\subset \mathbb R^q$ of type B. The associated double coset hypergroups on $ C_q^B$ can be embedded into a continuous family of commutative hypergroups $(C_q^B,*_p)$ with $p\in[2q-1,\infty[$ associated with these hypergeometric functions by Rösler. Several limit theorems for random walks on these hypergroups were recently derived by Voit (2017). We here present further limit theorems when the time as well as $p$ tend to $\infty$. For integers $p$, this admits interpretations for group-invariant random walks on the Grassmannians $G/K$.
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hypergeometric functions associated with root systems, Heckman-Opdam theory, noncompact Grassmann manifolds, spherical functions, random walks on symmetric spaces, random walks on hypergroups, moment functions, central limit theorems, dimension to infinity