Some central limit theorems for random walks associated with hypergeometric functions of type BC

TL;DR

This paper investigates limit theorems for random walks linked to hypergeometric functions of type BC, extending previous results to cases where both the time parameter and dimension parameter p tend to infinity, with applications to Grassmann manifolds.

Contribution

It introduces new limit theorems for hypergeometric function-based random walks, especially when both time and dimension parameters grow large, generalizing prior work.

Findings

Established limit theorems for large p and time parameters.

Connected hypergeometric functions to group-invariant random walks on Grassmann manifolds.

Extended previous results to broader parameter regimes.

Abstract

The spherical functions of the noncompact Grassmann manifolds over the real or complex numbers or the quaternions with rank q and dimension parameter p can be seen as Heckman-Opdam hypergeometric functions of type BC, when the double coset space is identified with some Weyl chamber of type B. The associated double coset hypergroups may be embedded into a continuous family of commutative hypergroups with these hypergeometric functions as multiplicative functions with p in some continuous parameter range by a result of R\"osler. Several limit theorems for random walks associated with these hypergroups were recently derived by the second author. We here present further limit theorems in particular for the case where the time parameter as well as p tend to infinity. For integers p, these results admit interpretations for group-invariant random walks on the Grassmann manifolds.