Central limit theorems for hyperbolic spaces and Jacobi processes on $$[0,\infty [$$

2015

SIGMA

Abstract. We consider compact Grassmann manifolds G/K over the real, complex or quaternionic numbers whose spherical functions are Heckman-Opdam polynomials of type BC. From an explicit integral representation of these polynomials we deduce a sharp Mehler-Heine formula, that is an approximation of the Heckman-Opdam polynomials in terms of Bessel functions, with a precise estimate on the error term. This result is used to derive a central limit theorem for random walks on the semi-lattice parametrizing the dual of G/K, which are constructed by successive decompositions of tensor powers of spherical representations of G. The limit is the distribution of a Laguerre ensemble in random matrix theory. Most results of this paper are established for a larger continuous set of multiplicity parameters beyond the group cases.

Section: Introductionmentioning

confidence: 99%

A Central Limit Theorem for Random Walks on the Dual of a Compact Grassmannian

Rösler

2015

SIGMA

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

2016

J Theor Probab

Self Cite

The spherical functions of the noncompact Grassmann manifolds Gp,q(F) = G/K over the (skew-)fields F = R, C, H with rank q ≥ 1 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 B q ⊂ 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 ∈ [2q − 1, ∞[, and that associated commutative convolution structures * p on C B q exist. In this paper we study the associated moment functions and the dispersion of probability measures on C B q 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 B q , * 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, 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, the results of this paper are well-known in the context of Jacobi-type hypergroups on [0, ∞[.

Integral representation and uniform limits for some Heckman-Opdam hypergeometric functions of type \textsl{BC}

Rösler

Trans. Amer. Math. Soc.

2015

Self Cite

The Heckman-Opdam hypergeometric functions of type BC extend classical Jacobi functions in one variable and include the spherical functions of non-compact Grassmann manifolds over the real, complex or quaternionic numbers. There are various limit transitions known for such hypergeometric functions, see e.g.[dJ], [RKV]. In the present paper, we use an explicit form of the Harish-Chandra integral representation as well as an interpolated variant, in order to obtain limit results for three continuous classes of hypergeometric functions of type BC which are distinguished by explicit, sharp and uniform error bounds. The first limit realizes the approximation of the spherical functions of infinite dimensional Grassmannians of fixed rank; here hypergeometric functions of type A appear as limits. The second limit is a contraction limit towards Bessel functions of Dunkl type.