Limit theorems for radial random walks on homogeneous spaces with growing dimensions

Voit, Michael

doi:10.1142/9789812832825_0020

Infinite Dimensional Harmonic Analysis IV 2008

DOI: 10.1142/9789812832825_0020

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Limit theorems for radial random walks on homogeneous spaces with growing dimensions

Michael Voit

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Cited by 4 publications

References 16 publications

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

Voit

2012

Monatsh Math

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We present a unified approach to a couple of central limit theorems for radial random walks on hyperbolic spaces and time-homogeneous Markov chains on [0, ∞[ whose transition probabilities are defined in terms of the Jacobi convolutions. The proofs of all results are based on limit results for the associated Jacobi functions. In particular, we consider α → ∞, the case ϕ (α,β) iρ−λ (t) for small λ, and ϕ (α,β) iρ−nλ (t/n) for n → ∞. The proofs of all these limit results are based on the known Laplace integral representation for Jacobi functions. Parts of the results are known, other improve known ones, and other are new.

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

Voit

2012

Monatsh Math

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Moment Functions and Central Limit Theorem for Jacobi Hypergroups on [ $$0,\infty $$ [

Grundmann

2012

J Theor Probab

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

Rösler

Voit

2015

Trans. Amer. Math. Soc.

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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.

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Limit theorems for radial random walks on homogeneous spaces with growing dimensions

Cited by 4 publications

References 16 publications

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

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

Moment Functions and Central Limit Theorem for Jacobi Hypergroups on [ $$0,\infty $$ [

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

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