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

TL;DR

This paper develops a unified framework for central limit theorems related to hyperbolic spaces and Jacobi processes, utilizing Jacobi functions and their limit behaviors, with some results improving existing knowledge and others being entirely new.

Contribution

It introduces new limit theorems for Jacobi functions and radial random walks on hyperbolic spaces, expanding understanding of their asymptotic behaviors.

Findings

New CLTs for hyperbolic space random walks.

Enhanced limit results for Jacobi functions.

Some results are completely novel, others improve existing theorems.

Abstract

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 the positive half line 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 the cases where the first parameter (i.e., the dimension of the hyperbolic space) tends to infinity as well as the cases $\phi_{i\rho-\lambda}^{(\alpha,\beta)}(t)$ for small $\lambda$, and $\phi_{i\rho-n\lambda}^{(\alpha,\beta)}(t/n)$ for $n\to\infty$. The proofs of all these limit results are based on the known Laplace integral representation for Jacobi functions. Parts of the limit results for Jacobi functions and of the CLTs are known, other improve known ones, and other are completely new.