  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.
