  Bessel processes $(X_{t,k})_{t\ge0}$ in $N$ dimensions are classified via
associated root systems and multiplicity constants $k\ge0$. They describe
interacting Calogero-Moser-Suther\-land particle systems with $N$ particles and
are related to $\beta$-Hermite and $\beta$-Laguerre ensembles. Recently,
several central limit theorems were derived for fixed $t>0$, fixed starting
points, and $k\to\infty$. In this paper we extend the CLT in the A-case from
start in 0 to arbitrary starting distributions by using a limit result for the
corresponding Bessel functions. We also determine the eigenvalues and
eigenvectors of the covariance matrices of the Gaussian limits and study
applications to CLTs for the intermediate particles for $k\to\infty$ and then
$N\to\infty$.
