  Multivariate Bessel processes describe the stochastic dynamics of interacting
particle systems of Calogero-Moser-Sutherland type and are related with
$\beta$-Hermite and Laguerre ensembles. It was shown by Andraus, Katori, and
Miyashita that for fixed starting points, these processes admit interesting
limit laws when the multiplicities $k$ tend to $\infty$, where in some cases
the limits are described by the zeros of classical Hermite and Laguerre
polynomials. In this paper we use SDEs to derive corresponding limit laws for
starting points of the form $\sqrt k\cdot x$ for $k\to\infty$ with $x$ in the
interior of the corresponding Weyl chambers. Our limit results are a.s. locally
uniform in time. Moreover, in some cases we present associated central limit
theorems.
