  We consider compact Grassmann manifolds $G/K$ over the real, complex or
quaternionic numbers whose spherical functions are Heckman-Opdam polynomials of
type $BC$. From an explicit integral representation of these polynomials we
deduce a sharp Mehler-Heine formula, that is an approximation of the
Heckman-Opdam polynomials in terms of Bessel functions, with a precise estimate
on the error term. This result is used to derive a central limit theorem for
random walks on the semi-lattice parametrizing the dual of $G/K$, which are
constructed by successive decompositions of tensor powers of spherical
representations of $G$. The limit is the distribution of a Laguerre ensemble in
random matrix theory. Most results of this paper are established for a larger
continuous set of multiplicity parameters beyond the group cases.
