  Given a classical symmetric pair, $(G,K)$, with $\mathfrak g = Lie(G)$, we
provide descriptions of the Hilbert series of the algebra of $K$-invariant
vectors in the associated graded algebra of $\mathcal U(\mathfrak g)$ viewed as
a $K$-representation under restriction of the adjoint representation. The
description illuminates a certain stable behavior of the Hilbert series, which
is investigated in a case-by-case basis. We note that the stable Hilbert series
of one symmetric pair often coincides with others. Also, for the case of the
real form $U(p,q)$ we derive a closed expression for the Hilbert series when
$\min(p,q) \to \infty$.
