  Let $X$ be a compact normal complex space of dimension $n$, and $L$ be a
holomorphic line bundle on $X$. Suppose $\Sigma=(\Sigma_1,\ldots,\Sigma_\ell)$
is an $\ell$-tuple of distinct irreducible proper analytic subsets of $X$,
$\tau=(\tau_1,\ldots,\tau_\ell)$ is an $\ell$-tuple of positive real numbers,
and consider the space $H^0_0 (X, L^p)$ of global holomorphic sections of
$L^p:=L^{\otimes p}$ that vanish to order at least $\tau_{j}p$ along
$\Sigma_{j}$, $1\leq j\leq\ell$. We find necessary and sufficient conditions
which ensure that $\dim H^0_0(X,L^p)\sim p^n$, analogous to Ji-Shiffman's
criterion for big line bundles. We give estimates of the partial Bergman
kernel, investigate the convergence of the Fubini-Study currents and their
potentials, and the equilibrium distribution of normalized currents of
integration along zero divisors of random holomorphic sections in $H^0_0 (X,
L^p)$ as $p\to\infty$. Regularity results for the equilibrium envelope are also
included.
