  Given a sequence of positive Hermitian holomorphic line bundles $(L_p,h_p)$
on a K\"ahler manifold $X$, we establish the asymptotic expansion of the
Bergman kernel of the space of global holomorphic sections of $L_p$, under a
natural convergence assumption on the sequence of curvatures $c_1(L_p,h_p)$. We
then apply this to study the asymptotic distribution of common zeros of random
sequences of $m$-tuples of sections of $L_p$ as $p\to\infty$.
