A necessary and sufficient condition for convergence in distribution of the quantile process in $L^1(0,1)$

We establish a necessary and sufficient condition for the quantile process based on iid sampling to converge in distribution in $L^1(0,1)$. The condition is that the quantile function is locally absolutely continuous and satisfies a slight strengthening of square integrability. If the quantile process converges in distribution then it may be approximated using the bootstrap.