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

Let $Q$ be the quantile function of a probability measure $P$ on the real line and let $Q_n$ be the empirical quantile function of an i.i.d.\ sample of size $n$ from $P$. For $1\le p<\infty$ we show that $\sqrt n(Q_n-Q)$ converges in distribution in $L^p(0,1)$ if and only if $Q$ is locally absolutely continuous on $(0,1)$ and $\int_0^1 Q'(t)^p\,[t(1-t)]^{p/2}\,dt<\infty$. The limit is $Q'B$, where $B$ is a Brownian bridge. No density is assumed and atoms are allowed. The case $p=1$ is a theorem of Beare and Kaji. For $p>1$ the sufficiency part relies on a moment bound of Bobkov and Ledoux for empirical Wasserstein distances, which we extend to signed perturbations of the quantile function, and the necessity part combines the result of Beare and Kaji with Fernique's theorem. This answers a question posed by del Barrio, González-Sanz, Loubes and Rodríguez-Vítores.