The quantile process converges in distribution in L^1(0,1) if and only if its quantile function is locally absolutely continuous and satisfies a strengthened square-integrability condition.
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A necessary and sufficient condition for convergence in distribution of the quantile process in $L^1(0,1)$
The quantile process converges in distribution in L^1(0,1) if and only if its quantile function is locally absolutely continuous and satisfies a strengthened square-integrability condition.