Convergence in distribution of the P-P process in $L^1[0,1]$

We show that the percentile-percentile (P-P) process constructed from an independent and identically distributed sample of pairs converges in distribution in $L^1[0,1]$ if and only if the associated P-P curve is absolutely continuous. When this condition holds, the limiting distribution is Gaussian and the process admits a valid bootstrap approximation.