Necessary and sufficient conditions for convergence in distribution of quantile and P-P processes in $L^1(0,1)$
Brendan K. Beare, Tetsuya Kaji
TL;DR
This work derives exact necessary and sufficient conditions for convergence in distribution in $L^1(0,1)$ of two fundamental empirical processes: the quantile process and the P-P process. It shows that the quantile process $\sqrt{n}(Q_n-Q)$ converges in $L^1(0,1)$ if and only if the quantile function $Q$ is locally absolutely continuous on $(0,1)$ and satisfies $\int_0^1 \sqrt{u(1-u)}\,dQ(u)<\infty$ (equivalently $\int_{-\infty}^{\infty}\sqrt{F(x)(1-F(x))}\,dx<\infty$), while the P-P process converges in $L^1(0,1)$ precisely when the P-P curve $R=F(Q)$ is locally absolutely continuous on $(0,1)$. The proofs leverage a delta-method variant based on quasi-Hadamard differentiability of the generalized inverse and Brownian-bridge limits, together with bootstrap validity results. The findings clarify when bootstrap approximations are valid for these processes and illuminate how regularity of the quantile and P-P curves governs weak convergence in $L^1$, with practical implications for inference on quantiles and percentiles across two-sample settings.
Abstract
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 on the open unit interval and satisfies a slight strengthening of square integrability. We further establish a necessary and sufficient condition for the P-P process based on iid sampling from two populations to converge in distribution in $L^1(0,1)$. The condition is that the P-P curve is locally absolutely continuous on the open unit interval. If either process converges in distribution then it may be approximated using the bootstrap.