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

TL;DR

This paper proves that the P-P process converges in distribution in L^1[0,1] if and only if its P-P curve is absolutely continuous, with the limit being Gaussian and allowing bootstrap approximation.

Contribution

It establishes a necessary and sufficient condition for the convergence of the P-P process in L^1[0,1], linking absolute continuity to Gaussian limits.

Findings

P-P process converges in distribution in L^1[0,1] if and only if the P-P curve is absolutely continuous.

When convergent, the limiting distribution is Gaussian.

The process admits a valid bootstrap approximation.

Abstract

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.