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A necessary and sufficient condition for convergence in distribution of the quantile process in $L^1(0,1)$
Brendan K. Beare, Tetsuya Kaji · 2025-02-03 · via math.ST updates on arXiv.org

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 and satisfies a slight strengthening of square integrability. If the quantile process converges in distribution then it may be approximated using the bootstrap.

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