A Note on the Cut-Locus-Corrected Fréchet Mean Central Limit Theorem
Abstract
We record two observations concerning \cite{HLW24}. First, the proof of its Theorem~1 yields the population linearization (13)--(14) at \textit{any critical point} of the Fréchet function Fμ, under its remaining geometric and measure-theoretic assumptions; neither isolation nor a nonsingular Hessian of Fμ is needed. Second, its Theorem~2 is given a shorter proof by applying the M-estimator theorem \cite[Theorem~5.23, pp.~53--54]{vdV98} to negative squared distance in normal coordinates, as opposed to the Donsker argument used there. We verify the pointwise differentiability, local square-integrable Lipschitz bound, second order expansion of population loss, consistency, and empirical maximization hypotheses. The resulting covariance retains the cut-locus correction. The same argument applies to suitably selected local sample Fréchet minima when the population Hessian is positive definite.
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BibTeX
@misc{pal2026note,
title = {{A Note on the Cut-Locus-Corrected Fréchet Mean Central Limit Theorem}},
author = {Susovan Pal},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07956},
eprint = {2611.07956},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07956},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Note on the Cut-Locus-Corrected Fréchet Mean Central Limit Theorem
AU - Susovan Pal
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07956
UR - https://arxiv.org/abs/2611.07956
AB - We record two observations concerning \cite{HLW24}. First, the proof of its Theorem~1 yields the population linearization (13)--(14) at \textit{any critical point} of the Fréchet function F_μ, under its remaining geometric and measure-theoretic assumptions; neither isolation nor a nonsingular Hessian of F_μ is needed. Second, its Theorem~2 is given a shorter proof by applying the M-estimator theorem \cite[Theorem~5.23, pp.~53--54]{vdV98} to negative squared distance in normal coordinates, as opposed to the Donsker argument used there. We verify the pointwise differentiability, local square-integrable Lipschitz bound, second order expansion of population loss, consistency, and empirical maximization hypotheses. The resulting covariance retains the cut-locus correction. The same argument applies to suitably selected local sample Fréchet minima when the population Hessian is positive definite.
ER -