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 $F_μ$, 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.