A Note on the Cut-Locus-Corrected Fréchet Mean Central Limit Theorem

We record two observations concerning {HLW24}. First, the proof of its Theorem~1 yields the population linearization (13)--(14) at 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 [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.