Asymptotically Anytime-valid Inference for Fréchet Variance

Fréchet variance quantifies dispersion for random objects in metric spaces. Existing inference is mainly fixed-sample and does not justify repeated inspection or data-dependent stopping. We construct positive asymptotic confidence sequences by applying time-uniform normal approximations to log Fréchet variance and then exponentiating. The offline procedure recomputes empirical Fréchet means, while the online procedure uses predictable squared-distance losses based on Riemannian stochastic-gradient centres. Under local smoothness, curvature and moment conditions, estimating or tracking the Fréchet mean has a uniformly lower-order effect, yielding asymptotic anytime-valid coverage and boundary-dependent width rates. Simulations and an EEG covariance example illustrate the effect of repeated monitoring and the computational gain from the online construction.