Decay rate under curvature beyond Hilbert spaces

Determine which additional geometric structure, beyond the Hadamard-space assumptions, is sufficient to recover the Hilbert-space decay-in-time rate for the autocorrelation of the geodesic exponential smoothing level process.

Background

The paper proves a decay-in-time law for the autocorrelation of the geodesic exponential smoothing level only when the data space is a Hilbert space. That proof uses the martingale property of the level and linear growth of its variance, features that do not generally hold in curved Hadamard spaces.

Although the quasi-autocovariance remains nonpositive throughout Hadamard spaces, the authors leave unresolved what additional geometric or structural assumptions would recover the Hilbert-space decay rate under curvature.

References

Second, the decay-in-time law of \Cref{cor:hilbert} is proved only in Hilbert spaces, although the quasi-autocovariance remains nonpositive in every Hadamard space by \Cref{thm:innov-centering}(ii); which additional structure recovers the rate under curvature remains open.

— Exponential Smoothing for Time Series of Random Objects  (2609.20274 - Matsubara et al., 17 Sep 2026) in Section 6, Conclusion, paragraph beginning “A couple of directions for further work follow from these restrictions”