Establish logarithmic-weight sequential monitoring theory

Establish whether the logarithmic weighting schemes based on rho_beta(x)=x^{1/2}\log^beta(e/x) can be used for sequential monitoring, including the resulting shorter detection delays for polynomially growing dimension.

Background

The paper develops Holder-type multiscale statistics for sequential change-point detection and retrospective segmentation. For retrospective segmentation in polynomially growing dimensions, it discusses replacing the polynomial weighting by logarithmic weights of the form h1/2log⁡β(eN/h)h^{1/2}\log^\beta(eN/h), which can improve detection of short-lived changes and narrow confidence intervals. The authors state that analogous logarithmic weighting could plausibly be applied to sequential monitoring, potentially yielding detection delays of logarithmic order, but they do not formally establish the corresponding sequential HolderGauss approximation or theoretical guarantees.

References

The open-end alternative theory treats changes at $k\ast=\lfloor nr_0\rfloor$ for fixed $r_0>1$. A distinct problem is to allow the scaled change location itself to diverge, $r_0=r_{0,n}\to\infty$; in that regime, detectability depends jointly on the change magnitude and the growth of the weight $g(r_{0,n})$.

— Online Change-Point Monitoring for Object-valued Time Series  (2609.10889 - Zhang et al., 9 Sep 2026) in Section 5, Conclusion

Even though we have not formally proved it, it seems clear that using the suggested weighting schemes based on \rho_\beta could also be used for the task of sequential monitoring.

— Change Point Detection and Localization in High-Dimensional Time Series  (2608.14344 - Bastian et al., 14 Aug 2026) in Section "Optimality and dependence" (Section 2.3 in the paper's theoretical discussion)

The weighted multiscale binomial scan of Algorithm~\ref{alg:test} closes both gaps, attaining every detection boundary and the optimal exponential rate without knowledge of the next-token distributions, the entropy budget, or the edit fraction (Theorems~\ref{thm:boundary_adaptive}, \ref{thm:activehigh_adaptive} and \ref{thm:efficiency_adaptive}), although extending these guarantees to the repeated contexts of the official implementation in remains open.

— textGrain watermarking is harder to detect: can sum-tests still do it optimally?  (2610.17655 - Bhattacharyya et al., 13 Oct 2026) in Section Discussion