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 $h^{1/2}\log^\beta(eN/h)$, 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.