Optimal one-sided logarithmic and finite-q rates for time-profile classes

Determine the optimal one-sided logarithmic and finite-q rates for the time-profile classes treated in Supplement Sections S.3 and S.6, where the existing results provide only one-sided Gaussian and stable bounds.

Background

The paper develops first- and second-order Wasserstein response formulas for finite-variance infinitely divisible laws with a Gaussian component, including stability estimates and approximation results. Its supplementary sections extend these methods to time-profile and related non-additive settings, establishing one-sided Gaussian and stable bounds as well as other comparison results.

The authors explicitly identify a remaining gap in those time-profile classes: the optimal one-sided logarithmic rates and finite-q rates have not been established. The problem is therefore to sharpen the supplementary bounds to optimal rates in both regimes.