Lower bounds for pointwise majorization envelopes

Establish matching lower bounds for the pointwise majorization envelopes under suitable nondegeneracy, small-ball, or two-sided increment assumptions, and determine whether the local ball-mass terms, anchor-dependent confidence scales, and their separation across distinct increment pseudo-metrics are unavoidable.

Background

The paper develops upper bounds for Banach-valued stochastic processes with sub-Weibull increments and two-metric mixed-tail increments. These bounds depend pointwise on fixed-measure ball-mass functionals and on confidence scales determined by the distance from each index to an anchor.

The authors note that an increment upper bound alone cannot yield a meaningful nontrivial lower bound because degenerate processes satisfy the same condition. They therefore identify the development of matching lower bounds as an open problem requiring additional assumptions such as nondegeneracy, small-ball conditions, or two-sided increment control. The unresolved issue is whether the specific components of the upper bounds—the local ball-mass terms, anchor-dependent confidence scales, and their separation across different metrics—are necessary under such strengthened assumptions.

References

Finally, matching lower bounds remain open. The Gaussian lower bounds of \citet{xu2026} provide a starting point, but an upper sub-Weibull or mixed-tail increment condition alone cannot imply a nontrivial lower bound, since it is also satisfied by degenerate processes. A meaningful lower-bound theory will therefore require suitable nondegeneracy, small-ball, or two-sided increment assumptions. Under such conditions, one may ask whether the local ball-mass terms, the anchor-dependent confidence scales, and their separation across distinct increment pseudo-metrics are unavoidable.

— Pointwise Majorization for sub-Weibull and Mixed Tail Processes with Applications in Quadratic Chaos and Ergodic Diffusions  (2609.01576 - Hu et al., 1 Sep 2026) in Section Discussion