Sharper upper bounds for the local entropy integral

Obtain a sharper upper bound for the local entropy integral in Condition A for the quadratic Wasserstein space, ideally one closer to the square-root logarithmic lower bound established in the paper, to support Fréchet regression beyond the global approach and other statistical problems relying on empirical-process theory.

Background

The paper studies a commonly used local entropy condition for empirical-process analyses of Fréchet regression in the quadratic Wasserstein space of univariate probability distributions. It proves that the entropy integral diverges as the local radius tends to zero, both on the full compactly supported Wasserstein space and on a regular subclass of distribution functions satisfying fixed two-sided Lipschitz bounds.

The discussion notes that weaker radius-dependent upper bounds are available and that global Fréchet regression can achieve the optimal parametric rate through a projection argument. However, a sharper upper bound for the entropy integral remains unresolved, with the paper identifying this as relevant to Fréchet regression methods beyond the global approach and to other Wasserstein-space statistical problems based on empirical-process theory.