A Caveat on Metrizing Convergence in Distribution on Hilbert Spaces

TL;DR

The paper shows that Sobolev-type integral probability distances on Hilbert-space-valued distributions, defined via the second Fréchet derivative and Schatten-class constraints, fail to metrize convergence in distribution in infinite dimensions for any finite Schatten index $p$. It clarifies that while the $ ho_ty$ distance does metrize distributional convergence, the family $\rho_p$ with $p\in[1,\infty)$ does not, resolving misconceptions linked to the related $d_2$ distance in prior literature. The authors prove this by a Gaussian-counterexample argument: constructing a sequence of Gaussian elements with covariances $S_n$ that converge in operator norm to a limit yet violate tight distributional convergence under $\rho_p$, contradicting metrizability. The results underscore a precise obstruction stemming from the density of Schatten classes in the compact operators, and hence explain why some infinite-dimensional CLT approaches based on $d_2$ or $\rho_p$ are not generally valid.

Abstract

We consider Sobolev-type distances on probability measures over separable Hilbert spaces involving the Schatten-$p$ norms, which include as special cases a distance first introduced by Bourguin and Campese (2020) when $p=2$, and a distance introduced by Giné and Leon (1980) when $p=\infty$. Our analysis shows that, unless $p=\infty$, these distances fail to metrize convergence in distribution in infinite dimensions. This clarifies several inconsistencies and misconceptions in the recent literature that arose from confusion between different types of distances.

Theorems & Definitions (12)

• Theorem 1
• Lemma 1: Representation of Bilinear Forms
• Definition 1
• Lemma 2: Smooth Radial Mappings
• Definition 2
• Remark 1
• Remark 2
• Proposition 1
• Proposition 2
• proof