A Caveat on Metrizing Convergence in Distribution on Hilbert Spaces

Federico Bassetti; Solesne Bourguin; Simon Campese; Giovanni Peccati

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arXiv · 2509.13427

A Caveat on Metrizing Convergence in Distribution on Hilbert Spaces

Federico Bassetti·Solesne Bourguin·Simon Campese·Giovanni Peccati

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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.

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Federico Bassetti, Solesne Bourguin, Simon Campese, Giovanni Peccati. 2026-01-31. A Caveat on Metrizing Convergence in Distribution on Hilbert Spaces. https://arxiv.org/abs/2509.13427

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