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A Caveat on Metrizing Convergence in Distribution on Hilbert Spaces
Federico Bassetti, Solesne Bourguin, Simon Campese, Giovanni Pec · 2025-09-17 · via math.PR updates on arXiv.org

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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