A caveat on metrizing convergence in distribution on Hilbert spaces

Publikationstyp
Journal Article
Date Issued
2026-02-03
Sprache
English
Author(s)
Bassetti, Federico
Bourguin, Solesne
Campese, Simon 
Mathematik E-10  
Peccati, Giovanni  
TORE-URI
https://hdl.handle.net/11420/61558
Journal
Statistics & probability letters  
Volume
233
Article Number
110671
Citation
Statistics and Probability Letters 233: 110671 (2026)
Publisher DOI
10.1016/j.spl.2026.110671
Scopus ID
2-s2.0-105029288775
Publisher
Elsevier
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=∞. Our analysis shows that, unless p=∞, 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.
Subjects
Convergence in distribution
Hilbert spaces
Probabilistic distances
DDC Class
519: Applied Mathematics, Probabilities