A Caveat on Metrizing Convergence in Distribution on Hilbert Spaces

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.

💡 Research Summary

The paper investigates a family of Sobolev‑type probabilistic distances ρₚ defined on the space P(K) of probability measures over a real separable Hilbert space K. Each distance is built by taking the supremum of differences of expectations of test functions f∈C²(K,ℝ) whose first Fréchet derivative is uniformly bounded and whose second derivative D²f(x) belongs to the Schatten‑p class Sₚ(K) with norm ≤1. The case p=2 corresponds to the distance d₂ introduced by Bourguin and Campese (2020); the case p=∞ coincides (up to minor variations) with the Zolotarev‑type distance ρ_∞ studied by Giné‑Leon (1980) and later by others.

The authors’ main result (Theorem 1) states that if K is infinite‑dimensional and p∈