{"paper":{"title":"Logarithmic energy distances and Gini covariance for Hilbert-valued random elements","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"stat.ME","authors_text":"M. Dolores Jim\\'enez-Gamero, Norbert Henze","submitted_at":"2026-06-16T18:14:09Z","abstract_excerpt":"For $\\alpha\\in(0,2)$, the generalized energy distance and the Gini covariance statistic are based on kernels of the form $(x,y)\\mapsto \\|x-y\\|^\\alpha$, where $\\|\\cdot\\|$ denotes the norm in a real separable Hilbert space. This paper investigates the boundary regime $\\alpha\\downarrow 0$. After suitable normalization, the corresponding energy distance converges to a logarithmic energy distance involving the kernel $(x,y)\\mapsto\\log\\|x-y\\|$. We establish that the resulting logarithmic energy distance retains the fundamental characterization property of ordinary energy distances in separable Hilbe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.18365","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.18365/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}