{"paper":{"title":"Exact Signature Tail Asymptotics for Pure Rough Paths","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Nannan Li, Xing Gao","submitted_at":"2026-06-29T09:45:41Z","abstract_excerpt":"We prove~\\cite[Conjecture 2.12]{BGS20} on the signature tail asymptotics of pure rough paths and extend it to arbitrary reasonable tensor norms. In more details, let \\[ \\mathbf X_t=\\exp(tl) \\,\\text{ with }\\, l=l_1+\\cdots+l_m\\,\\text{ and }\\, l_r\\in\\mathcal L_r(V), \\] be a pure $m$-rough path over a finite dimensional real or complex Banach space, and equip the tensor powers of $V$ with arbitrary reasonable tensor algebra norms. We prove that \\[ \\limsup_{n\\to\\infty}\\left(\\left(\\frac{n}{m}\\right)!\\left\\|\\pi_n(\\exp l)\\right\\|_n\\right)^{m/n}=\\|l_m\\|_m . \\] In particular, this identifies the signatu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.30055","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/2606.30055/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"}