{"paper":{"title":"Finite-Order Hilbertian Gaussian Random Tensor Estimates","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Guangqian Zhao","submitted_at":"2026-06-28T09:27:42Z","abstract_excerpt":"We prove fixed finite-chaos-order estimates for Hilbert-space-valued Gaussian random tensors. Given a finite-rank kernel \\[\n  K\\in\\cA_1\\otimes\\cdots\\otimes\\cA_m\\otimes\\cC\\otimes\\cE \\] and the associated decoupled homogeneous Gaussian chaos operator $\\cT_K^{(m)}:\\cC\\to\\cE$, we show that, for $p\\ge2$ and $2\\le r<\\infty$, \\[\n  \\|\\cT_K^{(m)}\\|_{L^p(\\Omega;\\mathfrak S_r(\\cC,\\cE))}\n  \\le C_m(p+r)^{m/2}\n  \\max_{S\\subset[m]}\\|\\cF_S(K)\\|_{\\mathfrak S_r}, \\] where $\\cF_S(K):\\cA_S\\otimes\\cC\\to\\cA_{S^c}\\otimes\\cE$ is the oriented input-output flattening. The proof is an induction on $m$ from the rectangul"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.29292","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/2606.29292/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"}