{"paper":{"title":"The Laplace asymptotic expansion in high dimensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.CA","authors_text":"Anya Katsevich","submitted_at":"2024-06-18T15:19:02Z","abstract_excerpt":"We prove that the classical Laplace asymptotic expansion (AE) of $\\int_{\\mathbb R^d} g(x)e^{-nu(x)}dx$, $n\\gg1$ extends to the high-dimensional regime in which $d$ may grow large with $n$. More specifically, we use new techniques suitable to high-$d$ to derive an AE which formally coincides with the classical one because the terms are the same, but which now has a new small parameter. Namely under classical assumptions on $z$ and $g$ and additional bounds on the growth of $\\|\\nabla^kz\\|$ and $\\|\\nabla^kg\\|$ with $d$, we show the new small parameter is $d^2/n$, in the sense that $|\\text{Rem}_L|"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2406.12706","kind":"arxiv","version":3},"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/2406.12706/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"}