{"paper":{"title":"Poincar\\'e Inequality for Dirichlet Distributions and Infinite-Dimensional Generalizations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"F.-Y. Wang, L. Miclo, S. Feng","submitted_at":"2015-04-11T03:50:00Z","abstract_excerpt":"For any $N\\ge 2$ and $\\aa:=(\\aa_1,\\cdots, \\aa_{N+1})\\in (0,\\infty)^{N+1}$, let $\\mu^{(N)}_{\\aa}$ be the corresponding Dirichlet distribution on $\\DD:= \\big\\{ x=(x_i)_{1\\le i\\le N}\\in [0,1]^N:\\ \\sum_{1\\le i\\le N} x_i\\le 1\\big\\}.$ We prove the Poincar\\'e inequality\n  $$\\mu^{(N)}_{\\aa}(f^2)\\le \\ff 1 {\\aa_{N+1}} \\int_{\\DD}\\Big\\{\\Big(1-\\sum_{1\\le i\\le N} x_i\\Big) \\sum_{n=1}^N x_n(\\pp_n f)^2\\Big\\}\\mu^{(N)}_\\aa(\\d x)+\\mu^{(N)}_{\\aa}(f)^2,\\ f\\in C^1(\\DD)$$ and show that the constant $\\ff 1 {\\aa_{N+1}}$ is sharp.\n  Consequently, the associated diffusion process on $\\DD$ converges to $\\mu^{(N)}_{\\aa}$ i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.02829","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}