文章基本信息

标题：Poincaré Inequality for Dirichlet Distributions and Infinite-Dimensional Generalizations
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作者：Shui Feng ; Laurent Miclo ; Feng-Yu Wang 等
期刊名称：Latin American Journal of Probability and Mathematical Statistics
电子版ISSN：1980-0436
出版年度：2017
卷号：XIV
页码：361-380
出版社：Instituto Nacional De Matemática Pura E Aplicada
摘要：For any N ≥ 2 and = (1, · · · , N+1) ∈ (0,∞)N+1, let μ(N) be thecorresponding Dirichlet distribution on (N) := x = (xi)1≤i≤N ∈ [0, 1]N : |x|1 :=P1≤i≤N xi ≤ 1 . We prove the Poincar´e inequalityμ(N) (f2) ≤1N+1 Z(N) n1 − |x|1NXn=1xn(@nf)2oμ(N) (dx) + μ(N) (f)2,for f ∈ C1((N)), and show that the constant 1N+1is sharp. Consequently, theassociated diffusion process on (N) converges to μ(N) in L2(μ(N) ) at the exponentiallyrate N+1. The whole spectrum of the generator is also characterized.Moreover, the sharp Poincar´e inequality is extended to the infinite-dimensionalsetting, and the spectral gap of the corresponding discrete model is derived.
关键词：Dirichlet distribution; Poincar´e inequality; diffusion process; spectral;gap.