Measure concentration for Euclidean distance in the case of dependent random variables

• 成果类型: Article
• 署名作者: Marton, K
• 署名单位: HUN-REN; HUN-REN Alfred Renyi Institute of Mathematics; Hungarian Academy of Sciences
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117904000000702
• 发表日期: 2004
• 页码: 2526-2544
• 关键词: logarithmic sobolev inequalities; transportation cost

摘要

Let q(n) be a continuous density function in n-dimensional Euclidean space. We think of q(n) as the density function of some random sequence X-n with values in R-n. For I subset of [1. n], let X-I denote the collection of coordinates X-i, i is an element of I, and let X-I, denote the collection of coordinates X-i, i is not an element of I. We denote by Q(I)(x(I)\x(I)) the joint conditional density function of X-I, given X-I. We prove measure concentration for q in the case when, for an appropriate class of sets I, (i) the conditional densities Q(I) (x(l) \x(I)), as functions of x(I), uniformly satisfy a logarithmic Sobolev inequality and (ii) these conditional densities also satisfy a contractivity condition related to Dobrushin and Shlosman's strong mixing condition.