Normal approximation for hierarchical structures

• 成果类型: Article
• 署名作者: Goldstein, L
• 署名单位: University of Southern California
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/105051604000000440
• 发表日期: 2004
• 页码: 1950-1969
• 关键词: independent random-variables; iterated functions; limit

摘要

Given F: [a,b](k) --> [a,b] and a nonconstant X-0 with P(X-0 is an element of [a,b]) = 1, define the hierarchical sequence of random variables {X-n}(ngreater than or equal to0) by Xn+1 = F(X-n,X-1,...,X-n,X-k), where X-n,X-i are i.i.d. as X-n. Such sequences arise from hierarchical structures which have been extensively studied in the physics literature to model, for example, the conductivity of a random medium. Under an averaging and smoothness condition on nontrivial F, an upper bound of the form Cgamma(n) for 0 < gamma < 1 is obtained on the Wasserstein distance between the standardized distribution of X-n and the normal. The results apply, for instance, to random resistor networks and, introducing the notion of strict averaging, to hierarchical sequences generated by certain compositions. As an illustration, upper bounds on the rate of convergence to the normal are derived for the hierarchical sequence generated by the weighted diamond lattice which is shown to exhibit a full range of convergence rate behavior.