Nonuniform random transformations

• 成果类型: Article
• 署名作者: O'Cinneide, CA; Pokrovskii, AV
• 署名单位: Purdue University System; Purdue University; University College Cork
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• 发表日期: 2000
• 页码: 1151-1181
• 关键词: chaotic dynamical-systems; continuum random tree; Random mappings; riffle shuffles; discretizations; cycles

摘要

With a given transformation on a finite domain, we associate a three-dimensional distribution function describing the component size, cycle length and trajectory length of each point in the domain. We then consider a random transformation on the domain, in which images of points are independent and identically distributed. The three-dimensional distribution function associated with this random transformation is itself random. We show that, under a simple homogeneity condition on the distribution of images, and with a suitable scaling, this random distribution function has a limit law as the number of points in the domain tends to oo. The proof is based on a Poisson approximation technique for matches in an urn model. The result helps to explain the behavior of computer implementations of chaotic dynamical systems.