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November 2000 Nonuniform random transformations
C. A. O'Cinneide, A. V. Pokrovskii
Ann. Appl. Probab. 10(4): 1151-1181 (November 2000). DOI: 10.1214/aoap/1019487611


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 $\infty$. 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.


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C. A. O'Cinneide. A. V. Pokrovskii. "Nonuniform random transformations." Ann. Appl. Probab. 10 (4) 1151 - 1181, November 2000.


Published: November 2000
First available in Project Euclid: 22 April 2002

zbMATH: 1073.60509
MathSciNet: MR1810869
Digital Object Identifier: 10.1214/aoap/1019487611

Primary: 60F05
Secondary: 60C05

Keywords: chaotic dynamical systems , Poisson approximations , Random mappings , urn models

Rights: Copyright © 2000 Institute of Mathematical Statistics


Vol.10 • No. 4 • November 2000
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