Nonuniform random transformations

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