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June 2000 Computing pointwise fractal dimension by conditioning in multivariate distributions and time series
Colleen D. Cutler
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Bernoulli 6(3): 381-399 (June 2000).


It is natural in many contexts to employ conditioning arguments in order to deduce properties of a multivariate distribution Pn from properties of a lower-order distribution Pn-1. In this paper we show that the pointwise (fractal) dimension of Pn can be updated from the pointwise dimension of Pn-1 via the one-step conditional distributions provided the latter satisfy certain Lipschitz-type properties. Specifically, we prove that pointwise dimension can be computed iteratively according to the conditional additivity rule α (x 1,dots,x n)=α(x 1)+α(x 2|x 1)+α(x 3|x 1,x 2)++α(x n|x 1,dots,x n -1). This approach is then used to analyse the behaviour of pointwise dimension for various stationary stochastic processes; the emphasis is on dynamical systems corrupted by noise. In particular, we show that for functionals of stochastic processes with discrete conditional distributions satisfying the necessary conditions (such as missing-data models of dynamical systems and randomly iterated function systems) pointwise dimension remains bounded over time just as in the strictly deterministic case. On the other hand, we prove that for stochastic dynamical systems with additive diffuse noise, pointwise dimension diverges to infinity over time. An example of a stationary dynamical system where the conditional additivity rule fails is also provided.


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Colleen D. Cutler. "Computing pointwise fractal dimension by conditioning in multivariate distributions and time series." Bernoulli 6 (3) 381 - 399, June 2000.


Published: June 2000
First available in Project Euclid: 10 April 2004

zbMATH: 0970.37039
MathSciNet: MR2001F:28009

Keywords: determinism , fractal dimension , Grassberger-Procaccia algorithm , iterated function system , Markov process , missing data , pointwise dimension , stochastic dynamical system , time series

Rights: Copyright © 2000 Bernoulli Society for Mathematical Statistics and Probability


Vol.6 • No. 3 • June 2000
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