Computing pointwise fractal dimension by conditioning in multivariate distributions and time series

Abstract

It is natural in many contexts to employ conditioning arguments in order to deduce properties of a multivariate distribution P_n from properties of a lower-order distribution P_n-1. In this paper we show that the pointwise (fractal) dimension of P_n can be updated from the pointwise dimension of P_n-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 $$\alpha (x_1,dots,x_n)=\alpha \left(x_1\right)+\alpha \left(x_2\right|x_1)+\alpha (x_3|x_1,x_2)+\dots +\alpha \left(x_n\right|x_1,dots,x_{n-1}).$$ 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.