Estimating the dimension of factors of diffusion processes

Summary We present consistency results for estimators of the box-counting dimension of the support of probability distributions. The box-counting dimension of the support E is defined via the covering number, i.e. the minimal cardinality of a cover of E consisting of cubes of fixed side-length. Accordingly the covering number of a sample allows the definition of an estimator of the box-counting dimension of E . Consistency results for arrays of probability distributions may be applied to the distributions of innovations of Itô processes and allow the construction of consistent estimators of the dimension of the factors, i.e. of the dimension of the Brownian motion driving the process.