Dimension spectra for multifractal measures with connections to nonparametric density estimation

We consider relations between Rényi's and Hentschel - Procaccia's definitions of generalized dimensions of a probability measure μ, and give conditions under which the two concepts are equivalent/different. Estimators of the dimension spectrum are developed, and strong consistency is established. Particular cases of our estimators are methods based on the sample correlation integral and box counting. Then we discuss the relation between generalized dimensions and kernel density estimators f̂. It was shown in Frigyesi and Hössjer (1998), that ∫ f̂^1+q(x)dx diverges with increasing sample size and decreasing bandwidth if the marginal distribution μ has a singular part and q > 0. In this paper, we show that the rate of divergence depends on the qth generalized Rényi dimension of μ.