Dimension spectra for multifractal measures with connections to nonparametric density estimation

Attila Frigyesi, Ola Hössjer

Research output: Contribution to journalArticlepeer-review


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 ∫ f̂1+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.

Original languageEnglish
Pages (from-to)351-395
Number of pages45
JournalJournal of Nonparametric Statistics
Issue number3
Publication statusPublished - 2001 Jan 1

Subject classification (UKÄ)

  • Probability Theory and Statistics


  • Box counting
  • Correlation integral
  • Fractal dimension estimation
  • Generalized dimensions
  • Hentschel-Proccacia dimension
  • Kernel density estimates
  • Rényi dimension


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