Mathematical Notes

, Volume 71, Issue 5, pp 634-648

First online:

Statistical Estimation of Generalized Dimensions

  • V. V. MaiorovAffiliated withYaroslavl State University
  • , E. A. TimofeevAffiliated withYaroslavl State University

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A statistical estimate for generalized dimensions of a set \(A \subset \mathbb{R}^m\) based on the computation of average distances to the closest points in a sample of elements of A is given. For smooth manifolds with Lebesgue measures and for self-similar fractals with self-similar measures, the estimate is proved to be consistent.

generalized dimensions Rényi dimensions consistent estimates smooth manifolds self-similar fractals random points