Convergence rate of Sevcik’s fractal dimension estimator to the Hausdorff–Besicovitch dimension

Determine the rate at which Sevcik’s fractal dimension estimator for waveforms, defined by D_S = 1 + ln(L)/ln(2 N') after linear normalization to the unit square (with L the polyline length and N' the number of segments), converges to the Hausdorff–Besicovitch dimension D_HB as the number of sampled points N increases; derive finite-sample error bounds or asymptotic convergence rates under explicit regularity classes of curves or signals.

Background

The paper introduces Sevcik’s fractal dimension estimator D_S for planar waveforms via a double linear normalization and shows, for the Koch triadic curve, that D_S converges to the Hausdorff–Besicovitch dimension D_HB as the number of samples grows. While convergence is analytically demonstrated for this specific self-similar fractal, the authors emphasize that the speed of convergence is not characterized in general.

Because D_S is widely applied to finite, noisy empirical data, quantifying the rate of convergence and the associated finite-sample error is crucial for practical use. The authors highlight this as a central limitation that affects inference from real-world datasets.