Finite-size occupancy scaling of apparent fractal dimensions in stochastic trajectories

Abstract

Estimating a fractal dimension from a finite stochastic trajectory is a finite-size scaling problem: the apparent box-counting exponent is shaped by an occupancy crossover between the resolved range of scales and the finite number of sampled points, and need not equal the dimension of the limiting process. We model this crossover with a balls-in-boxes occupancy law, which predicts the box-count curve, the finite-size saturation scale, and a scaling function for the normalized local slope. Across random-walk traces, fractional Brownian graphs, and Levy flights, the normalized local slope collapses onto a single crossover curve, while the windowed box-counting bias collapses when the regression window is positioned relative to the saturation scale. Inverting the occupancy model gives a finite-size bias correction that reduces error on controlled stochastic trajectories and transfers across held-out model classes. Comparisons with correlation dimension, detrended fluctuation analysis, the variogram, and Higuchi's method show that the dominant bias is specific to point-sampled box-counting over finite scale windows, and that local-slope stability alone is not a reliable diagnostic. A DNA-walk example illustrates the workflow on measured data, and all figures, tables, and in-text numbers are regenerated from released single-seed code.