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.

翻译：从有限随机轨迹估计分形维数是一个有限尺寸标度问题：表观盒计数指数受限于解析尺度范围与有限采样点之间的占据交叉，未必等于极限过程的分维。我们利用球入盒占据定律模拟这一交叉，预测了盒计数曲线、有限尺寸饱和标度以及归一化局部斜率的标度函数。对于随机游走轨迹、分数布朗图与列维飞行，归一化局部斜率塌缩至单一交叉曲线，而当回归窗口相对于饱和标度定位时，窗口化盒计数偏差发生塌缩。逆推占据模型可得有限尺寸偏差校正，该校正可降低受控随机轨迹的误差，并泛化至保留模型类别。与关联维数、去趋势波动分析、变异函数及Higuchi方法的比较表明，主导偏差为有限尺度窗口上点采样盒计数所特有，且仅凭局部斜率稳定性并非可靠诊断指标。DNA游走示例展示了测量数据上的工作流程，所有图表及正文数字均基于开源单种子代码重新生成。