Multifractal analysis of the galaxy distribution: Reliability of results from finite data sets

We test the reliability of the different generalized fractal dimension estimators, when applied to point distributions with a priori known scaling properties. We consider the effects of varying the amount of available data and the dimensionality of the distribution. The present work is motivated by the growing interest in cosmological context to safely analyze the scale-invariant properties of the observed galaxy distribution; these results may also be of value in all physical situations where the statistical analysis of a fractal ‘‘dust’’ is required. We consider (a) a monofractal structure with dimension D=1, (b) a multifractal structure, and (c) a scale-dependent structure, behaving like a D=1 monofractal at small scales and an homogeneous dust at large scales. For this structure, the clustering strength and the point number density have been chosen as to be similar to those observed for the galaxy distribution. Although the different methods display different advantages and pitfalls, we find that the presently available galaxy samples can be usefully employed to trace the scaling properties generated by nonlinear clustering.