Near neighbor distribution in fractal and finite sets

Distances of several nearest neighbors of a given point in a multidimensional space play important role in some tasks of data mining. Here we analyze these distances analyzed as random variables defined to be functions of a given point and its k-th nearest neighbor. We prove that if there is a constant q such that the mean k-th neighbor distance to this constant power is proportional to the near neighbor index k then its distance to this constant power converges to Erlang distribution of order k. We also show that constant q is the scaling exponent known from the theory of multifractals.