Deterministic sampling from uniform distributions with Sierpiński space-filling curves

In this paper the problem of sampling from uniform probability distributions is approached by means of space-filling curves, a topological concept that has found a number of important applications in recent years. Departing from the theoretical fact that they are surjective but not necessarily injective, the investigation focused upon the structure of the distributions obtained when their domains are swept in a uniform and discrete manner, and the corresponding values used to build histograms, that are approximations of their true PDFs. This work concentrates on the real interval [0,1] and the Sierpiński space-filling curve was chosen because of its favorable computational properties. In order to validate the results, the Kullback–Leibler and other divergence measures are used when comparing the obtained distributions in several levels of granularity with other already established sampling methods. In truth, the generation of uniform random numbers is a deterministic simulation of randomness using numerical operations. In this fashion, sequences resulting from this sort of process are not truly random. Despite this, and to be coherent with the literature, the expression “random number” will be used along the text to mean “pseudo-random number”.