Daniel Fulger

Daniel Fulger, Guido Germano

We present a rejection method based on recursive covering of the probability density function with equal tiles. The concept works for any probability density function that is pointwise computable or representable by tabular data. By the implicit construction of piecewise constant majorizing and minorizing functions that are arbitrarily close to the density function the production of random variates is arbitrarily independent of the computation of the density function and extremely fast. The method works unattended for probability densities with discontinuities (jumps and poles). The setup time is short, marginally independent of the shape of the probability density and linear in table size. Recently formulated requirements to a general and automatic non-uniform random number generator are topped. We give benchmarks together with a similar rejection method and with a transformation method.

Daniel Fulger, Enrico Scalas, Guido Germano

The speed of many one-line transformation methods for the production of, for example, Levy alpha-stable random numbers, which generalize Gaussian ones, and Mittag-Leffler random numbers, which generalize exponential ones, is very high and satisfactory for most purposes. However, for the class of decreasing probability densities fast rejection implementations like the Ziggurat by Marsaglia and Tsang promise a significant speed-up if it is possible to complement them with a method that samples the tails of the infinite support. This requires the fast generation of random numbers greater or smaller than a certain value. We present a method to achieve this, and also to generate random numbers within any arbitrary interval. We demonstrate the method showing the properties of the transform maps of the above mentioned distributions as examples of stable and geometric stable random numbers used for the stochastic solution of the space-time fractional diffusion equation.