Generalized Box–MÜller Method for Generating $q$ -Gaussian Random Deviates

The q-Gaussian distribution is known to be an attractor of certain correlated systems and is the distribution which, under appropriate constraints, maximizes a generalization of the familiar Shannon entropy. This generalized entropy, or q-entropy, provides the basis of nonextensive statistical mechanics, a theory which is postulated as a natural extension of the standard (Boltzmann-Gibbs) statistical mechanics, and which may explain the ubiquitous appearance of heavy-tailed distributions in both natural and man-made systems. The q-Gaussian distribution is also used as a numerical tool, for example as a visiting distribution in Generalized Simulated Annealing. A simple, easy to implement numerical method for generating random deviates from a q-Gaussian distribution based upon a generalization of the well known Box-Miiller method is developed and presented. This method is suitable for a larger range of q values, -infin < q < 3, than has previously appeared in the literature, and can generate deviates from q-Gaussian distributions of arbitrary width and center. MATLAB code showing a straightforward implementation is also included.