A model for the frequency distribution of multi-scale phenomena

Frequency analysis is often used to investigate the structure of systems representing multi-scale real-world phenomena. In many different environments, functional relationships characterized by a power law have been recognized, but, in many cases this simple model has turned out to be absolutely inadequate and other models have been proposed. In this paper, we propose a general abstract model which constitutes a unifying framework, including many models found in literature, like the mixed model, the exponential cut-off and the log-normal. It is based on a discrete-time stochastic process, which leads to a recurrence relation describing the temporal evolution of the system. The steady state solution of the system highlights the probability distribution, which underlies the frequency behavior. A particular instance of the general model, called cubic-cut-off, was analyzed and tested in a number of experiments, producing good answers in difficult cases, even in the presence of peculiar behaviors.