Mellin transform for the probabilistic characterization of random variables and stochastic processes

The probabilistic characterization of random variables and stochastic processes involves the evaluation of the probability density function or characteristic function. The latter is typically obtained by using integer-order statistical moments, that could lead to divergence problem for high-order moments especially in case of heavy-tailed distributions, such as the distribution of the α-stable random variables. On the other hand, recent approaches that use complex fractional moments, offer a more robust probabilistic description, but for particular cases. In this paper, a novel approach based on Mellin transform for the probabilistic characterization of random variables is proposed. Starting from numerical data, this approach is effective for the evaluation of both the probability density function and the characteristic function, and then is valid for a wide class of random variables. Further, an extension of the approach from random variables to stochastic processes is proposed. The reliability of the proposed approach is assessed through several numerical simulations involving α-stable distributions, Gaussian distributions and α-stable stochastic processes.