Generalized complex fractional moment for the probabilistic characteristic of random vectors

A definition of multi-variate complex quantities called Generalized Complex Fractional Moment (GCFM) based on the multi-dimensional Mellin transform is proposed in this paper, which is also related to the multi-dimensional Riesz fractional integral evaluated in zero. The equivalence property between GCFM, for both multi-dimensional probability density functions and multi-dimensional characteristic functions is established. Furthermore, a method for obtaining marginal probability distributions from GCFM is presented. The validity of the GCFM method is verified through an example involving α-stable random vectors. Additionally, another example using GCFM to reconstruct the non-stationary PDF of the stochastic dynamic system highlights the prospect of applying the GCFM method in engineering.