A generalization of the classic Gaussian random variable to the family of multi-Gaussian (MG) random variables characterized by shape parameter M > 0, in addition to the mean and the standard deviation, is introduced. The probability density function (PDF) of the MG family members is an alternating series of Gaussian functions with suitably chosen heights and widths. In particular, for integer values of M, the series has a finite number of terms and leads to flattened profiles, while reducing to the classic Gaussian PDF for M = 1. For non-integer, positive values of M, a convergent infinite series of Gaussian functions is obtained that can be truncated in practical problems. For all M > 1, the MG PDF has flattened profiles, while for 0 < M < 1, the MG PDF has cusped profiles. Moreover, the multivariate extension of the MG random variable is obtained and the log-multi-Gaussian random variable is introduced. In order to illustrate the usefulness of these new random variables for optics, the application of MG random variables to the characterization of novel speckle fields is discussed, both theoretically and via numerical simulations.
Olga Korotkova and Milo W. Hyde, "Multi-Gaussian random variables for modeling optical phenomena," Opt. Express 29, 25771-25799 (2021). https://doi.org/10.1364/OE.432227