Multi-Gaussian random variables

Abstract: 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 of the MG family members is the alternating series of the Gaussian functions with the suitably chosen heights and widths. In particular, for the integer values of M the series has finite number of terms and leads to flattened profiles, while reducing to classic Gaussian density 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. While for all M > 1 the MG PDF has attened profiles, for 0 < M < 1 it leads to cusped profiles. Moreover, the multivariate extension of the MG random variable is obtained and the Log-Multi-Gaussian (LMG) random variable is introduced.