Multi-Gaussian random variables

TL;DR

This paper introduces Multi-Gaussian (MG) random variables, a flexible generalization of Gaussian variables characterized by a shape parameter, allowing for flattened or cusped profiles and extending to multivariate and Log-Multi-Gaussian forms.

Contribution

The paper presents the definition, properties, and multivariate extension of MG random variables, including their probability density functions and special cases, expanding Gaussian models.

Findings

MG PDFs are series of Gaussian functions with adjustable profiles.

For integer M, MG PDFs have finite terms and flattened shapes.

For 0 < M < 1, MG PDFs exhibit cusped profiles.

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.