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.

中文翻译：

---

用于模拟光学现象的多高斯随机变量

介绍了经典高斯随机变量到以形状参数M > 0 为特征的多高斯 (MG) 随机变量族的推广，以及均值和标准差。The probability density function (PDF) of the MG family members is an alternating series of Gaussian functions with suitably chosen heights and widths. 特别地，对于整数值中号，该系列具有条款和导致扁平轮廓的有限数，同时减小对经典高斯PDF对于中号= 1。对于非整数，的正值中号，收敛无穷级数的获得了在实际问题中可以截断的高斯函数。对于所有M> 1，MG PDF 具有扁平轮廓，而对于 0 < M < 1，MG PDF 具有尖头轮廓。此外，获得了MG随机变量的多元扩展，并引入了log-multi-Gaussian随机变量。为了说明这些新随机变量对光学的有用性，从理论上和通过数值模拟讨论了 MG 随机变量在新散斑场表征中的应用。