The family of Generalized Gaussian (GG) distributions has received considerable attention from the engineering community, due to the flexible parametric form of its probability density function, in modeling many physical phenomena. However, very little is known about the analytical properties of this family of distributions, and the aim of this work is to fill this gap. Roughly, this work consists of four parts. The first part of the paper analyzes properties of moments, absolute moments, the Mellin transform, and the cumulative distribution function. For example, it is shown that the family of GG distributions has a natural order with respect to second-order stochastic dominance. The second part of the paper studies product decompositions of GG random variables. In particular, it is shown that a GG random variable can be decomposed into a product of a GG random variable (of a different order) and an independent positive random variable. The properties of this decomposition are carefully examined. The third part of the paper examines properties of the characteristic function of the GG distribution. For example, the distribution of the zeros of the characteristic function is analyzed. Moreover, asymptotically tight bounds on the characteristic function are derived that give an exact tail behavior of the characteristic function. Finally, a complete characterization of conditions under which GG random variables are infinitely divisible and self-decomposable is given. The fourth part of the paper concludes this work by summarizing a number of important open questions.

中文翻译：

---

广义高斯分布的解析性质

广义高斯（GG）分布族在建模许多物理现象时由于其概率密度函数的灵活参数形式而受到了工程界的广泛关注。但是，对该分布族的分析特性了解甚少，这项工作的目的是填补这一空白。大致上，这项工作包括四个部分。本文的第一部分分析了矩，绝对矩，Mellin变换和累积分布函数的属性。例如，表明GG分布族相对于二阶随机优势具有自然顺序。本文的第二部分研究GG随机变量的乘积分解。尤其是，结果表明，GG随机变量可以分解为GG随机变量（不同顺序）和独立正随机变量的乘积。仔细分析了这种分解的性质。本文的第三部分考察了GG分布的特征函数的性质。例如，分析特征函数的零点的分布。此外，推导了特征函数的渐近严格边界，这些边界给出了特征函数的精确尾部行为。最后，给出了GG随机变量可无限分解且可自分解的条件的完整表征。本文的第四部分通过总结一些重要的开放性问题来总结这项工作。