TY - JOUR

T1 - Analytical properties of generalized Gaussian distributions

AU - Dytso, Alex

AU - Bustin, Ronit

AU - Poor, H. Vincent

AU - Shamai, Shlomo

N1 - Funding Information:
The work of A. Dytso and H.V. Poor was supported by the U.S. National Science Foundation under Grant CNS-1702808. The work of S. Shamai and R. Bustin was supported by the European Union’s Horizon 2020 Research and Innovation Programme Grant 694630.

PY - 2018/12/1

Y1 - 2018/12/1

N2 - 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.

AB - 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.

KW - Characteristic function

KW - Generalized Gaussian distribution

KW - Infinite divisibility

KW - Mellin transform

KW - Self-decomposition

UR - http://www.scopus.com/inward/record.url?scp=85062698524&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85062698524&partnerID=8YFLogxK

U2 - 10.1186/s40488-018-0088-5

DO - 10.1186/s40488-018-0088-5

M3 - Article

AN - SCOPUS:85062698524

VL - 5

JO - Journal of Statistical Distributions and Applications

JF - Journal of Statistical Distributions and Applications

SN - 2195-5832

IS - 1

M1 - 6

ER -