关于自由广义逆高斯分布
On free Generalized Inverse Gaussian distributions
摘要：
本文研究了自由广义逆高斯分布在自由概率下的性质。我们表明，在许多情况下，fGIG股票与经典的GIG分布相似的属性。特别是，我们证明了fGIG是自由无限可分的，自由正则和单峰的，而且我们确定在这个类中的分布是自由自分解。本文的第二部分证明了对于自由随机变量$X,Y$，其中$Y$具有自由泊松分布，有$X\stackrel{d}{=}\frac{1}{X+Y}$当且仅当$X$具有fGIG分布，并对参数进行了特殊选择.我们还指出，自由GIG分布最大化相同的自由熵功能的经典GIG的经典熵。
英文摘要：
We study here properties of free Generalized Inverse Gaussian distributions (fGIG) in free probability. We show that in many cases the fGIG shares similar properties with the classical GIG distribution. In particular we prove that fGIG is freely infinitely divisible, free regular and unimodal, and moreover we determine which distributions in this class are freely selfdecomposable. In the second part of the paper we prove that for free random variables $X,Y$ where $Y$ has a free Poisson distribution one has $X\stackrel{d}{=}\frac{1}{X+Y}$ if and only if $X$ has fGIG distribution for special choice of parameters. We also point out that the free GIG distribution maximizes the same free entropy functional as the classical GIG does for the classical entropy.
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