论文标题
无限划分的分布和交换图
Infinitely Divisible Distributions and Commutative Diagrams
论文作者
论文摘要
我们在非负半行$ \ mathbb {r} _+$上研究无限分配（ID）分布。 lévy-khintchine表示这种分布是众所周知的。我们的主要贡献是将概率对象和它们之间的关系施放为统一的视觉形式，我们将其称为Lévy-Khintchine通勤图（LKCD）。虽然将其作为代表性工具引入，但LKCD促进了对新ID分布的探索，因此至少可以部分地将其视为一种发现工具。该研究的基本对象是伽马分布。与此密切相关的是$ \ Mathbb {r} _+$ for $ 0 <α<1 $上的$α$ - 稳定分布，我们认为这是由伽马分布而不是单独的对象引起的。它的特征是其拉普拉斯变换$ \ exp（-s^α）$，价格为$ 0 <α<1 $。确实，它通常被视为一类ID分布的实例，称为广义伽玛卷积（GGC）。我们利用伽马和稳定密度的卷积和混合物来产生其他GGC分布的密度，其中涉及贝塞尔，汇合超几何，mittag-leffler和抛物线缸功能。我们将所有实例表示为LKCD表示。
We study infinitely divisible (ID) distributions on the nonnegative half-line $\mathbb{R}_+$. The Lévy-Khintchine representation of such distributions is well-known. Our primary contribution is to cast the probabilistic objects and the relations amongst them in a unified visual form that we refer to as the Lévy-Khintchine commutative diagram (LKCD). While it is introduced as a representational tool, the LKCD facilitates the exploration of new ID distributions and may thus also be looked upon, at least in part, as a discovery tool. The basic object of the study is the gamma distribution. Closely allied to this is the $α$-stable distribution on $\mathbb{R}_+$ for $0<α<1$, which we regard as arising from the gamma distribution rather than as a separate object. It is characterised by its Laplace transform $\exp(-s^α)$ for $0<α<1$. It is indeed often characterised as an instance of a class of ID distributions known as generalised gamma convolutions (GGCs). We make use of convolutions and mixtures of gamma and stable densities to generate densities of other GGC distributions, with particular cases involving Bessel, confluent hypergeometric, Mittag-Leffler and parabolic cylinder functions. We present all instances as LKCD representations.
