凸锥上的严格稳定分布
Strictly stable distributions on convex cones
摘要：
利用LePage表示，具有$α\in(0,2)$的Banach空间中的严格稳定随机元可以表示为泊松过程的点之和.这个点过程是并稳定的，即它的两个独立副本的并与重新标度的原点过程在分布上重合。这些概念在任何凸锥中都是有意义的，即在一个交换半群中配备了数乘，并导致通过LePage级数在一般锥中构造稳定律。相应的极限定理表明随机样本（或二项式点过程）按分布收敛于并稳定的Poisson点过程，并由此得到了具有$α$ -稳定极限的随机元的正规化和的极限定理，其中$α$是$α\in(0,1)$的稳定极限。利用半群上的调和分析技术，我们给出了$α$ -稳定随机元的分布，并说明了$α$的可能值与半群的性质和相应的标度运算，特别是它们的分布性性质的关系.本文所发展的方法不仅可以处理相当一般的锥（如集或测度空间）中的稳定分布，而且为证明经典极限定理和推导Banach空间中严格稳定随机向量的LePage表示提供了一种可供选择的方法。
英文摘要：
Using the LePage representation, a strictly stable random element in a Banach space with $α\in(0,2)$ can be represented as a sum of points of a Poisson process. This point process is union-stable, i.e. the union of its two independent copies coincides in distribution with the rescaled original point process. These concepts makes sense in any convex cone, i.e. in a commutative semigroup equipped with multiplication by numbers, and lead to a construction of stable laws in general cones by means of the LePage series. The corresponding limit theorem shows that random samples (or binomial point processes) converge in distribution to the union-stable Poisson point process, and so yields a limit theorem for normalised sums of random elements with $α$-stable limit for $α\in(0,1)$. By using the technique of harmonic analysis on semigroups we characterise distributions of $α$-stable random elements and show how possible values of $α$ relate to the properties of the semigroup and the corresponding scaling operation, in particular, their distributivity properties. The approach developed in the paper not only makes it possible to handle stable distributions in rather general cones (like spaces of sets or measures), but also provides an alternative way to prove classical limit theorems and deduce the LePage representation for strictly stable random vectors in Banach spaces.
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