Limit Property for Regular and Weak Generalized Convolution

作者：Jasiulis Barbara H^*

来源：Journal of Theoretical Probability, 2010, 23(1): 315-327.

DOI：10.1007/s10959-009-0238-2

• Weakly stable distribution
• Generalized weak convolution
• Generalized convolution
• Factor of strictly stable distribution

We denote by P (P(+)) the set of all probability measures defined on the Borel subsets of the real line (the positive half-line [0, infinity)). K. Urbanik defined the generalized convolution as a commutative and associative P(+)-valued binary operation center dot on P(+)(2) which is continuous in each variable separately. This convolution is distributive with respect to convex combinations and scale changes T(a) (a > 0) with delta(0) as the unit element. The key axiom of a generalized convolution is the following: there exist norming constants c(n) and a measure. other than delta(0) such that T(cn)delta(center dot n)(1) -> nu.
In Sect. 2 we discuss basic properties of the generalized convolution on P which hold for the convolutions without the key axiom. This rather technical discussion is important for the weak generalized convolution where the key axiom is not a natural assumption. In Sect. 4 we show that if the weak generalized convolution defined by a weakly stable measure mu has this property, then mu is a factor of strictly stable distribution.

• 出版日期2010-3