Limit Property for Regular and Weak Generalized Convolution

Abstract

We denote by ℘ $(\mathcal{P_{+}})$ the set of all probability measures defined on the Borel subsets of the real line (the positive half-line [0,∞)). K. Urbanik defined the generalized convolution as a commutative and associative ℘₊-valued binary operation • on ℘ 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 δ ₀ as the unit element. The key axiom of a generalized convolution is the following: there exist norming constants c _n and a measure ν other than δ ₀ such that $T_{c_{n}}\delta_{1}^{\bullet n}\to\nu$ .

In Sect. 2 we discuss basic properties of the generalized convolution on ℘ 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 μ has this property, then μ is a factor of strictly stable distribution.