Limit Property for Regular and Weak Generalized Convolution

We denote by ℘ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\mathcal{P_{+}})$\end{document} 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 Ta (a>0) with δ0 as the unit element. The key axiom of a generalized convolution is the following: there exist norming constants cn and a measure ν other than δ0 such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T_{c_{n}}\delta_{1}^{\bullet n}\to\nu$\end{document} .