On the Uniqueness of the Kendall Generalized Convolution

Kendall (Foundations of a theory of random sets, in Harding, E.F., Kendall, D.G. (eds.), pp. 322–376, Willey, New York, 1974) showed that the operation $\diamond_{1}\colon \mathcal{P}_{+}^{2}\rightarrow \mathcal{P}_{+}$ given by $$\delta_x\diamond_1\delta_1=x\pi_2+(1-x)\delta_1,$$ where x∈[0,1] and π β is the Pareto distribution with the density function β s −β−1 on the set [1,∞), defines a generalized convolution on ℘+. Kucharczak and Urbanik (Quasi-stable functions, Bull. Pol. Acad. Sci., Math. 22(3):263–268, 1974) noticed that also the following operation $$\delta_x\diamond_{\alpha}\delta_1=x^{\alpha}\pi_{2\alpha}+\bigl(1-x^{\alpha}\bigr)\delta_1$$ defines generalized convolutions on ℘+. In this paper, we show that ⋄ α convolutions are the only possible convolutions defined by the convex linear combination of two fixed measures. To be precise, we show that if ⋄ :℘2→℘ is a generalized convolution defined by $$\delta_x\diamond \delta_1=p(x)\lambda_1+\bigl(1-p(x)\bigr)\lambda_2,$$ for some fixed probability measures λ 1,λ 2 and some continuous function p :[0,1]→[0,1], p(0)=0=1−p(1), then there exists an α>0 such that p(x)=x α , ⋄=⋄ α , λ 1=π 2α and λ 2=δ 1. We present a similar result also for the corresponding weak generalized convolution.