How exceptional is the extremal Kendall and Kendall-type convolution

Abstract

In the paper we describe several important properties of the Kendall convolution at the same time pointing to these generalized convolutions which have the same property. For example the monotonic property is necessary to build a renewal process with respect to generalized convolution, lack of memory property is needed for the construction of the Poisson process with respect to generalized convolution. Another valuable property is the simplicity of inverting the corresponding generalized characteristic function e.g. inverting the Williamson transform in the case of the Kendall convolution. The convex linear combination property makes calculations easier and the representation as a weak convolution with respect to max-convolution allows describing extreme phenomena.In the last section we discuss the representable generalized convolutions in the sense similar to the one defined in [6], i.e. such convolutions ⋄ for which the random variable Z satisfies the condition L(θ 1 ) ⋄ L(θ 1 ) = L(Z) (L(X) denotes the distribution of X) can be explicitly given by some function on variables X, Y and maybe some additional variables. This property enables the computer simulation of stochastic processes with respect to representable generalized convolution and makes possible studying path properties of such processes. Contents 1. Motivations 2 2. A primer on generalized convolutions 5