How Exceptional is the Extremal Kendall and Kendall-Type Convolution

Abstract This paper deals with the generalized convolutions connected Williamson transform and maximum operation. We focus on such which can define transition probabilities of renewal processes. They should be monotonic since described time or destruction does not go back, it admit existence a distribution lack memory property because analog Poisson process shall exist. Another valuable is simplicity calculating inverting corresponding characteristic function (in particular transform) so that technique used in description our The convex linear combination (the convolution two point measures several fixed measures), representability (which means easily written language independent random variables)—they also facilitate modeling real processes language. describe examples having required properties ranging from its simplest generalization—the Kendall (associated transform), up to most complicated here—Kingman convolution. It novel approach apply extreme value theory. Stochastic representation Kucharczak-Urbanik order statistics terms proved, open new paths investigate Archimedean copulas. door solve an old problem relationship between copulas mentioned by B. Schweizer A. Sklar 1983. indicates path further research towards extremes dependency modelling.