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