Research goals of this talk are focused on the renewal theory, some aspects of the fluctuation theory and asymptotic properties for the first order maximal autoregressive processes of the Kendall type, which are discrete time Lévy processes under the Kendall convolution. Since most of the considered processes have heavy tailed distributions, then we rely on the use of results in the theory of extreme event modeling.

We use generalized convolutions to construct new stochastic processes.  Using generalized convolutions, we create new mathematical objects that have potential in applications. It is enough to look at the case of the maximum convolution corresponding to the Extreme Value Theory. Currently, limit distributions of extremes , i.e generalized extreme value distributions (Fréchet, Gumbel, Weibull) is commonly used for modeling rainfall, floods, drought, extreme air pollutants, etc. Random walks with respect to generalized convolutions form a class of extremal Markov chains and studying them in the appropriate algebras will be a meaningful contribution to extreme value theory.

We also investigate the dependence between air pollution indicators, meteorological data and the numer of ambulance calls to model influence of air pollutants on civilization diseases.