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Smog is a serious problem in most big urban areas. We rarely realize the consequences of being in polluted air, while mixture of air pollutants can seriously endanger human health. Bronchitis, pneumonia and asthma are only some of the respiratory diseases that are associated with the effects of smog. Polluted air also makes it difficult for people to breathe properly. We analyze the relationship between respiratory diseases and smog based on data from Wrocław (Poland) regarding calls for ambulance services of 15107 individuals and indicators of air pollution and meteorological data in 2016. The results of our analyzes are optimized for the selection of explanatory variables for models using the Spearman coefficient values. A novel approach proposed here is to use generalized linear models by optimizing shifted air pollution data to predict the number of ambulance calls on a given day. Finally, the best generalized linear model with logarithm linking function was fitted to analyzed data.

Kingman, in his seminal work [13], introduced a new type of convolution of distributions that is naturally related to spherically symmetric random walks. Motivated by this paper, Urbanik in a series of papers [17] established a theory of generalized convolutions ⋄ as certain binary commutative and associative operations that include classical and Kingman's convolutions as a special case. This theory was further developed by Bingham ([2, 3]) in the context of regularly varying functions. There is a rich class of examples of generalized convolutions that are motivated by problems in applications of probability theory. For instance, the distribution of the maximum of two independent random variables is a generalized convolution fundamentally associated with the extreme value theory, and extensively applied to model events that rarely occur, but the appearance of which causes large losses. Similarly, to the classical theory, we define infinite divisibility with respect to generalized convolution ⋄ and establish Lévy-Khintchine representation [11]. Lévy and additive stochastic processes under generalized convolutions are constructed as the Markov processes in ([5]). In this paper we survey examples of generalized convolutions and related Lévy-Khintchine representation. Results on Kendall convolution and extreme Markov chains driven by the Kendall convolution ([1, 5, 10]) using Williamson transform ([18]) are also presented.

We consider a class of max-AR(1) sequences connected with the Kendall convolution. For a large class of step size distributions we prove that the one dimensional distributions of the Kendall random walk with any unit step distribution, are regularly varying. The finite dimensional distributions for Kendall convolutions are given. We prove convergence of a continuous time stochastic process constructed from the Kendall random walk in the finite dimensional distributions sense using regularly varying functions.

We consider here the Cramer-Lundberg model based on generalized convolutions. In our model the insurance company invests at least part of its money, have employees, shareholders. The financial situation of the company after paying claims can be even better than before. We compute the ruin probability for $\alpha$-convolution case, maximal convolution and the Kendall convolution case, which is formulated in the Williamson transform terms. We also give some new results on the Kendall random walks.

The paper deals with fluctuations of Kendall random walks, which are extremal Markov chains. We give the joint distribution of the first ascending ladder epoch and height over any level $a \geq 0$ and distribution of maximum and minimum for these extremal Markovian sequences. We show that distribution of the first crossing time of level $a \geq0$ is a mixture of geometric and negative binomial distributions. The Williamson transform is the main tool for considered problems connected with the Kendall convolution.

The paper deals with the renewal theory for a class of extremal Markov sequences connected with the Kendall convolution. We consider here some particular cases of the Wold processes connected with generalized convolutions. We prove an analogue of the Fredholm theorem for all generalized convolutions algebras. Using the technique of regularly varying functions we prove the Blackwell theorem for renewal processes defined by the Kendall random walks.