First order Kendall maximal autoregressive processes and their applications

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.