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The paper deals with fluctuations of Kendall random walks, which are extremalMarkov chains and iterated integral transforms with the Williamson kernel$\Psi(t) = \left(1-|t|^{\alpha}\right)_+$, $\alpha>0$. We obtain the jointdistribution of the first ascending ladder epoch and height over any level $a\geq 0$ and distribution of maximum and minimum for these extremal Markoviansequences solving recursive integral equations. We show that distribution ofthe first crossing time of level $a \geq0$ is a mixture of geometric andnegative binomial distributions. The Williamson transform is the main tool forconsidered problems connected with the Kendall convolution. All results aredescribed by the Williamson transform of the unit step distribution of Kendallrandom walks. Using regular variation, we investigate the asymptotic propertiesof the maximum distribution.

Fluctuations of extremal Markov chains driven by the Kendall convolution