  The paper deals with fluctuations of Kendall random walks, which are extremal
Markov chains and iterated integral transforms with the Williamson kernel
$\Psi(t) = \left(1-|t|^{\alpha}\right)_+$, $\alpha>0$. We obtain 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 solving recursive integral equations. 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. All results are
described by the Williamson transform of the unit step distribution of Kendall
random walks. Using regular variation, we investigate the asymptotic properties
of the maximum distribution.
