# Fluctuations of extremal Markov chains driven by the Kendall convolution

## TL;DR

This paper studies the fluctuations of extremal Markov chains called Kendall random walks, deriving their joint distributions, crossing times, and asymptotic behaviors using the Williamson transform and regular variation techniques.

## Contribution

It introduces new analytical results for Kendall random walks, including joint distributions and crossing time distributions, utilizing the Williamson transform.

## Key findings

- Distribution of first crossing time is a mixture of geometric and negative binomial distributions.
- Joint distribution of the first ladder epoch and height is explicitly derived.
- Asymptotic properties of the maximum distribution are characterized using regular variation.

## Abstract

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.