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<p>The paper deals with fluctuations of extremal Markov chains connected with Kendall convolution and the Williamson kernel <i>Ψ</i>(<i>t</i>) = (1 <i>− |t|</i><sup>α</sup>)+, α<i> &gt;</i> 0. The joint distribution of the first ascending ladder epoch and height over any level <i>a</i> ≥ 0, maximum and minimum distributions are calculated. We show that the distribution of the first crossing time of level <i>a</i> ≥ 0 is a mixture of geometric and negative binomial distributions. Using regular variation, we investigate the asymptotic properties of the maximum distribution. The results are supported by examples of novel probability distributions, which have the potential to model the phenomenon of crossing the barriers.</p>

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Fluctuations of extremal Markov chains driven by the Kendall convolution*

  • 期刊论文

  • 发表日期：

    2026年4月1日

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  • Barbara Jasiulis-Gołdyn,
  • Edward Omey,
  • Mateusz Staniak

摘要

The paper deals with fluctuations of extremal Markov chains connected with Kendall convolution and the Williamson kernel Ψ(t) = (1 − |t|α)+, α > 0. The joint distribution of the first ascending ladder epoch and height over any level a ≥ 0, maximum and minimum distributions are calculated. We show that the distribution of the first crossing time of level a ≥ 0 is a mixture of geometric and negative binomial distributions. Using regular variation, we investigate the asymptotic properties of the maximum distribution. The results are supported by examples of novel probability distributions, which have the potential to model the phenomenon of crossing the barriers.