Asymptotic properties of extremal Markov processes driven by Kendall convolution

Abstract

This paper is devoted to the analysis of the finite-dimensional distributions and asymptotic behavior of extremal Markov processes connected to the Kendall convolution. In particular, based on its stochastic representation, we provide general formula for finite dimensional distributions of the random walk driven by the Kendall convolution for a large class of step size distributions. Moreover, we prove limit theorems for random walks and connected continuous time stochastic process.

Asymptotics for Kendall’s renewal function

“…(5) Let F X (x) = (1 + x −α ) e −x −α , x ≥ 0, where α > 0. Notice that it is the limit distribution (see Arendarczyk et al (2019)) in the Kendall convolution algebra corresponding to normal distribution in the classical case. Then the Williamson transform and truncated α-moment for X are given by…”