Asymptotics of the order statistics for a process with a regenerative structure

TL;DR

This paper investigates the asymptotic behavior of the order statistics, specifically the q-th largest values, in a regenerative process with finite mean cycle length, providing convergence results and illustrative examples.

Contribution

It establishes the asymptotic distribution of order statistics for regenerative processes, extending extreme value theory to this class of stochastic processes.

Findings

Convergence of the distribution of the q-th largest value to a specified limit

Explicit formulas for the limit distribution involving cycle maxima

Illustrative examples demonstrating the theoretical results

Abstract

In the paper, a regenerative process $\{X_n:n\in\mathbb{N}\}$ with finite mean cycle length is considered. For~$M_n^{(q)}$ denoting the $q$-th largest value in $\{X_k : 1\leqslant k \leqslant n\}$, we prove that \begin{equation*} \sup_{x\in\mathbb{R}} \left|P\left(M^{(q)}_n\leqslant x\right) - G(x)^n \sum_{k=0}^{q-1}\frac{\left(-\log G(x)^n\right)^k}{k!}\gamma_{q,k}(x)\right| \to 0,\quad \text{as} \quad n\to\infty, \end{equation*} for $G$ and $\gamma_{q,k}$ expressed in terms of maxima over the cycle. The result is illustrated with examples.