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

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.