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

In the paper, a regenerative process {Xn:n∈N} with finite mean cycle length is considered. For Mn(q) denoting the qth largest value in {Xk:1⩽k⩽n}, we prove that supx∈R|PMn(q)⩽x−G(x)n∑k=0q−1−logG(x)nkk!γq,k(x)|→0,asn→∞,for G and γq,k expressed in terms of maxima over the cycle. The result is illustrated with examples.