Max-semistable laws in extremes of stationary random sequences

In this paper we consider stationary sequences under the validity of an extension of Leadbetter's condition $D(u_n)$. For these sequences we prove that, if $\{k_n\}$ is a nondecreasing integer sequence satisfying $\lim_{n\to+\infty}k_{n+1}/k_n=r\ge 1$, then the limit law for the maximum of the first $k_n$ variables is a max-semistable law. This generalizes the corresponding result for sequences of independent identically distributed random variables of Grinevich [Theory Probab. Appl., 38 (1993), pp. 640–650] and the extremal types theorem of Leadbetter [Z. Wahrsch. Verw. Gebiete, 28 (1974), pp. 289–303]. We also prove that the limiting behavior of this maximum can be inferred from the limiting behavior of the corresponding maximum of the associated independent sequence, and we extend the well-known notion of extremal index. An illustrative example is given.