The results of Dwass (1964) [Ann. Math. Statist. 35 1718-1725] and Lamperti (1964) [Ann. Math. Statist. 35 1726-1737] on the weak convergence of extremal processes (in the Skorokhod $J_1$-topology) to appropriate Markov processes are extended here for random sample sizes.
Pranab Kumar Sen. "On Weak Convergence of Extremal Processes for Random Sample Sizes." Ann. Math. Statist. 43 (4) 1355 - 1362, August, 1972. https://doi.org/10.1214/aoms/1177692490