For a wide class of (dependent) random variables $X_1, X_2, \cdots, X_n$, a limit law is proved for the maximum, with suitable normalization, of $X_1, X_2, \cdots, X_n$. The results are more general in two aspects than the ones obtained earlier by several authors, namely, the stationary of the $X$'s is not assumed and secondly, the assumptions on the dependence of the $X$'s are weaker than those occurring in previous papers. A generalization of the method of inclusion and exclusion is one of the main tools.
Janos Galambos. "On the Distribution of the Maximum of Random Variables." Ann. Math. Statist. 43 (2) 516 - 521, April, 1972. https://doi.org/10.1214/aoms/1177692632