An absolutely continuous distribution $F$ is said to be in the domain of uniform local attraction of the absolutely continuous distribution $H$ if the density of the normalized maximum of an independent sample of size $n$ converges locally uniformly to the density of $H$ as $n \rightarrow \infty$. Under the sole restriction that $F$ is eventually increasing, the domains of uniform local attraction to the three types of extreme value distribution are shown to be characterized by the usual Von Mises' conditions. The equivalent form of conditions used here greatly simplifies and shortens proofs of known results. In particular, $L_p$ convergence and convergence of the $k$ upper sample extremes are investigated and extensions to known results derived.
T. J. Sweeting. "On Domains of Uniform Local Attraction in Extreme Value Theory." Ann. Probab. 13 (1) 196 - 205, February, 1985. https://doi.org/10.1214/aop/1176993075