A local limit theorem for maxima of i.i.d. random variables is proved. Also it is shown that under the so-called von Mises' conditions the density of the normalized maximum converges to the limit density in $L_p(0 < p \leq \infty)$ provided both the original density and the limit density are in $L_p$. Finally an occupation time result is proved. The methods of proof are different from those used for the corresponding results concerning partial sums.
L. de Haan. S. I. Resnick. "Local Limit Theorems for Sample Extremes." Ann. Probab. 10 (2) 396 - 413, May, 1982. https://doi.org/10.1214/aop/1176993865