Several estimates of the rate of convergence of normalized maxima of random processes are exhibited by using the theory of probability metrics. In contrast to the summation scheme, uniform estimates for normalized maxima of random sequences will be derived.
L. de Haan. S. T. Rachev. "Estimates of the Rate of Convergence for Max-Stable Processes." Ann. Probab. 17 (2) 651 - 677, April, 1989. https://doi.org/10.1214/aop/1176991420