The sequence of partial maxima for i.i.d. random variables is considered. Two theorems concerning the sample behaviour of the maxima are proved. Also a large deviations result is given, connected with weak convergence to the double exponential distribution.
Laurens de Haan. Arie Hordijk. "The Rate of Growth of Sample Maxima." Ann. Math. Statist. 43 (4) 1185 - 1196, August, 1972. https://doi.org/10.1214/aoms/1177692470