A Donsker and Glivenko‐Cantelli theorem for random measures linked to extreme value theory

Abstract We consider a class of random point measures that share properties with empirical measures when conditioned to another exogenous random phenomenon. We investigate the validity of some Glivenko‐Cantelli and Donsker theorems for such random measures. In this setup, we prove that the usual conditions on uniform entropy numbers are strong enough to derive these two theorems. A bootstrap Donsker theorem is also proved. Some applications of these results are also presented in the framework of extreme value theory and nearest‐neighbor rules.