Entropy Conditions for $L_r$-Convergence of Empirical Processes

The law of large numbers (LLN) over classes of functions is a classical topic of empirical processes theory. The properties characterizing classes of functions on which the LLN holds uniformly (i.e. Glivenko\u2013Cantelli classes) have been widely studied in the literature. An elegant suf\ufb01 cient condition for such a property is \ufb01niteness of the Koltchinskii\u2013Pollard entropy integral, and other conditions have been formulated in terms of suitable combinatorial complexities (e.g. the Vapnik\u2013 Chervonenkis dimension). In this paper, we endow the class of functions F with a probability measure and consider the LLN relative to the associated Lr metric. This framework extends the case of uniform convergence over F , which is recovered when r goes to in\ufb01nity. The main result is a Lr -LLN in terms of a suitable uniform entropy integral which generalizes the Koltchinskii\u2013Pollard entropy integra