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–Cantelli classes) have been widely studied in the literature. An elegant suffi cient condition for such a property is finiteness of the Koltchinskii–Pollard entropy integral, and other conditions have been formulated in terms of suitable combinatorial complexities (e.g. the Vapnik– 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 infinity. The main result is a Lr -LLN in terms of a suitable uniform entropy integral which generalizes the Koltchinskii–Pollard entropy integral