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 sufficient 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 L _r metric. This framework extends the case of uniform convergence over F, which is recovered when r goes to infinity. The main result is a L _r -LLN in terms of a suitable uniform entropy integral which generalizes the Koltchinskii-Pollard entropy integral. © 2008 Springer Science+Business Media, LLC.