TY - JOUR
T1 - Entropy conditions for L r -convergence of empirical processes
AU - Caponnetto, Andrea
AU - De Vito, Ernesto
AU - Pontil, Massimiliano
PY - 2009/5
Y1 - 2009/5
N2 - 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.
AB - 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.
KW - Empirical processes
KW - Glivenko-Cantelli classes
KW - Rademacher averages
KW - Uniform entropy
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U2 - 10.1007/s10444-008-9072-9
DO - 10.1007/s10444-008-9072-9
M3 - 21_Publication in refereed journal
VL - 30
SP - 355
EP - 373
JO - Advances in Computational Mathematics
JF - Advances in Computational Mathematics
SN - 1019-7168
IS - 4
ER -