A local probability exponential inequality for the large deviation of an empirical process indexed by an unbounded class of functions and its application (Q2386554)

This article contains `localized' exponential inequalities for an empirical process based on independent but not identically distributed random variables \(X_i\), uniform over classes of functions \({\mathcal F}\) that may not be uniformly bounded. `Localized' inequalities means inequalities for \({\mathbf P}_S(\cdot)= \text{Pr}(S\cap\cdot)/\text{Pr}(S)\), for \(\text{Pr}(S)\geq 1-\delta\) for some \(\delta>0\). The hypotheses consist of the SLLN uniform in \(\{f^2:f\in{\mathcal F}\}\), \(L_2\)-boundedness and a uniform \(L_1(P_n)\) random entropy bound for \({\mathcal F}\), where \(P_n\) is the empirical measure. The proofs are elementary given the modern developments in this theory (basically, they only use symmetrization and Hoeffding's inequality). Some applications are presented (SLLN, `laws of the logarithm'). The results may be considered as extensions to the unbounded non-i.d. case of an early law of large numbers of Vapnik and Červonenkis under a random \(L_1(P_n)\)-entropy condition and its proof, but are not related to the incomparably deeper exponential inequalities of Alexander, Massart and Talagrand.