Concentration inequalities and asymptotic results for ratio type empirical processes

Let $\mathcal{F}$ be a class of measurable functions on a measurable space $(S,\mathcal{S})$ with values in $[0,1]$ and let $$P_n=n^{-1}\sum_{i=1}^n\delta_{X_i}$$ be the empirical measure based on an i.i.d. sample $(X_1,...,X_n)$ from a probability distribution $P$ on $(S,\mathcal{S})$. We study the behavior of suprema of the following type: $$\sup_{r_n<\sigma_Pf\leq \delta_n}\frac{|P_nf-Pf|}{\phi(\sigma_Pf)},$$ where $\sigma_Pf\ge\operatorname {Var}^{1/2}_Pf$ and $\phi$ is a continuous, strictly increasing function with $\phi(0)=0$. Using Talagrand's concentration inequality for empirical processes, we establish concentration inequalities for such suprema and use them to derive several results about their asymptotic behavior, expressing the conditions in terms of expectations of localized suprema of empirical processes. We also prove new bounds for expected values of sup-norms of empirical processes in terms of the largest $\sigma_Pf$ and the $L_2(P)$ norm of the envelope of the function class, which are especially suited for estimating localized suprema. With this technique, we extend to function classes most of the known results on ratio type suprema of empirical processes, including some of Alexander's results for VC classes of sets. We also consider applications of these results to several important problems in nonparametric statistics and in learning theory (including general excess risk bounds in empirical risk minimization and their versions for $L_2$-regression and classification and ratio type bounds for margin distributions in classification).