  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).
