Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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).

28
The categories of this paper are: alg-geom
4
The categories of this paper are: cond-mat
32
The categories of this paper are: atom-ph
30
The categories of this paper are: q-alg
0
The categories of this paper are: hep-ph
14
The categories of this paper are: math-ph
3
The categories of this paper are: physics
27
The categories of this paper are: comp-gas
18
The categories of this paper are: econ
17
The categories of this paper are: q-fin
33
The categories of this paper are: plasm-ph
11
The categories of this paper are: hep-lat
34
The categories of this paper are: chem-ph
9
The categories of this paper are: nlin
10
The categories of this paper are: q-bio
35
The categories of this paper are: mtrl-th