Publication Date:
2012-10-08
Description:
We introduce two new concepts designed for the study of empirical processes. First, we introduce a new Orlicz norm which we call the Bernstein–Orlicz norm. This new norm interpolates sub-Gaussian and sub-exponential tail behavior. In particular, we show how this norm can be used to simplify the derivation of deviation inequalities for suprema of collections of random variables. Secondly, we introduce chaining and generic chaining along a tree. These simplify the well-known concepts of chaining and generic chaining. The supremum of the empirical process is then studied as a special case. We show that chaining along a tree can be done using entropy with bracketing. Finally, we establish a deviation inequality for the empirical process for the unbounded case. Content Type Journal Article Pages 1-26 DOI 10.1007/s00440-012-0455-y Authors Sara van de Geer, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland Johannes Lederer, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland Journal Probability Theory and Related Fields Online ISSN 1432-2064 Print ISSN 0178-8051
Print ISSN:
0178-8051
Electronic ISSN:
1432-2064
Topics:
Mathematics
