Title: The Bennett-Orlicz Norm

Author(s): Jon A. Wellner
Issue: Volume 79 Series A Part 2 Year 2017
Pages: 355 -- 383
van de Geer and Lederer ({\em Probab. Theory Related Fields {\bf 157} (1-2), 225–250, 2013) introduced a new Orlicz norm, the Bernstein-Orlicz norm, which is connected to Bernstein type inequalities. Here we introduce another Orlicz norm, the Bennett-Orlicz norm, which is connected to Bennett type inequalities. The new Bennett-Orlicz norm yields inequalities for expectations of maxima which are potentially somewhat tighter than those resulting from the Bernstein-Orlicz norm when they are both applicable. We discuss cross connections between these norms, exponential inequalities of the Bernstein, Bennett, and Prokhorov types, and make comparisons with results of Talagrand ({\em Ann. Probab., {\bf 17}(4), 1546–1570, 1989, 1991), and Boucheron et al. (2013).
AMS (2000) subject classification . Primary: 60E15, 60F10; Secondary: 60G50, 33E20.
Keywords and phrases: Bennett’s inequality, Exponential bound, Maximal inequality, Orlicz norm, Poisson, Prokhorov’s inequality.