van de Geer and Lederer (Probab. Theory Related Fields 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 (Ann. Probab., 17(4), 1546-1570, 1989, 1991), and Boucheron et al. (2013).
Keywords: Bennett’s inequality; Exponential bound; Maximal inequality; Orlicz norm; Poisson; Primary: 60E15, 60F10; Prokhorov’s inequality; Secondary: 60G50, 33E20.