On norms in some class of exponential type Orlicz spaces of random variables

A new characterization of the exponential type Orlicz spaces generated by the functions $\exp (\mathrm{\mid }x{\mathrm{\mid }}^p)-1$ ($p\ge 1$) is given. We define norms for centered random variables belonging to these spaces. We show equivalence of these norms with the Luxemburg norms. On the example of the Hoeffding inequality we present some application of these norms in a probabilistic context.