Empirical Processes: Maximal Inequalities and Chaining

It can be checked that this is a valid norm (on the set of random variables for which the left side of the above display is finite). Of special interest to us will be the Orlicz norms corresponding to the functions {ψp : p ≥ 1} where ψp(x) = exp(xp) − 1. Lemma 8.1 of Kosorok (2008) provides a necessary and sufficient condition for the ψp Orlicz norm to be finite in terms of the tail-behavior of X. As a consequence of this lemma, if P (|X| > x) ≤ K exp(−C xp) for some constants C,K > 0, then ‖X‖ψp ≤ ( 1 +K C )1/p . (1.1)