Suprema of compound Poisson processes with light tails

It is known that if the Lévy measure of a Lévy process X(t), 0⩽t⩽1, is “heavy tailed”, then the right tails of sup0⩽t⩽1X(t) and X(1) are of the same rate of decay. One of the results of this note is a description of a class of compound Poisson processes with negative drift and “light” tails (which is a subclass of Lévy processes) such that these tails are incomparable.