Maxima over random time intervals for heavy-tailed compound renewal and Lévy processes

We derive subexponential tail asymptotics for the distribution of the maximum of a compound renewal process with linear component and of a Lévy process, both with negative drift, over random time horizon $\tau$ that does not depend on the future increments of the process. Our asymptotic results are uniform over the whole class of such random times. Particular examples are given by stopping times and by $\tau$ independent of the processes. We link our results with random walk theory.