  We derive subexponential tail asymptotics for the distribution of the maximum
of a compound renewal process with linear component and of a L\'evy 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.
