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We derive subexponential tail asymptotics for the distribution of the maximumof 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 dependon the future increments of the process. Our asymptotic results are uniformover the whole class of such random times. Particular examples are given bystopping times and by $\tau$ independent of the processes. We link our resultswith random walk theory.

Maxima over random time intervals for heavy-tailed compound renewal and L\'evy processes