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We study the tail asymptotics of two functionals (the maximum and the sum ofthe marks) of a generic cluster in two sub-models of the marked Poisson clusterprocess, namely the renewal Poisson cluster process and the Hawkes process.Under the hypothesis that the governing components of the processes areregularly varying, we extend results due to [18] and [5] notably, relying onKaramata's Tauberian Theorem to do so. We use these asymptotics to deriveprecise large deviation results in the fashion of [30] for the above-mentionedprocesses.

Tail asymptotics and precise large deviations for some Poisson cluster processes