In this paper we study the probability $��_n(u):={\mathbb P}\left(C_n\geqslant u n \right)$, with $C_n:=A(��_n B(��_n))$ for L��vy processes $A(\cdot)$ and $B(\cdot)$, and $��_n$ and $��_n$ non-negative sequences such that $��_n ��_n =n$ and $��_n\to\infty$ as $n\to\infty$. Two timescale regimes are distinguished: a `fast' regime in which $��_n$ is superlinear and a `slow' regime in which $��_n$ is sublinear. We provide the exact asymptotics of $��_n(u)$ (as $n\to\infty$) for both regimes, relying on change-of-measure arguments in combination with Edgeworth-type estimates. The asymptotics have an unconventional form: the exponent contains the commonly observed linear term, but may also contain sublinear terms (the number of which depends on the precise form of $��_n$ and $��_n$). To showcase the power of our results we include two examples, covering both the case where $C_n$ is lattice and non-lattice. Finally we present numerical experiments that demonstrate the importance of taking into account the doubly stochastic nature of $C_n$ in a practical application related to customer streams in service systems; they show that the asymptotic results obtained yield highly accurate approximations, also in scenarios in which there is no pronounced timescale separation.

25 pages, no figures