Exact asymptotics for a multi-timescale model, with applications in modeling overdispersed customer streams

In this paper we study the probability $\xi_n(u):={\mathbb P}\left(C_n\geqslant u n \right)$, with $C_n:=A(\psi_n B(\varphi_n))$ for L\'{e}vy processes $A(\cdot)$ and $B(\cdot)$, and $\varphi_n$ and $\psi_n$ non-negative sequences such that $\varphi_n \psi_n =n$ and $\varphi_n\to\infty$ as $n\to\infty$. Two timescale regimes are distinguished: a `fast' regime in which $\varphi_n$ is superlinear and a `slow' regime in which $\varphi_n$ is sublinear. We provide the exact asymptotics of $\xi_n(u)$ (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 $\varphi_n$ and $\psi_n$). 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.