The present paper is concerned with the local times of a Lé process reflected at two barriers $0$ and $K>0$. The reflected process is decomposed into the original process plus local times at $0$ and $K$ and a starting condition, and we study $\ell^{K,n}$, the mean rate of increase of the local time at $K$ when the reflected process is started in stationarity. We derive asymptotics $(K \to \infty)$ for $\ell^{K,n}$ when the Lé process has mean zero. The precise form of the asymptotics depends on the existence or non-existence of a finite second moment, paralleling the difference between the normal and the stable central limit theorem. To achieve the asymptotic results, we prove a uniform integrability criterion for Lé processes and a continuity result for $\ell^{K,n}$, which are of independent interest.

Keywords continuity of the local time, finite buffer, Lévy process, reflection, loss rate, Skorokhod problem, stable central limit theorem, stable distribution, uniform integrability