Local Asymptotics for Entrance Laws of Reflected Lévy Excursions

Abstract

This paper establishes uniform local large-deviation asymptotics for the entrance law ${\underline{n}}_t(dx)$ of excursions of a Lévy process with negative drift and locally regularly varying Lévy measure $\nu$, reflected at its running infimum. In the fixed-time regime, we prove that as $x\to \infty$, $${\underline{n}}_t((x,x+\delta ])\sim \underline{n}(\zeta \wedge t)\nu ((x,x+\delta ]),$$ uniformly in $\delta \in [{\delta }_0,\infty ]$ for every ${\delta }_0>0$. Our principal result concerns the large-time regime and shows that, as $t\to \infty$, $${\underline{n}}_t((x,x+\delta ])\sim {\int }_x^{x+\delta }V(z)\nu (\beta t+dz),$$ uniformly in $x\ge 0$ and $\delta \in [{\delta }_0,\infty ]$, where $V$ denotes the renewal function. Using a fluctuation-theoretic decomposition of the killed semigroup, we further derive corresponding uniform local large-deviation asymptotics for the Lévy process killed upon its first passage below zero.

arXiv comment34 pages