Local Limit Theorems for Heavy-Tailed Lévy Processes with Negative Drift

This paper establishes uniform local asymptotics for the first-passage time below zero and the stopped maximum of a Lévy process with negative drift and a locally regularly varying Lévy measure. We show that their probabilities of lying in distant intervals are asymptotic to the expected first-passage time multiplied by the corresponding Lévy masses, uniformly over interval widths bounded away from zero, including unbounded intervals. Although the two local events have asymptotically equivalent probabilities, their conditional overlap vanishes for fixed-width intervals. Nevertheless, we establish conditional concentration of the first-passage time and stopped maximum around their deterministic linear relation on scales dominating the fluctuations of the centered process. We further prove conditional functional limit theorems yielding a single-step process with random jump time on the original time scale and a deterministic linear descent on the accelerated scale. Finally, we identify the joint limiting distribution of the unscaled pre-passage position and terminal overshoot, distinguishing continuous passage from passage by a downward jump.