Random walk with heavy tail and negative drift conditioned by its minimum and final values

Abstract

We consider random walks with finite second moment which drifts to $-\infty$ and have heavy tail. We focus on the events when the minimum and the final value of this walk belong to some compact set. We first specify the associated probability. Then, conditionally on such an event, we finely describe the trajectory of the random walk. It yields a decomposition theorem with respect to a random time giving a big jump whose distribution can be described explicitly.