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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 finalvalue of this walk belong to some compact set. We first specify the associatedprobability. Then, conditionally on such an event, we finely describe thetrajectory of the random walk. It yields a decomposition theorem with respectto a random time giving a big jump whose distribution can be describedexplicitly.

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