  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.
