  Let $(S_n)_{n \geq 0}$ be a transient random walk in the domain of attraction
of a stable law and let $(\xi(s))_{s \in \mathbb{Z}}$ be a stationary sequence
of random variables. In a previous work, under conditions of type $D(u_n)$ and
$D'(u_n)$, we established a limit theorem for the maximum of the first $n$
terms of the sequence $(\xi(S_n))_{n\geq 0}$ as $n$ goes to infinity. In this
paper we show that, under the same conditions and under a suitable scaling, the
point process of exceedances converges to a Poisson point process. We also give
some properties of $(\xi(S_n))_{n\geq 0}$.
