  Let $\{\xi(k), k \in \mathbb{Z} \}$ be a stationary sequence of random
variables and let $\{S_n, n \in \mathbb{N}_+ \}$ be a transient random walk in
the domain of attraction of a stable law. In the previous work
\cite{Nicolas_Ahmad}, under conditions of type $D(u_n)$ and $D'(u_n)$ we
provided a limit theorem for the maximum of the first $n$ terms of the sequence
$\{\xi(S_n), n \in \mathbb{N} \}$. In this paper, under the same conditions we
will see that, the limit of the process which counts the numbers of the
exceedances of the form $\{\xi(S_k)>u_n\}, k\geq 1$, is a compound Poisson
point process. We also deal with the so-called extremal index for the sequence
$\{\xi(S_n), n \in \mathbb{N} \}$ and we discuss some weak mixing properties.
