  Let $(\xi_1,\eta_1),(\xi_2,\eta_2),...$ be a sequence of i.i.d.\ copies of a
random vector $(\xi,\eta)$ taking values in $\R^2$, and let $S_n :=
\xi_1+...+\xi_n$. The sequence $(S_{n-1} + \eta_n)_{n \geq 1}$ is then called
perturbed random walk.
  We study random quantities defined in terms of the perturbed random walk:
$\tau(x)$, the first time the perturbed random walk exits the interval
$(-\infty,x]$, $N(x)$, the number of visits to the interval $(-\infty,x]$, and
$\rho(x)$, the last time the perturbed random walk visits the interval
$(-\infty,x]$. We provide criteria for the a.s.\ finiteness and for the
finiteness of exponential moments of these quantities. Further, we provide
criteria for the finiteness of power moments of $N(x)$ and $\rho(x)$.
