  We consider shot noise processes $(X(t))_{t \geq 0}$ with deterministic
response function $h$ and the shots occurring at the renewal epochs $0= S_0 <
S_1 < S_2 ...$ of a zero-delayed renewal process. We prove convergence of the
finite-dimensional distributions of $(X(ut))_{u \geq 0}$ as $t \to \infty$ in
different regimes. If the response function $h$ is directly Riemann integrable,
then the finite-dimensional distributions of $(X(ut))_{u \geq 0}$ converge
weakly as $t \to \infty$. Neither scaling nor centering are needed in this
case. If the response function is eventually decreasing, non-integrable with an
integrable power, then, after suitable shifting, the finite-dimensional
distributions of the process converge. Again, no scaling is needed. In both
cases, the limit is identified. If the distribution of $S_1$ is in the domain
of attraction of an $\alpha$-stable law and the response function is regularly
varying at $\infty$ with index $\beta$ (with $\beta < 1/\alpha$ or $\beta \leq
1/\alpha$, depending on whether $\mathbb{E} S_1 < \infty$ or $\mathbb{E} S_1 =
\infty$), then scaling is needed to obtain weak convergence of the
finite-dimensional distributions of $(X(ut))_{u \geq 0}$. The limiting
processes are fractionally integrated stable L\'{e}vy motions if $\mathbb{E}
S_1 < \infty$ and fractionally integrated inverse stable subordinators if
$\mathbb{E} S_1 = \infty$.
