  We investigate weak convergence of finite-dimensional distributions of a
renewal shot noise process $(Y(t))_{t\geq 0}$ with deterministic response
function $h$ and the shots occurring at the times $0 = S_0 < S_1 < S_2<\ldots$,
where $(S_n)$ is a random walk with i.i.d.\ jumps. There has been an outbreak
of recent activity around this topic. We are interested in one out of few cases
which remained open: $h$ is regularly varying at $\infty$ of index $-1/2$ and
the integral of $h^2$ is infinite. Assuming that $S_1$ has a moment of order
$r>2$ we use a strong approximation argument to show that the random
fluctuations of $Y(s)$ occur on the scale $s=t+g(t,u)$ for $u\in [0,1]$, as
$t\to\infty$, and, on the level of finite-dimensional distributions, are well
approximated by the sum of a Brownian motion and a Gaussian process with
independent values (the two processes being independent). The scaling function
$g$ above depends on the slowly varying factor of $h$. If, for instance,
$\lim_{t\to\infty}t^{1/2}h(t)\in (0,\infty)$, then $g(t,u)=t^u$.

Probability