 Solution

For large $\sigma$, neighboring peaks overlap and the density has a broad, smooth, symmetric central hump, approximately $(\sqrt{2\pi }\sigma {)}^{-1}{e}^{-z^2/(2{\sigma }^2)}$ on the scale $z=O(\sigma )$. A precise central-profile statement follows fromfor each fixed $t$, $X/\sigma \to 0$ almost surely and the integrand is bounded, so dominated convergence applies. The right panel of the preceding figure shows this Gaussian-looking core.

This approximation does not remove the heavy-tailed distribution. Here $\mathbb{E}|X|=(6/{\pi }^2){\sum }_{n\ge 1}1/n=\infty$, so the symmetric distribution has no ordinary mean and no finite variance. Adding the independent Gaussian, which has finite absolute moments, leaves these conclusions unchanged. In particular a central Gaussian sketch cannot justify assigning variance ${\sigma }^2$ to $Z$.