  We study the large-width asymptotics of random fully connected neural
networks with weights drawn from $\alpha$-stable distributions, a family of
heavy-tailed distributions arising as the limiting distributions in the
Gnedenko-Kolmogorov heavy-tailed central limit theorem. We show that in an
arbitrary bounded Euclidean domain $\mathcal{U}$ with smooth boundary, the
random field at the infinite-width limit, characterized in previous literature
in terms of finite-dimensional distributions, has sample functions in the
fractional Sobolev-Slobodeckij-type quasi-Banach function space
$W^{s,p}(\mathcal{U})$ for integrability indices $p < \alpha$ and suitable
smoothness indices $s$ depending on the activation function of the neural
network, and establish the functional convergence of the processes in
$\mathcal{P}(W^{s,p}(\mathcal{U}))$. This convergence result is leveraged in
the study of functional posteriors for edge-preserving Bayesian inverse
problems with stable neural network priors.
