In this talk, I consider random neural networks with Gaussian weights and biases. A well-known result (proved under various assumptions in several references) concerns the convergence in distribution of the network output, in the infinite-width limit, to a centered Gaussian process with i.i.d. components (with the depth $L$ kept fixed). I will present some large deviation results describing a collapse of the network output (which, as expected, converges to the zero vector) when it is multiplied by a scaling factor tending to zero; see [1], where moderate deviations are also studied. In the final part, I will briefly outline some results in a very recent work (see [2]) concerning the deep limit (that is, as $L\to\infty$) for a generic  component of the Gaussian process mentioned above.

REFERENCES
[1] C. Macci, B. Pacchiarotti, G.L. Torrisi. Journal of Applied Probability (2026), in press.
[2] S. Di Lillo, C. Macci, B. Pacchiarotti. Submitted, and available on arxiv.