In modern deep learning, there is a recent and growing literature on the interplay between large-width asymptotics for deep Gaussian neural networks (NNs), i.e. deep NNs with Gaussian-distributed weights, and classes of Gaussian stochastic processes (SPs). Such an interplay has proved to be critical in several contexts of practical interest, e.g. Bayesian inference under Gaussian SP priors, kernel regression for infinite-wide deep NNs trained via gradient descent, and information propagation within infinite-wide NNs. Motivated by empirical analysis, showing the potential of replacing Gaussian distributions with Stable distributions for the NNs weights, in this paper we investigate large-width asymptotics for (fully connected) feed-forward deep Stable NNs, i.e. deep NNs with Stable-distributed weights. First, we show that as the width goes to infinity jointly over the NNs layers, a suitable rescaled deep Stable NN converges weakly to a Stable SP whose distribution is characterized recursively through the NNs layers. Because of the non-triangular NNs structure, this is a non-standard asymptotic problem, to which we propose a novel and self-contained inductive approach, which may be of independent interest. Then, we establish sup-norm convergence rates of a deep Stable NN to a Stable SP, quantifying the critical difference between the settings of ``joint growth and ``sequential growth of the width over the NNs layers. Our work extends recent results on infinite-wide limits for deep Gaussian NNs to the more general deep Stable NNs, providing the first result on convergence rates for infinite-wide deep NNs.
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