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There is a recent and growing literature on large-width asymptotic propertiesof Gaussian neural networks (NNs), namely NNs whose weights are initialized asGaussian distributions. Two popular problems are: i) the study of thelarge-width distributions of NNs, which characterizes the infinitely wide limitof a rescaled NN in terms of a Gaussian stochastic process; ii) the study ofthe large-width training dynamics of NNs, which characterizes the infinitelywide dynamics in terms of a deterministic kernel, referred to as the neuraltangent kernel (NTK), and shows that, for a sufficiently large width, thegradient descent achieves zero training error at a linear rate. In this paper,we consider these problems for $\alpha$-Stable NNs, namely NNs whose weightsare initialized as $\alpha$-Stable distributions with $\alpha\in(0,2]$. First,for $\alpha$-Stable NNs with a ReLU activation function, we show that if theNN's width goes to infinity then a rescaled NN converges weakly to an$\alpha$-Stable stochastic process. As a difference with respect to theGaussian setting, our result shows that the choice of the activation functionaffects the scaling of the NN, that is: to achieve the infinitely wide$\alpha$-Stable process, the ReLU activation requires an additional logarithmicterm in the scaling with respect to sub-linear activations. Then, we study thelarge-width training dynamics of $\alpha$-Stable ReLU-NNs, characterizing theinfinitely wide dynamics in terms of a random kernel, referred to as the$\alpha$-Stable NTK, and showing that, for a sufficiently large width, thegradient descent achieves zero training error at a linear rate. The randomnessof the $\alpha$-Stable NTK is a further difference with respect to the Gaussiansetting, that is: within the $\alpha$-Stable setting, the randomness of the NNat initialization does not vanish in the large-width regime of the training.

Large-width asymptotics for ReLU neural networks with $\alpha$-Stable initializations