We consider fully connected and feedforward deep neural networks with dependent and possibly heavy-tailed weights, as introduced in Lee et al. 2023, to address limitations of the standard Gaussian prior. It has been proved in Lee et al. 2023 that, as the number of nodes in the hidden layers grows large, according to a sequential and ordered limit, the law of the output converges weakly to a Gaussian mixture. Among our results, we present sufficient conditions on the model parameters (the activation function and the associated Lévy measures) which ensure that the sequential limit is independent of the order. Next, we study the neural network through the lens of the posterior distribution with a Gaussian likelihood. If the random covariance matrix of the infinite-width limit is positive definite under the prior, we identify the posterior distribution of the output in the wide-width limit according to a sequential regime. Remarkably, we provide mild sufficient conditions to ensure the aforementioned invertibility of the random covariance matrix under the prior, thereby extending the results in L. Carvalho et al. 2025. We illustrate our findings using numerical simulations.

翻译：本文研究具有依赖且可能重尾权重的全连接前馈深度神经网络，该模型由Lee等人于2023年提出，旨在解决标准高斯先验的局限性。Lee等人（2023）已证明：当隐藏层节点数依序递增至无穷时，网络输出的分布弱收敛于高斯混合分布。我们的研究成果包括：提出了关于模型参数（激活函数及相关Lévy测度）的充分条件，以确保该序列极限与顺序无关。进一步，我们通过结合高斯似然的贝叶斯后验分布来研究该神经网络。若无限宽度极限的随机协方差矩阵在先验下正定，我们基于序列极限框架确定了宽宽度极限下输出的后验分布。值得注意的是，我们给出了较弱的充分条件以保证该随机协方差矩阵在先验下的可逆性，从而扩展了L. Carvalho等人（2025）的研究结果。最后通过数值模拟验证了我们的理论发现。