We establish central and non-central limit theorems for sequences of functionals of the Gaussian output of an infinitely-wide random neural network on the d-dimensional sphere . We show that the asymptotic behaviour of these functionals as the depth of the network increases depends crucially on the fixed points of the covariance function, resulting in three distinct limiting regimes: convergence to the same functional of a limiting Gaussian field, convergence to a Gaussian distribution, convergence to a distribution in the Qth Wiener chaos. Our proofs exploit tools that are now classical (Hermite expansions, Diagram Formula, Stein-Malliavin techniques), but also ideas which have never been used in similar contexts: in particular, the asymptotic behaviour is determined by the fixed-point structure of the iterative operator associated with the covariance, whose nature and stability governs the different limiting regimes.

翻译：我们针对d维球面上无限宽随机神经网络高斯输出的泛函序列，建立了中心极限定理与非中心极限定理。研究表明，随着网络深度增加，这些泛函的渐近行为关键取决于协方差函数的不动点，从而产生三种不同的极限状态：收敛至某高斯场的同一泛函、收敛至高斯分布、收敛至第Q阶Wiener混沌分布。我们的证明既运用了经典工具（Hermite展开、图表公式、Stein-Malliavin方法），也引入了此前未被应用于类似场景的创新思想：特别地，渐近行为由协方差关联迭代算子的不动点结构决定，该算子的性质与稳定性主导了各极限状态之间的相变。