Assess optimality of the Lipschitz regularity assumption (H4)

Determine whether the uniform Lipschitz continuity assumption on the activation function (H4) is optimal (in the Gaussian case) for establishing the functional large deviation principle for the covariance process, or whether weaker growth/regularity conditions suffice to obtain the LDP in the infinite-dimensional setting.

Background

To lift the operator-space LDP to the kernel-space topology ^{+,s}, the paper imposes an additional Lipschitz regularity condition on the activation function (H4). This is stronger than assumptions used for other types of convergence results (e.g., functional CLTs), and the authors question whether it is necessary.

Clarifying the optimal regularity requirements on the activation function would sharpen the theory and potentially broaden applicability of the functional LDP to a larger class of activations.