Removing the sub-Gaussian assumption in the dependent CGMT

Determine whether Theorem CGMT_2 (the dependent Convex Gaussian Min-Max Theorem under low-rank dependence) holds without the sub-Gaussian assumption on the covariate vectors, by establishing the theorem under weaker tail conditions (e.g., sub-exponential or heavy-tailed distributions) and precisely characterizing any additional requirements needed for the result.

Background

The theoretical development of the dependent CGMT (Theorem CGMT_2) and the universality results assume sub-Gaussian covariates for technical convenience. However, the authors’ simulations show universality behavior across distributions with heavier tails (e.g., exponential, gamma, and t-distribution with 3 degrees of freedom), suggesting the sub-Gaussian assumption may not be essential.

The authors explicitly conjecture that sub-Gaussianity is not necessary for the dependent CGMT, motivating a formal extension of Theorem CGMT_2 to broader distribution classes and a precise delineation of conditions under which the theorem remains valid.