Upper bounds for the L^q empirical process via generic chaining

Using the generic chaining method, we derive upper bounds for the $L^q$ process of sub-Gaussian classes when $1\le q\le 2$, thereby resolving an open problem posed by Al-Ghattas, Chen, and Sanz-Alonso in arXiv:2502.16916. Combined with the results of arXiv:2502.16916, this yields upper bounds for the $L^q$ process for all $1\le q<\mathrm{\infty }$. We also present corollaries of this result in the geometry of Banach spaces, including high-probability bounds on the ${\mathrm{\ell }}_q$ norm diameter of random hyperplane sections of convex bodies where the subspaces are not necessarily uniformly distributed on the Grassmannian manifold and the restricted isomorphic property for ${\mathrm{\ell }}_q$ norm.