Weak-lq covariance class with directional sub-Gaussian tails

Let $p\ge 2$ be an integer, $0\le q<1$, $c_{n,p}\ge 0$, and $\tau >0$. For a symmetric covariance matrix $\mathrm{\Sigma }=({\sigma }_{ij}{)}_{1\le i,j\le p}$, write $\mathrm{\mid }{\sigma }_{-j,j}{\mathrm{\mid }}_{(1)}\ge \cdots \ge \mathrm{\mid }{\sigma }_{-j,j}{\mathrm{\mid }}_{(p-1)}$ for the magnitudes of the off-diagonal entries in column $j$, arranged in nonincreasing order. Define $G_q(c_{n,p})=\{\mathrm{\Sigma }:{\max }_{1\le j\le p}{\max }_{1\le k\le p-1}k\mathrm{\mid }{\sigma }_{-j,j}{\mathrm{\mid }}_{(k)}^q\le c_{n,p}\}(0<q<1)\mathrm{.}$ For $q=0$, define $G_0(c_{n,p})$ to be the symmetric covariance matrices with at most $c_{n,p}$ nonzero off-diagonal entries in each column. The corresponding law class ${\mathcal{P}}_q(\tau ,c_{n,p})$ consists of iid laws of $X\in {\mathbb{R}}^p$ whose covariance lies in $G_q(c_{n,p})$ and which satisfy

$$$$

for every unit vector $v$ and every $t>0$. This is a clean sub-Gaussian model; it assumes neither coordinate independence nor robustness to contamination.