Sharp Convex Concentration for Symmetric Random Tensors with Subgaussian Coordinates

Let $X=(X_1,\dots ,X_n)$ have independent coordinates with mean zero, variance one, and $\mathrm{\parallel }{X}_i{\mathrm{\parallel }}_{{\psi }_2}\le K$, and let $H_d=({\mathbb{R}}^n{)}^{{\otimes }_{2}d}$. Let $L>0$ and let $f:H_d\to \mathbb{R}$ be convex and $L$-Lipschitz. We prove that, for $0\le t\le c_{K}L{n}^{d\mathrm{/}2}$, $$\mathsf{\text{P}}\{\mid f(X^{\otimes d})-\mathsf{\text{E}}f(X^{\otimes d})\mid >t\}\le C\exp [-c_K{\mathcal{I}}_{n,d}\left(\frac{t}{L{n}^{(d-1)\mathrm{/}2}}\right)],$$ where $${\mathcal{I}}_{n,d}(s)=\min \{\frac{s^2}{d^2},\frac{s^2}{d\log (e+nd\mathrm{/}{s}^2)}\},s>0,{\mathcal{I}}_{n,d}(0)=0.$$ The first rate is forced by changes in $\mathrm{\parallel }X\mathrm{\parallel }$. The second comes from changes of $X$ when its norm is nearly fixed. The proof constructs one coupling that controls both the coordinatewise conditional displacement and the mean squared Euclidean distance, and combines these bounds with a second-order estimate for $x\mapsto x^{\otimes d}$. The rate is minimax sharp, scale by scale, even when the subgaussian norms are bounded by an absolute constant. For bounded coordinates the logarithm in the second rate disappears.