Anticoncentration and Berry–Esseen bounds for random tensors

We obtain estimates for the Kolmogorov distance to appropriately chosen gaussians, of linear functions $$\begin{aligned} \sum _{i\in [n]^d} \theta _i X_i \end{aligned}$$ random tensors $$\varvec{X}=\langle X_i:i\in [n]^d\rangle $$ which are symmetric and exchangeable, whose entries have bounded third moment vanish on diagonal indices. These expressed in terms intrinsic (and easily computable) parameters associated with tensor $$\varvec{X}$$ given coefficients $$\langle _i:i\in , they optimal various regimes. The key ingredient—which is independent interest—is a combinatorial CLT high-dimensional provides quantitative non-asymptotic normality under suitable conditions, statistics form _{(i_1,\dots ,i_d)\in \varvec{\zeta }\big (i_1,\dots ,i_d,\pi (i_1),\dots ,\pi (i_d)\big ) where $$\varvec{\zeta }:[n]^d\times [n]^d\rightarrow \mathbb {R}$$ deterministic real tensor, $$\pi permutation uniformly distributed group $$\mathbb {S}_n$$ . Our results extend, any dimension d, classical work Bolthausen who covered one-dimensional case, more recent Barbour/Chen treated two-dimensional case.