论文标题
抗浓缩和浆果 - 随机张量的界限
Anticoncentration and Berry--Esseen bounds for random tensors
论文作者
论文摘要
We obtain estimates for the Kolmogorov distance to appropriately chosen gaussians, of linear functions \[ \sum_{i\in [n]^d} θ_i X_i \] of random tensors $\boldsymbol{X}=\langle X_i:i\in [n]^d\rangle$ which are symmetric and exchangeable, and whose entries have bounded third时刻并在对角线指数上消失。这些估计值是根据与随机张量$ \ boldsymbol {x} $和给定系数$ \langleθ_i：i \ in [n]^d \ rangle $相关的固有（易于计算）参数表示的，它们在各种方面都是最佳的。 关键成分 - 具有独立感兴趣的主要成分是用于高维张量的组合CLT，可在适当条件下，形式的统计量\ [\ sum _ {（i_1，\ dots，i_d，i_d）\ in [n]^d} In [n]^d} In [n]^d} \ \boldsymbolζ\ big（i_1，\ dots，i_d，π（i_1），\ dots，π（i_d）\ big）\]其中$ \boldsymbolζ\ colon [n]^d \ times [n]^d \ times [n]^n]对称组$ \ mathbb {s} _n $。我们的结果在任何维度$ d $上扩展了Bolthausen的经典作品，他们涵盖了一维案例，以及最近对二维案件的Barbour/Chen的工作。
We obtain estimates for the Kolmogorov distance to appropriately chosen gaussians, of linear functions \[ \sum_{i\in [n]^d} θ_i X_i \] of random tensors $\boldsymbol{X}=\langle X_i:i\in [n]^d\rangle$ which are symmetric and exchangeable, and whose entries have bounded third moment and vanish on diagonal indices. These estimates are expressed in terms of intrinsic (and easily computable) parameters associated with the random tensor $\boldsymbol{X}$ and the given coefficients $\langle θ_i:i\in [n]^d\rangle$, and they are optimal in various regimes. The key ingredient -- which is of independent interest -- is a combinatorial CLT for high-dimensional tensors which provides quantitative non-asymptotic normality under suitable conditions, of statistics of the form \[ \sum_{(i_1,\dots,i_d)\in [n]^d} \boldsymbolζ\big(i_1,\dots,i_d,π(i_1),\dots,π(i_d)\big) \] where $\boldsymbolζ\colon [n]^d\times [n]^d\to\mathbb{R}$ is a deterministic real tensor, and $π$ is a random permutation uniformly distributed on the symmetric group $\mathbb{S}_n$. Our results extend, in any dimension $d$, classical work of Bolthausen who covered the one-dimensional case, and more recent work of Barbour/Chen who treated the two-dimensional case.
