论文标题
从浆果 - 埃斯尼到超指数
From Berry-Esseen to super-exponential
论文作者
论文摘要
对于任何整数$ m <n $，其中$ m $都可以取决于$ n $，我们研究$ \ frac {1} {\ sqrt {\ sqrt {m}} \ mathrm {tr} \ mathbf {tr} \ mathbf {u}^m $随机分布$ n \ of fors oortal and haar and gutty和haar nog，矩阵$ \ mathbf {u} $ size $ n $。在单一的情况下，我们证明总变化距离小于$γ（\ lfloor n/m \ rfloor+2）^{ - 1} m^{ - \ lfloor n/m \ rfloor} \ rfloor} \ lfloor n/m \ lfloor n/m \ rfloor n/m \ rfloor n/m \ rfloor^rfloor^{1/4} {1/4} {1/4}。该结果插值在固定$ M $获得的超指定约束与$ 1/n $绑定的$ 1/n $绑定来自浆果 - 埃森定理时适用于$ m \ ge n $，这是由于降雨而适用的。我们获得了正交和符号组的类似结果。在这些情况下，我们的总变化上限为$γ（2 \ lfloor n/m \ rfloor +1）^{ - 1/2} m^{ - \ lfloor n/m \ rfloor +1}（\ log n）（\ log n）对于$ m = 1 $，我们获得了$ l^2 $ distances $ n \ to \ infty $的互补下限和精确的渐近级，这表明我们的结果有多敏锐。
For any integer $m<n$, where $m$ can depend on $n$, we study the rate of convergence of $\frac{1}{\sqrt{m}}\mathrm{Tr} \mathbf{U}^m$ to its limiting Gaussian as $n\to\infty$ for orthogonal, unitary and symplectic Haar distributed random matrices $\mathbf{U}$ of size $n$. In the unitary case, we prove that the total variation distance is less than $Γ(\lfloor n/m \rfloor+2)^{-1} m^{- \lfloor n/m\rfloor} \lfloor n/m \rfloor^{1/4}\sqrt{\log n}$ times a constant. This result interpolates between the super-exponential bound obtained for fixed $m$ and the $1/n$ bound coming from the Berry-Esseen theorem applicable when $m\ge n$ by a result of Rains. We obtain analogous results for the orthogonal and symplectic groups. In these cases, our total variation upper bound takes the form $Γ(2\lfloor n/m\rfloor+1)^{-1/2}m^{-\lfloor n/m\rfloor +1}(\log n)^{1/4}$ times a constant and the result holds provided $n \geq 2m$. For $m=1$, we obtain complementary lower bounds and precise asymptotics for the $L^2$-distances as $n\to\infty$, which show how sharp our results are.
