论文标题
高尺寸的中央限制定理：最佳结合尺寸增长率
Central Limit Theorem in High Dimensions : The Optimal Bound on Dimension Growth Rate
论文作者
论文摘要
在本文中，我们试图回答一个简单的问题：``\ textit {dimension $ p $的关键增长率是样本大小$ n $的函数的函数，而中央限制定理在收集$ p $ p $ diper-diber-dipermensional hyperementional hypereftangles y}中均均匀地保持？具体而言，我们对$ \ sum_ {i = 1}^{n} x_i $的正常缩放版本的正常近似感兴趣$ \ Mathcal {a}^{re} = \ {\ prod_ {j = 1}^{p} [a_j，b_j，b_j] \ cap \ mathcal {r}： - \ \ \ iffty \ leq a_j \ leq a_j \ leq b_j \ leq b_j \ leq \ leq \ leq \ leq \ leq \ eftty，j $ x_1，\ dots，x_n $是独立的$ p- $尺寸随机向量，每个向量具有独立且相同分布的（IID）组件。我们调查了$ \ log p $的关键截止率，以下统一中央限制定理（CLT）所持并在其失败之上。根据Chernozukov等人的最新结果。 （2017年），众所周知，如果$ \ log p = o \ big（n^{1/7} \ big）$，CLT均超过$ \ MATHCAL {A}^{RE} $均匀地保持。他们还推测，要使CLT均匀地固定在$ \ Mathcal {a}^{re} $上，最佳速率为$ \ log P = o \ big（n^{1/3} \ big）$。相反，在某些情况下，当$ \ log P = o \ big（n^{1/2} \ big）$时，CLT在$ \ Mathcal {a}^{re} $上均匀地保持。更准确地说，我们表明，如果$ \ log p =ε\ sqrt {n} $对于某些足够小的$ε> 0 $，则正常近似值在错误$ε$的情况下是有效的，均超过$ \ Mathcal {a} a}^{re} $。此外，我们以一个示例表明，如果$ \ limsup_ {t \ rightArrow \ infty} n^{ - （1/2+δ）} \ log p> 0 $ for Some $δ> 0 $。因此，$ \ log P = o \ big（n^{1/2} \ big）$给出了CLT有效性的$ P $的关键率。
In this article, we try to give an answer to the simple question: ``\textit{What is the critical growth rate of the dimension $p$ as a function of the sample size $n$ for which the Central Limit Theorem holds uniformly over the collection of $p$-dimensional hyper-rectangles ?''}. Specifically, we are interested in the normal approximation of suitably scaled versions of the sum $\sum_{i=1}^{n}X_i$ in $\mathcal{R}^p$ uniformly over the class of hyper-rectangles $\mathcal{A}^{re}=\{\prod_{j=1}^{p}[a_j,b_j]\cap\mathcal{R}:-\infty\leq a_j\leq b_j \leq \infty, j=1,\ldots,p\}$, where $X_1,\dots,X_n$ are independent $p-$dimensional random vectors with each having independent and identically distributed (iid) components. We investigate the critical cut-off rate of $\log p$ below which the uniform central limit theorem (CLT) holds and above which it fails. According to some recent results of Chernozukov et al. (2017), it is well known that the CLT holds uniformly over $\mathcal{A}^{re}$ if $\log p=o\big(n^{1/7}\big)$. They also conjectured that for CLT to hold uniformly over $\mathcal{A}^{re}$, the optimal rate is $\log p = o\big(n^{1/3}\big)$. We show instead that under some conditions, the CLT holds uniformly over $\mathcal{A}^{re}$, when $\log p=o\big(n^{1/2}\big)$. More precisely, we show that if $\log p =ε\sqrt{n}$ for some sufficiently small $ε>0$, the normal approximation is valid with an error $ε$, uniformly over $\mathcal{A}^{re}$. Further, we show by an example that the uniform CLT over $\mathcal{A}^{re}$ fails if $\limsup_{t\rightarrow \infty} n^{-(1/2+δ)} \log p >0$ for some $δ>0$. Hence the critical rate of the growth of $p$ for the validity of the CLT is given by $\log p=o\big(n^{1/2}\big)$.
