Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Central Limit Theorem in High Dimensions : The Optimal Bound on
  Dimension Growth Rate   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 $\log p =\epsilon \sqrt{n}$ for some sufficiently small $\epsilon>0$,
the normal approximation is valid with an error $\epsilon$, uniformly over
$\mathcal{A}^{re}$. Further, we show by an example that the uniform CLT over
$\mathcal{A}^{re}$ fails if $\limsup_{t\rightarrow \infty} n^{-(1/2+\delta)}
\log p >0$ for some $\delta>0$. Hence the critical rate of the growth of $p$
for the validity of the CLT is given by $\log p=o\big(n^{1/2}\big)$.

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The categories of this paper are: cmp-lg
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: comp-gas
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The categories of this paper are: stat
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The categories of this paper are: nucl-th
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The categories of this paper are: patt-sol
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The categories of this paper are: cond-mat
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The categories of this paper are: econ
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The categories of this paper are: alg-geom
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The categories of this paper are: eess
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The categories of this paper are: adap-org
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The categories of this paper are: nlin