Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: How many entries of a typical orthogonal matrix can be approximated by
  independent normals?   We solve an open problem of Diaconis that asks what are the largest orders of
$p_n$ and $q_n$ such that $Z_n,$ the $p_n\times q_n$ upper left block of a
random matrix $\boldsymbol{\Gamma}_n$ which is uniformly distributed on the
orthogonal group O(n), can be approximated by independent standard normals?
This problem is solved by two different approximation methods. First, we show
that the variation distance between the joint distribution of entries of $Z_n$
and that of $p_nq_n$ independent standard normals goes to zero provided
$p_n=o(\sqrt{n})$ and $q_n=o(\sqrt{n})$. We also show that the above variation
distance does not go to zero if $p_n=[x\sqrt{n} ]$ and $q_n=[y\sqrt{n} ]$ for
any positive numbers $x$ and $y$. This says that the largest orders of $p_n$
and $q_n$ are $o(n^{1/2})$ in the sense of the above approximation. Second,
suppose $\boldsymbol{\Gamma}_n=(\gamma_{ij})_{n\times n}$ is generated by
performing the Gram--Schmidt algorithm on the columns of
$\bold{Y}_n=(y_{ij})_{n\times n}$, where $\{y_{ij};1\leq i,j\leq n\}$ are
i.i.d. standard normals. We show that $\epsilon_n(m):=\max_{1\leq i\leq n,1\leq
j\leq m}|\sqrt{n}\cdot\gamma_{ij}-y_{ij}|$ goes to zero in probability as long
as $m=m_n=o(n/\log n)$. We also prove that $\epsilon_n(m_n)\to 2\sqrt{\alpha}$
in probability when $m_n=[n\alpha/\log n]$ for any $\alpha>0.$ This says that
$m_n=o(n/\log n)$ is the largest order such that the entries of the first $m_n$
columns of $\boldsymbol{\Gamma}_n$ can be approximated simultaneously by
independent standard normals.

11
The categories of this paper are: hep-lat
9
The categories of this paper are: nlin
0
The categories of this paper are: hep-ph
32
The categories of this paper are: atom-ph
28
The categories of this paper are: alg-geom
23
The categories of this paper are: solv-int
24
The categories of this paper are: dg-ga
15
The categories of this paper are: nucl-ex
25
The categories of this paper are: patt-sol
35
The categories of this paper are: mtrl-th
30
The categories of this paper are: q-alg
33
The categories of this paper are: plasm-ph
27
The categories of this paper are: comp-gas
26
The categories of this paper are: bayes-an
34
The categories of this paper are: chem-ph
8
The categories of this paper are: hep-ex