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.Comment: Published at http://dx.doi.org/10.1214/009117906000000205 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org