How many entries of a typical orthogonal matrix can be approximated by independent normals?

Abstract

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.