How many entries of a typical orthogonal matrix can be approximated by independent normals? (English) Zbl 1107.15018

The goal of the paper is to find the largest orders such that the left upper block of a random matrix uniformly distributed on \(O(n)\), can be approximated by standard normals. Given two absolutely continuous probability measures \(\mu\) and \(\nu\) on \(\mathbb{R}^m\), the variation distance between \(\mu\) and \(\nu\) is defined as \[ \| \mu-\nu\| =2\cdot\sup_A\,| \mu(A)-\nu(A)| . \] Consider random matrices \(\Gamma_n\) uniformly distributed on \(O(n)\), and let \(Z_n\) be the \(p_n\times q_n\) upper left block of \(\Gamma_n\). Let \(G_n\) be the distribution of \(p_nq_n\) independent standard normals, and let \(\mathcal{L}(\sqrt{n}\,Z_n)\) denote the joint probability distribution of the \(p_nq_n\) entries of \(\sqrt{n}Z_n\). The author proves that if \(p_n=o(\sqrt{n})\) and \(q_n=o(\sqrt{n})\) then \[ \lim_{n\rightarrow\infty}\| \mathcal{L}(\sqrt{n}Z_n)-G_n\| =0, \] and that if \(x,y>0\) are fixed numbers such that \(p_n=[x\sqrt{n}]\), \(q_n=[y\sqrt{n}]\), then \[ \liminf_{n\rightarrow\infty}\| \mathcal{L}(\sqrt{n}Z_n)-G_n\| \geq E\biggl| \exp\biggl(-\frac{x^2y^2}8+\frac{xy}4\,\xi \biggr)-1\biggr| >0, \] where \(\xi\) is a standard normal.
For the case of variation distance, the above results show that the largest orders of \(p_n\) and \(q_n\) such that \(Z_n\) can be approximated by standard normals are \(o(\sqrt{n})\).
Let \(Y_n\) be \(n\times n\) random matrices with independent standard normal entries, and obtain \(\Gamma_n\) from \(Y_n\) by doing Gram-Schmidt on the columns of \(Y_n\). Then, if \[ \varepsilon_n(m)=\max_{1\leq i\leq n,1\leq j\leq m}\,| \sqrt{n}\gamma_{ij}-y_{ij}| , \] the author proves that \(\Gamma_n\) is Haar invariant on \(O(n)\), that \(\varepsilon_n(m_n)\rightarrow 0\) in probability provided that \(m_n=o(n/\log n)\), and that given \(\alpha>0\), \(\varepsilon_n([n\alpha/\log n])\rightarrow 2\sqrt\alpha\) in probability.
The above result shows that \(m_n=o(n/\log n)\) is the largest order such that the entries of the first \(m_n\) columns of \(\Gamma_n\) can be approximated simultaneously by independent standard normals.


15B52 Random matrices (algebraic aspects)
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60B10 Convergence of probability measures
60F05 Central limit and other weak theorems
62H10 Multivariate distribution of statistics
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