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×q_n upper left block of a random matrix Γ_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 Γ_n=(γ_ij)_n×n is generated by performing the Gram–Schmidt algorithm on the columns of Y_n=(y_ij)_n×n, where {y_ij;1≤i,j≤n} are i.i.d. standard normals. We show that $\varepsilon _{n}(m):=\max_{1\leq i\leq n,1\leq j\leq m}|\sqrt{n}\cdot\gamma_{ij}-y_{ij}|$ goes to zero in probability as long as m=m_n=o(n/logn). We also prove that $\varepsilon _{n}(m_{n})\to 2\sqrt{\alpha}$ in probability when m_n=[nα/logn] for any α>0. This says that m_n=o(n/logn) is the largest order such that the entries of the first m_n columns of Γ_n can be approximated simultaneously by independent standard normals.