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 pn and qn such that Zn, the pn×qn upper left block of a random matrix Γn 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 Zn and that of pnqn 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 pn and qn are o(n1/2) in the sense of the above approximation. Second, suppose Γn=(γij)n×n is generated by performing the Gram–Schmidt algorithm on the columns of Yn=(yij)n×n, where {yij;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=mn=o(n/logn). We also prove that $\varepsilon _{n}(m_{n})\to 2\sqrt{\alpha}$ in probability when mn=[nα/logn] for any α>0. This says that mn=o(n/logn) is the largest order such that the entries of the first mn columns of Γn can be approximated simultaneously by independent standard normals.

Locations

• The Annals of Probability - View - PDF
• arXiv (Cornell University) - View - PDF
• DataCite API - View