The Annals of Probability

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

Tiefeng Jiang

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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,jn} are i.i.d. standard normals. We show that $\varepsilon _{n}(m):=\max_{1\leq i\leq n,1\leq j\leq m}|\sqrt{n}\gamma_{ij}-y_{ij}|$ 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.

Article information

Source
Ann. Probab. Volume 34, Number 4 (2006), 1497-1529.

Dates
First available: 19 September 2006

Permanent link to this document
http://projecteuclid.org/euclid.aop/1158673325

Digital Object Identifier
doi:10.1214/009117906000000205

Mathematical Reviews number (MathSciNet)
MR2257653

Zentralblatt MATH identifier
1107.15018

Subjects
Primary: 15A52 60B10: Convergence of probability measures 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 60F05: Central limit and other weak theorems 60F99: None of the above, but in this section 62H10: Distribution of statistics

Keywords
Haar measure Gram–Schmidt algorithm large deviation maxima product distribution random matrix theory variation distance

Citation

Jiang, Tiefeng. How many entries of a typical orthogonal matrix can be approximated by independent normals?. The Annals of Probability 34 (2006), no. 4, 1497--1529. doi:10.1214/009117906000000205. http://projecteuclid.org/euclid.aop/1158673325.


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