## 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*_{n}*q*_{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*/log*n*). We also prove that $\varepsilon _{n}(m_{n})\to 2\sqrt{\alpha}$ in probability when *m*_{n}=[*n**α*/log*n*] for any *α*>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 **Γ**_{n} can be approximated simultaneously by independent standard normals.

## Citation

Tiefeng Jiang. "How many entries of a typical orthogonal matrix can be approximated by independent normals?." Ann. Probab. 34 (4) 1497 - 1529, July 2006. https://doi.org/10.1214/009117906000000205

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