distinguishing random orthogonal matrix from Gaussian random matrix

Jiang's paper (http://projecteuclid.org/euclid.aop/1158673325) shows the following: Suppose that $G$ is an $n\times n$ random matrix with entries i.i.d. $N(0,1/n)$, and $Z$ is a random $n\times n$ orthogonal matrix (uniform w.r.t. Haar measure). Let $D_1$ be the distribution of topleft $m \times m$ block of $G$, and $D_2$ be the distribution of topleft $m\times m$ block of $Z$. Then the total variance distance between $D_1$ and $D_2$ goes to 0 if $m=o(\sqrt n)$ and the t.v.d. is at least a constant if $m$ is at the order of $\sqrt n$.

This suggests that, given a random matrix which comes from the distribution of $G$ or the distribution of $Z$, we can tell which case it is with probability, say, $\geq 3/4$, by reading its top-left $\sqrt{n}\times\sqrt{n}$ block (dimension up to a constant factor).

My question is, What property should we examine on the top-left $\sqrt{n}\times\sqrt{n}$ block? Operator norm? Have there been any results on this?

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Just because the distance goes to zero for $m=o(\sqrt{n})$ does not mean that it does NOT go to zero for bigger $m$ (for example, any $m=o(n)$). – Igor Rivin Dec 2 '12 at 18:37
A nonzero total variation distance doesn't necessarily mean that the distributions can be told apart from samples. The place where the distributions differ substantially might take up a vanishingly small fraction of the probability weight of the distributions. – Carl Feynman Dec 2 '12 at 21:05
Igor Rivin: The paper proved that it doesn't go to $0$ when m is a the order of $\sqrt n$. – user14432 Dec 3 '12 at 0:58
Carl Feynman: Sorry, I should say that they could be told apart with probability at least a constant. – user14432 Dec 3 '12 at 0:59
If that's true [the response to @Carl Feynman], don't you have an answer to your question? – Igor Rivin Dec 3 '12 at 2:04

1 Answer

The answer is implicit in Jiang's paper. There is a formula for the Radon-Nykodym derivative of the two measures, and you just check whether this is more or less than 1. This would precisely achieve the total variation distance which is at least some constant.

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