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Oct 23, 2013 at 8:02 comment added Mark Meckes No, I was confused. I thought the OP meant for $G$ to be symmetric matrix with N(0,1) entries on and above the diagonal, which is not quite the GOE and is not orthogonally invariant. But I see now that the OP didn't say that $G$ should be symmetric, so there's no problem with what you said.
Oct 23, 2013 at 7:30 vote accept Lior Eldar
Oct 22, 2013 at 21:05 comment added Carlo Beenakker @MarkMeckes --- you mean that the analysis depends on whether $G$ is symmetric or not? I'm probably confused, but where did I make that assumption?
Oct 22, 2013 at 20:42 comment added Mark Meckes Although the OP may have meant for $G$ to be a GOE matrix, that's not what the question actually said. That shouldn't actually change the answer, but it makes the analysis less trivial.
Oct 22, 2013 at 13:03 history answered Carlo Beenakker CC BY-SA 3.0