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We provide an elementary proof for a theorem due to Petz and R\'effy whichstates that for a random $n\times n$ unitary matrix with distribution given bythe Haar measure on the unitary group U(n), the upper left (or any other)$k\times k$ submatrix converges in distribution, after multiplying by anormalization factor $\sqrt{n}$ and as $n\to\infty$, to a matrix of independentcomplex Gaussian random variables with mean 0 and variance 1.

Elementary Proof for Asymptotics of Large Haar-Distributed Unitary Matrices