We provide an elementary proof for a theorem due to Petz and Réffy which states that for a random
n ×
n unitary matrix with distribution given by the Haar measure on the unitary group
U(
n), the upper left (or any other)
k ×
k submatrix converges in distribution, after multiplying by a normalization factor
Ön\sqrt{n} and as
n®¥n\to\infty , to a matrix of independent complex Gaussian random variables with mean 0 and variance 1.
Keywords random matrices - Haar measure on the unitary group - Gaussian matrices
Mathematics Subject Classification (2000) 15A52 - 60B10