Exercise 7, § 4

For every integer $n\geqslant 1$, let $\mu_n$ denote the positive measure on $\mathbf{C}$ which is the image of the normalized Haar measure of the group $\mathbf{U}(\mathbf{C}^n)\subset$ GL($\mathbf{C}^n$) under the trace mapping $g\mapsto$ Tr($g$); it is a measure with compact support and total mass 1.

a) Let $a$ and $b$ be natural integers such that $a+b\leqslant n$. Then

$\int_{ab}\{0$ if $a\not =b$

$_{\mathbf{C}}a$! if $a=b$.

b) The sequence of measures $(\mu_n)_{n\geqslant 1}$ converges narrowly to the Gaussian measure on the real vector space $\mathbf{C}$ whose covariance is $Q(z) =|z|^2$.