as observed in [@Meckes].

\[T:C-Ginibre-eigenvalues\] Let $G$ be a finite abelian group and let $\{Y_a \mid a \in G\}$ be independent, standard complex Gaussian random variables. Then the eigenvalues $\bigl\{ \lambda_\chi \mid \chi \in \widehat{G} \bigr\}$ of $M$ given by are independent, standard complex Gaussian random variables.

The random matrix ensemble in Proposition \[T:C-Ginibre-eigenvalues\] is the $G$-circulant analogue of the complex Ginibre ensemble $X$, which consists of a square matrix with independent, standard complex Gaussian entries.

\[T:C-Ginibre-limit\] Suppose that for each $n$, $\bigl\{Y_a^{(n)} \mid a \in G^{(n)}
  \bigr\}$ are independent, standard complex Gaussian random variables. Then ${\mathbb{E}}\mu^{(n)} = \gamma_{\mathbb{C}}$ for each $n$, and $\mu^{(n)} \to \gamma_{\mathbb{C}}$ weakly in probability. Furthermore, if ${\left\vert G^{(n)} \right\vert} = \Omega(n^{{\varepsilon}})$ for some ${\varepsilon}> 0$, then $\mu^{(n)} \to \gamma_{\mathbb{C}}$ weakly almost surely.

For each, say, Lipschitz $f: {\mathbb{C}}\to {\mathbb{R}}$, $$({\mathbb{E}}\mu)(f) := {\mathbb{E}}\bigl(\mu(f)\bigr) 
  = \frac{1}{{\left\vert G \right\vert}} \sum_{\chi \in \widehat{G}} {\mathbb{E}}f(\lambda_\chi),$$ where the $(n)$ superscripts are omitted for simplicity. By Proposition \[T:C-Ginibre-eigenvalues\], each $\lambda_\chi$ is distributed according to $\gamma_{\mathbb{C}}$, and so $ ({\mathbb{E}}\mu)(f) =
  \gamma_{\mathbb{C}}(f) $. Thus ${\mathbb{E}}\mu = \gamma_{\mathbb{C}}$.

By the concentration properties of Gaussian measure (see [@Ledoux]), since the $\lambda_\chi$ are distributed as independent standard complex Gaussian random variables, if $f$ is $1$-Lipschitz, then $${\mathbb{P}}\bigl[{\left\vert \mu(f) - \gamma_{\mathbb{C}}(f) \right\vert} \ge t \bigr] 
  \le 2 e^{-{\left\vert G \right\vert} t^2}$$ for each $t > 0$. If ${\left\vert G^{(n)} \right\vert} = \Omega(n^{{\varepsilon}})$, then the Borel–Cantelli lemma implies that $\mu^{(n)}(f) \to \gamma_{\mathbb{C}}(f)$ almost surely. Applying 